diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdvlr" "b/data_all_eng_slimpj/shuffled/split2/finalzzdvlr" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdvlr" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nWaves can exhibit wavefront dislocations or vortices, i.e. phase \nsingularities of the complex wavefuntion, where the modulus of the wave\nvanishes, and around which the phase of the wave changes by a multiple of \n2$\\pi$~\\cite{Berry:royal:1974}. \nIn~\\cite{Irvine:NatPhys:2008}, it has been shown that Maxwell's equations in fact admit solutions where \nthese lines of dislocation or vortex lines are tied into knots embedded in light fields.\nHere, the term ``knot'' refers to an embedding of a circle $S^1$ into a three-sphere $S^3$~\\cite{knot_book}.\nBoth theoretical~\\cite{Shabtay:OC:2003} as well as \nexperimental advances~\\cite{Padgett:NJP:2005,Whyte:NJP:2005,Dennis:Nature:2010,Shanblatt:OE:11} in three-dimensional light shaping allowed to \nthe experimental realization of knotted vortex lines embedded in optical fields. \nApart from optics, the study of knotted topological defect lines and their dynamics has fascinated scientists from diverse settings, including\nclassical fluid dynamics~\\cite{Moffatt:JFM:1969,Moffatt:nature:1990}, excitable media~\\cite{Paul:PRE:2003,Sutcliffe:PRL:2016}, \nchiral nematic colloids~\\cite{Tkalec:science:2011} to semiconductors~\\cite{Babaev:PRL:2002,Babaev:PRB:2009}.\n\nLord Kelvin speculated in 1867~\\cite{Kelvin:1867} that atoms can be described by vortex tubes in the aether.\nIn fact, almost two decades ago it was suggested that (nontrivial) knots might exist as stable solitons or Hopfions\nin three-dimensional field theories~\\cite{Faddeev:Nature:1997}. A Skyrme model served as an example for a field theory \nwhich admits stable knot solitons~\\cite{Paul:PRL:1998}. \n\nIn the context of spinor Bose-Einstein condensates (BECs)~\\cite{TinLun:PRL:98,Machida:JphysJpn:1998}, the\nexistence of knots with nonzero Hopf charge was studied~\\cite{Niemi:PRB:2002,Ueda:PRL:08,Ueda:PTPS:2010} and recently realized\nexperimentally~\\cite{Mottonen:NatPhys:2016}. However, there is work~\\cite{Speight:JGP:2010} that casts doubt on existence of\nstable Hopfions in two-component Ginzburg-Landau type systems. \n\nContrasting these studies, \nnumerical investigations on the robustness and centre of mass motion of {\\em knotted vortex lines} embedded in a {\\em single}\ncomponent BEC have been undertaken~\\cite{Barenghi:PRE:2012,Barenghi:JOP:2014}. \nSuch formations are typically unstable, since there is no topological stabilization mechanism and reconnections of \nvortex lines are allowed due to the occurrence of the quantum stress tensor in the hydrodynamic formulation of the Gross-Pitaevskii Equation. \nNucleation~\\cite{Frisch:PRL:1992} and reconnection of vortex lines~\\cite{Koplik:PRL:1993,Bewley:PNAS:2008,Barenghi:PhysFluid:2012} and vortex line bundles~\\cite{Barenghi:PRL:2008} \nand subsequent emission of sound waves~\\cite{Leadbeater:PRL:2001} have been theoretically studied extensively. \n\nIn this paper, we follow up on the idea formulated in~\\cite{Ruostekoski:PRA:2005}, and \ndiscuss a general experimental scheme to create a two-component BEC which contains a knotted vortex line in one \nof its components. \nWe suggest using a light field containing a knotted vortex line as probe field of a Raman-pulse that drives a \ncoherent two-photon Raman transition of three-level atoms with $\\Lambda$-level configuration [cf.~\\reffig{fig:lambda_scheme}(a)]. \nPreviously, similar methods have been used to create dark solitons and vortices~\\cite{Wright:PRL:1997,Dum:prl:98,Ruostekoski:PRL:2004,Ruostekoski:PRA:2005,Andersen:PRL:2006} \nand have been experimentally realized using microwave~\\cite{Cornell:PRL::1999_2,Cornell:PRL::1999} and more recently optical coupling~\\cite{Schmiegelow:ARXIV:2015}. \nThe pump beam will be the mentioned knotted (stationary) light beam, whereas \nthe control beam is a co-propagating plane wave to remove fast oscillations in the $z$-direction~\\cite{Ruostekoski:PRA:2005}.\nThe large controllability of the pulse parameters (i.e. strength and duration of the Rabi-pulse) \nallows for a large controllability of the excitation. \nThe dynamics of two-component BECs can be monitored in real time via in situ measurements without ballistic expansion~\\cite{Cornell:PRL:2000}. \nUsing numerical methods, we study excitation and subsequent dynamics of specific examples of knotted matter waves for experimentally feasible parameters. \n\nThis paper thus paves the way for experimentally accessing many of the \nphenomena discussed only theoretically in, e.g.~\\cite{Barenghi:PRE:2012,Barenghi:JOP:2014,Irvine:arXiv:2015},\nand if such system were realized experimentally, it would give controlled experimental access to reconnection of vortex lines, \nsubsequent emission of sound waves and more generally quantum turbulence. \n\n\\section{Model}\\label{sec:artificial_light}\n\\subsection{Equations of motion for light-matter wave coupling}\n\nConsider a three-level atom in $\\Lambda$-configuration with ground states $\\ket{a}$ and $\\ket{b}$, which \nare off-resonantly coupled to an excited state $\\ket{e}$ with detuning $\\Delta$ and spatially dependent Rabi-frequencies\n$\\Omega_a({\\bf r})$ and $\\Omega_b(z)$. Assuming $\\Delta\\gg\\Omega_a,\\Omega_b$, the excited state can be \nadiabatically eliminated, and the dynamics of the condensate wave function confined in a trap $V({\\bf r})$ \ncan be described~\\cite{Dum:prl:98,Ruostekoski:PRA:2005} using~\\refeqs{eq:eqmo1}{eq:eqmo2}\n\\begin{eqnarray}\n\\fl i\\hbar\\partial_t\\psi_a&=\\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V({\\bf r}) + Ng_{aa} |\\psi_a|^2+Ng_{ab}|\\psi_b|^2\\right)\\psi_a+\\frac{\\hbar\\Omega_a({\\bf r})\\Omega_b^*(z)}{8\\Delta}\\psi_b\\label{eq:eqmo1}\\\\\n\\fl i\\hbar\\partial_t\\psi_b&=\\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V({\\bf r}) + Ng_{ba} |\\psi_a|^2+Ng_{bb}|\\psi_b|^2 -\\hbar\\delta \\right)\\psi_b+\\frac{\\hbar\\Omega_a^*({\\bf r})\\Omega_b(z)}{8\\Delta}\\psi_a\n \\label{eq:eqmo2}\n\\end{eqnarray}\nwith $g_{kj}=4\\pi\\hbar^2 a_{kj}\/m$, $a_{kj}$ denoting the scattering length between the species and \n\\begin{equation}\nV({\\bf r})= \\frac{1}{2}m\\omega^2 r_\\perp^2+\\frac{1}{2}m\\omega_z^2 z^2.\n\\end{equation}\n\\reffig{fig:lambda_scheme}(a) depicts the level scheme of three-level atoms in $\\Lambda$-type configuration.\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\textwidth]{fig1.eps}\n\\caption{\n(color online) (a) Schematic sketch of a Raman-type transition with $\\Lambda$-configuration. The state $\\ket{a}$ is\noff-resonantly coupled to state $\\ket{b}$ with two-photon detuning $\\delta$.\nThe knotted light field $\\mathcal{E}$ proportional to $\\Omega_a({\\bf r})$ off-resonantly couples state $\\ket{a}$ to $\\ket{e}$ with detuning $\\Delta$. \nThe final state $\\ket{b}$ then reflects the involved structure of the light field $\\mathcal{E}$ associated with $\\Omega_a({\\bf r})$.\n(b) depicts an isosurface of the intensity (purple) of the light-field $\\mathcal{E}$ at a low isointensity value and a slice of its \nphase in the $(x,y)$-plane. Additional to the vortex lines in the center of the beam, the usual diffraction cones become visible.\n(c) Illustration of the situation at $t=0$ for the parameters discussed in~\\ref{sec:trefoil}.\nThe isodensity of the cigar-shaped wave function $\\psi_a$ (yellow) is subject to the knotted light field (purple) from Fig. (b).\n\\label{fig:lambda_scheme}}\n\\end{center}\\end{figure}\nThe Rabi-frequencies $\\Omega_k$, $k=a,b$ relate to their corresponding electric fields via\n\\begin{equation}\n \\Omega_k=\\frac{dE_k}{\\hbar},\n \\label{eq:Omega_a}\n\\end{equation}\nwhere the transition dipole element $d$ has been introduced. \nAs electric fields, we consider a monochromatic, quasi-linearly polarized light field with \nwavevector $k_0$. \nThen, its slowly varying envelope~$\\mathcal{E}$ is defined by \n\\begin{equation}\n E({\\bf r},t)=\\sqrt{\\frac{\\omega_0\\mu_0}{2k_0}}\\mathcal{E}({\\bf r})e^{i\\left(k_0z-\\omega_0t\\right)}+{\\rm c.c.},\n \\label{eq:sve}\n\\end{equation}\nwhere ${\\rm c.c}$ denotes the complex conjugate. \nWithin the paraxial approximation, the slowly varying envelope $\\mathcal{E}$ of the beam is governed by \n\\begin{equation}\n2ik_0\\partial_z \\mathcal{E}({\\bf r}_{\\perp},z) = -\\nabla^2_{\\perp}\\mathcal{E}({\\bf r}_{\\perp},z),\n\\label{eq:paraxial}\n\\end{equation}\nwhere ${\\bf r}_\\perp=(x,y)$, and $\\nabla^2_\\perp$ denotes the transverse Laplacian.\n\nSuch a system can be realized by considering the two hyperfine ground states $\\ket{a}=\\ket{S_{1\/2},F=2,M_F=-1}$, $\\ket{b}=\\ket{S_{1\/2},F=2,M_F=1}$ \nof ${}^{87}$Rb, that are off-resonantly coupled to an excited state manifold. \nThen, the scattering lengths between the species $a_{kj}$ are given by $a_{ba}=5.5$nm, and the ratio $a_{aa}$:$a_{ba}$:$a_{bb}$ is given by $1.03$:$1$:$0.97$~\\cite{Cornell:PRL:1998}. \nThe motivation for assuming cylindrical symmetry of the trapping potential $V({\\bf r})$ is associated with \nthe specific form of the paraxial wave equation~\\refeq{eq:paraxial}. \nWhereas we use different beams for trapping and Raman transition, the aspect ratio $\\omega_z\/\\omega$ \nfor our otherwise independent cigar shaped trap cannot be chosen arbitrarily.\nInstead, in order to accommodate the optical knot in the BEC (see~\\reffig{fig:lambda_scheme}(c)), \nwe have to make sure that extent of the BEC in the z-direction is large enough. \nThe aspect ratio of the optical beam (and the optical knot) can be expressed using the ratio between the Rayleigh \nlength $z_r=k_0\\sigma^2$ and the width $\\sigma$ of the light field in the \n$(x,y)$-plane:\n\\begin{equation}\n \\frac{z_r}{\\sigma}=\\frac{k_0\\sigma^2}{\\sigma}=k_0\\sigma.\n \\label{eq:aspect_ratio}\n\\end{equation}\nThe basic idea towards imprinting knotted vortex lines into BECs is depicted in~\\reffig{fig:lambda_scheme}(a).\nConsider the case where all atoms are initially in state $\\ket{a}$ [$\\psi_b(t=0)=0$].\nThen, assuming $\\Omega_i\\ll \\Delta$ and $\\delta\\approx 0$ the \nRabi pulse coherently transfers population from $\\ket{a}$ to $\\ket{b}$ as \nillustrated in according to~\\refeqs{eq:eqmo1}{eq:eqmo2}. The light field is depicted in~\\reffig{fig:lambda_scheme}(b) and together with the density of $\\psi_a$ in~\\reffig{fig:lambda_scheme}(c).\nThe laser associated with $\\Omega_b$ is chosen to be a simple plane wave, which coherently co-propagates with $\\mathcal{E}$, \nso that the fast variations $\\sim e^{ik_0z}$ in~\\refeq{eq:sve} are canceled whenever products $\\Omega_a\\Omega_b^*$, as in~\\refeqs{eq:eqmo1}{eq:eqmo2}, occur.\nThe explanation as to why such a setup allows to imprint the phase of the structured light field $\\Omega_a$ onto the condensate is the following.\nFor short time and two-photon detuning $\\delta=0$, the small change $\\delta\\psi_b$ to $\\psi_b$ is \ngiven by \n\\begin{equation}\n \\delta\\psi_b \\approx-\\frac{i}{\\hbar}\\frac{\\hbar\\Omega_a^*({\\bf r})\\Omega_b(z)}{8\\Delta}\\psi_a(t=0)\\delta t.\n \\label{eq:simple_argument}\n\\end{equation}\nThus, we may conclude that at least for small times, it should be possible to populate state $\\ket{b}$ \nwith a given phase profile using our light field. \nOnce the phase profile $\\theta$ has been imprinted ($\\mathcal{E}=|\\mathcal{E}|e^{i\\theta}$) onto \nthe condensate, the atoms will display motion according to ${\\bf v}=\\hbar\\nabla\\theta\/m$~\\cite{Ruostekoski:PRA:2005}. \nTo realize such a scenario, the intensity of the involved \nlaser beams must be sufficiently strong, such that the time-scales of the imprinting are small compared to\ntime-scales of the dynamics of the condensate.\nWe will use numerical methods and realistic experimental \nparameters to extend this simple idea beyond the perturbative limit and \nstudy its subsequent dynamics. \n\n\n\\subsection{Knotted Light Field}\\label{sec:knotted_light_field}\n\nLaguerre-Gaussian (LG) functions $\\mathrm{LG}_{l,p}$ \nform a basis set for solutions for the paraxial wave equation~\\refeq{eq:paraxial}, and are given by the expression\n\\begin{eqnarray}\n \\mathrm{LG}_{l,p}^{\\sigma,z_r}({\\bf r}_\\perp,z)=&\\sqrt{\\frac{p!}{\\pi (|l|+p)!}}\\frac{r_\\perp^{|l|}e^{il\\varphi}}{\\sigma^{|l|+1}}\\frac{(1-iz\/z_r)^p}{(1+iz\/z_r)^{p+|l|+1}}\\nonumber\\\\\n & \\times e^{-r_{\\perp}^{2}\/2\\sigma^2(1+iz\/z_r)} L^{|l|}_p \\left( \\frac{r_{\\perp}^{2}}{\\sigma^2\\left[1+\\left(z\/z_r\\right)^2\\right]} \\right)\n \\label{eq:LG_modes}\n\\end{eqnarray}\nwith $r_{\\perp}^2=x^2+y^2$ and $L^{|l|}_p$ being the associated Laguerre polynomials. \n\nWe seek a linear superposition of LG-modes, $\\sum_{l,p}a_{lp}\\mathrm{LG}_{l,p}$, \nthat describe light-beams containing knotted vortex lines. \nTo this end, we will review the method proposed in~\\cite{Dennis:Nature:2010}, which uses Milnor polynomials~\\cite{Milnor_book} as an ansatz for complex light fields\nin the shape of torus knots to determine appropriate amplitudes $a_{lp}$, and rescale for application to our setup.\n\nThe basic idea~\\cite{King:thesis,Dennis:Nature:2010} is to parametrize an $N$-strand braid as the roots of the polynomial \n\\begin{equation}\n p_h^{N,n}(u)=\\prod_{j=0}^{N-1} \\left[u-s_j(h)\\right]\n\\end{equation}\nHere, $h$ denotes the height of the periodic braid and $N$ denotes the number of strands or roots of $p_h(u)$ in $u$.\nLet us choose $N=2$ and~\\cite{King:thesis} \n\\begin{equation}\n s_j(h)=\\cos(h_j)+i\\sin(h_j), \\quad h_j=(h-2\\pi j\/n)n\/2.\n\\end{equation}\nThe projection of the braid onto the $(h=0)$-plane leads to a circle. The parameter $n$ represents the number of braid crossings. \nWe will consider the cases $n=2,3$ in the following. \nA small computation allows us to find an explicit expression for the Milnor polynomial $p_h$\nin the variables $u$ and $\\exp(ih)=:v$,\n\\begin{equation}\n p_h^{2,n}=u^2-v^n\n \\label{eq:milnor}\n\\end{equation}\nWe can now imagine a cylinder containing the braid, and ``glue'' top and bottom surfaces together to obtain a knot. In fact, one can show~\\cite{Alexander:PNAS:1923} that\nany knot can be represented as a closure of a braid. \nAn easy way to deform our cylinder into a torus and to make it explicitly dependent on ${\\bf{r}^\\prime}=(x^\\prime,y^\\prime,z^\\prime)$ \nis to write $u$ and $v$ as an inverse \nstereographic projection from three-dimensional space to a \nunit three-sphere, $\\mathbb{R}^3\\rightarrow\\mathbb{S}^3$, i.e.\n\\begin{eqnarray}\nu&=\\frac{r^{\\prime 2}-1+2iz^{\\prime}}{r^{\\prime 2}+1},\\\\\nv&=\\frac{2(x^{\\prime}+iy^{\\prime})}{r^{\\prime 2}+1}\\label{eq:stereo}\n\\end{eqnarray}\nHere, $r^\\prime=\\sqrt{x^{\\prime 2}+y^{\\prime 2}+z^{\\prime 2}}$ and the units $\\bf{r}^\\prime$ are non-dimensional. \nOne can easily show that $|u|^2+|v|^2=\\Re(u)^2+\\Im(u)^2+\\Re(v)^2+\\Im(v)^2=1$, \nand thus~\\refeq{eq:stereo} indeed represents a parametrisation of a three-sphere. \nSince we aim to describe an actual light field, i.e. a solution to the paraxial wave equation, \ninstead of considering the Milnor polynomial~\\refeq{eq:milnor} $p_h$ as it is, it is reasonable to get rid of the denominator in \n$p_h$ and to consider~\\cite{Dennis:Nature:2010,King:thesis} \n\\begin{eqnarray}\n\\xi_a({\\bf r}^\\prime)&=\\left(u^2-v^n\\right)\\left(r^{\\prime 2}+1\\right)^n.\n\\label{eq:ansatz_light}\n\\end{eqnarray}\nFor $n=2,{\\ }3$,~\\refeq{eq:ansatz_light} \ndescribes a Hopf link and a trefoil knot, respectively, which \nrepresent the simplest non-trivial examples of a link and a knot, respectively.\nThe latter two will be used as exemplary fields in the following. \n\nIn order to use~\\refeq{eq:ansatz_light} to find appropriate coefficients $a_{l,p}$, \nlet us rescale the light field~\\refeq{eq:LG_modes} by $R_s$ to nondimensional units $(x^\\prime,y^\\prime,z^\\prime)$, such that \n\\begin{equation}\n {\\bf r}_\\perp={\\bf r}^\\prime_\\perp R_s,\\quad\n \\frac{z}{z_r}=\\frac{z\/R_s}{k_0R_s\\sigma^2\/R_s^2}=\\frac{z^\\prime}{(k_0R_s) (\\sigma^2\/R_s^2)},\n\\end{equation}\nto equate the light field to our knot in the same dimensionless units. \nEffectively, $R_s$ scales the abstract unit sphere to have a transverse extent of approximately $R_s$ with respect to the laser beam.\nHence, there are two length scales of our system, $R_s$ and $\\sigma$, that are associated with the nodal lines of the knot and the transverse extent of the beam. \nAs we will see, the ratio $w=\\sigma\/R_s$ will play a crucial role as an important degree of freedom, that allows us to change the width of the \nbeam relative to the positions of the nodal lines of the knot. \nTo find appropriate superpositions of ${\\mathrm{LG}}$-modes, it is sufficient to restrict considerations to the $(x,y)$-plane only.\nLet us equate~\\refeq{eq:ansatz_light} with a superposition of the above-mentioned rescaled version of~\\refeq{eq:LG_modes}: \n\\begin{eqnarray}\n\\xi_a({\\bf r}_\\perp^\\prime,z^\\prime=0) = \\sum_{l,p} a_{l,p}(w)\\mathrm{LG}_{l,p}^{w,k_0R_sw^2}\\left({\\bf r}_\\perp^\\prime,z^\\prime=0\\right)\\sqrt{\\pi}e^{r_{\\perp}^{\\prime 2}\/2w^2}w.\n\\label{eq:determine_alp}\n \\end{eqnarray}\nComparing different powers in $r_\\perp^\\prime$ allows us to determine the finite number of coefficients $a_{l,p}$ uniquely,\nwhich depend only on the real number $w$. Whereas this ansatz can be used for some knots, there are counterexamples, and \nthere is no rigorous general proof as to why and in which cases this ansatz leads to success. \nA large value of $w$ ensures that the vortex lines \nare actually embedded in the beam, and not chopped off.\nFurthermore, for finite $w$, additional vortex lines in the shape of hairpins appear for larger $z$-values, which leads to the fact that \n$w$ must be chosen large enough. On the other hand, if $w$ is too large, \nthe polynomial increase in the polynomials describing the knots will not be attenuated quickly enough\nby the Gaussian, and thus intensity variations become huge, which is undesirable for our setup. \nIt is possible~\\cite{Dennis:Nature:2010} to further optimize these coefficients to separate vortices with regions of \nlarger intensity. \nOnce appropriate coefficients $a_{l,p}$ have been found, the light field can be \nwritten down as superposition of LG modes by rescaling back\ninto physical units. For the sake of clarity, let us introduce an auxiliary function $f$ defined as \n$f(r_\\perp\/\\sigma,z\/z_r)\/\\sigma:=\\mathrm{LG}_{l,p}^{\\sigma,z_r}({\\bf r}_\\perp,z)$. Then, we find \n\\begin{eqnarray}\n \\mathcal{E}_a({\\bf r}_\\perp,z)&=\\frac{A}{R_s}\\sum_{l,p}a_{l,p}(w)\\mathrm{LG}_{l,p}^{w,k_0R_sw^2}({\\bf r}_\\perp^\\prime,z^\\prime),\\\\\n &=\\frac{A}{R_s}\\sum_{l,p}a_{l,p}(w)\\frac{1}{w}f\\left(\\frac{r_\\perp^\\prime}{w},\\frac{z^\\prime}{k_0R_sw^2}\\right),\\\\\n &=A\\sum_{l,p}a_{l,p}(w)\\frac{1}{\\sigma}f\\left(\\frac{r_\\perp}{\\sigma},\\frac{z}{z_r}\\right)\\\\\n &=A\\sum_{l,p}a_{l,p}(w)\\mathrm{LG}_{l,p}^{\\sigma,z_r}({\\bf r}_\\perp,z).\n \\label{eq:superpos}\n\\end{eqnarray}\nHere, the amplitude $A$ of the light field has been introduced, which gives the intensity of the beam the right value in \nappropriate units. \nNote, that~\\refeq{eq:superpos} is no longer dependent on the choice of $R_s$. \n\n\\subsection{Rescaling}\n\nLet us rescale~\\refeqs{eq:eqmo1}{eq:eqmo2} by \nintroducing spatial $r^\\prime$ and time $t^\\prime$ coordinates rescaled \nby oscillator length $a_0=\\sqrt{\\hbar\/m\\omega}$ and trapping frequency $\\omega$, respectively,\n\\begin{eqnarray}\n t^\\prime&=\\omega t,\\\\\n r^\\prime&=\\frac{r}{a_0},\\\\\n \\psi^\\prime&=a_0^{3\/2}\\psi.\n\\end{eqnarray}\nThen, after multiplying~\\refeqs{eq:eqmo1}{eq:eqmo2} by $1\/m\\omega^2\\sqrt{a_0}$, ~\\refeqs{eq:eqmo1}{eq:eqmo2} become\n\\begin{eqnarray}\n \\fl i\\partial_{t^\\prime}\\psi_a^\\prime&=\\left(-\\frac{1}{2}\\nabla^{\\prime 2} + \\frac{r_\\perp^{\\prime 2}}{2} + \\gamma_z^2 \\frac{z^{\\prime 2}}{2} + \\kappa_{aa} |\\psi_a^\\prime|^2+\\kappa_{ab}|\\psi_b^\\prime|^2\\right)\\psi_a^\\prime\n +\\frac{\\Omega_a({\\bf r}^\\prime)\\Omega_b^*(z^\\prime)}{8\\omega\\Delta}\\psi_b^\\prime\\label{eq:rescaled_eqmo1}\\\\\n \\fl i\\partial_{t^\\prime}\\psi_b^\\prime&=\\frac{\\Omega_a^*({\\bf r}^\\prime)\\Omega_b(z^\\prime)}{8\\omega\\Delta}\\psi_a^\\prime+ \n \\left(-\\frac{1}{2}\\nabla^{\\prime 2} + \\frac{r_\\perp^{\\prime 2}}{2} + \\gamma_z^2 \\frac{z^{\\prime 2}}{2} + \\kappa_{ba} |\\psi_a^\\prime|^2+\\kappa_{bb}|\\psi_b^\\prime|^2-\\frac{\\delta}{\\omega}\\right)\\psi_b^\\prime\n \\label{eq:rescaled_eqmo2}\n\\end{eqnarray}\nwhere we left away the prime for our non-dimensional time and space variables and introduced $\\gamma_z=\\omega_z\/\\omega$, and $\\kappa_{kj}=4\\pi a_{kj}N\/a_0$.\nWe can express as $\\kappa_{kj}=a_0^2\/2a_h^2$ using the healing length $a_h=\\sqrt{8\\pi a_{kj} N\/a_0^3}$. \nSince the aspect ratio $\\gamma_z$ of the trapping frequencies is typically small, it is more convenient for numerical studies to \nrescale the elongated $z$-axis as follows:\n\\begin{eqnarray}\n z^{\\prime\\prime}&=\\gamma_z z^\\prime,\\\\\n \\psi^{\\prime\\prime}_i&=\\frac{1}{\\sqrt{\\gamma_z}}\\psi_i^\\prime.\n\\end{eqnarray}\nThis rescaling finally yields our equations of motion:\n\\begin{eqnarray}\n\\fl i\\partial_t\\psi_a&=\\left(-\\frac{1}{2}\\left[\\nabla^2_\\perp + \\gamma_z^2\\partial_z^2 \\right] + \\frac{r^2}{2} + \\gamma_z\\kappa_{ja} |\\psi_a|^2+\\gamma_z\\kappa_{jb}|\\psi_b|^2\\right)\\psi_a\n +\\frac{\\Omega_a({\\bf r})\\Omega_b^*(z)}{8\\omega\\Delta}\\psi_b\\label{eq:rescaled_eqmo_final1}\\\\\n\\fl i\\partial_t\\psi_b&=\\left(-\\frac{1}{2}\\left[\\nabla^2_\\perp + \\gamma_z^2\\partial_z^2 \\right] + \\frac{r^2}{2} + \\gamma_z\\kappa_{ja} |\\psi_a|^2+\\gamma_z\\kappa_{jb}|\\psi_b|^2-\\frac{\\delta}{\\omega}\\right)\\psi_b\n +\\frac{\\Omega_a^*({\\bf r})\\Omega_b(z)}{8\\omega\\Delta}\\psi_a\n \\label{eq:rescaled_eqmo_final2}\n\\end{eqnarray}\nwhere we have dropped the primes for convenience. \nFurthermore, applying the same rescaling to the paraxial wave equation~[\\refeq{eq:paraxial}], we find the following expression in \nour rescaled units:\n\\begin{equation}\n2ik_0\\partial_z \\mathcal{E}({\\bf r}_{\\perp},z) = -\\nabla^2_{\\perp}\\mathcal{E}({\\bf r}_{\\perp},z).\n\\label{eq:paraxial_rescaled}\n\\end{equation}\nHere, $k_0$ has been redefined and stands for $k_0=2\\pi a_0\\gamma_z\/\\lambda$. \nThen, the functions defined in~\\refeq{eq:LG_modes} remain solutions to~\\refeq{eq:paraxial_rescaled} in the new coordinates, \nand using our modified $k_0$, we can still write the Rayleigh length as $z_r=k_0\\sigma^2$, where \n$z_r$ is measured in units of $a_0\/\\gamma_z$ and $\\sigma$ in units of $a_0$.\n\nWe have $m=1.44\\times 10^{-25}$kg for ${}^{87}$Rb, so that a trapping frequency of $\\omega=2\\pi\\times 10$Hz leads to \nan order of magnitude of $a_0=3.4\\mu$m for the typical transverse extent of the BEC. \nAn order of magnitude estimate for the off-diagonal coupling terms is given by \n\\begin{equation}\n \\frac{\\Omega_a({\\bf r})\\Omega_b^*(z)}{8\\omega\\Delta}\\approx \\frac{\\Omega_a({\\bf r})\\Omega_b^*(z)}{\\Delta}0.016 {\\mathrm s} \\approx \\Range{2d2}{2d6}\n\\end{equation}\nfor $\\Omega_i\\approx \\Range{1}{10}$MHz and $\\Omega_i\/\\Delta=\\Range{0.01}{0.1}$. \nThe required tight focusing of the beam leads to the fact, that we only need \nmoderate beam powers to achieve adequate Rabi-frequencies. \nIn the following, we set the two-photon detuning $\\delta=0$ to achieve optimal transfer. \n\nFinally, we need to find an estimate for the Rabi pulse duration $t_d$. \nTo this end, consider the simple case, where $\\Omega_a$ and $\\Omega_b$ describe driving by plane waves. \nWe seek to find the optimal time for maximal transfer of atoms from $\\ket{a}$ to $\\ket{b}$. \nGiven that the ``off-diagonal'' terms in the coupled equations~\\refeqs{eq:rescaled_eqmo_final1}{eq:rescaled_eqmo_final2} are much larger than the diagonals, we can approximate \nthe population $N_b$ in state $\\ket{b}$ as \n\\begin{equation}\nN_b\\sim 1-\\cos^2\\left(\\frac{\\Omega_a\\Omega_b^*}{8\\Delta \\omega}t\\right).\n\\end{equation}\nThis expression allows us to find the optimal (non-dimensional) time duration of the Rabi pulse $t_d$ that leads to a complete transfer of population:\n\\begin{equation}\n t_d=\\frac{\\pi\/2}{\\Omega_a\\Omega_b^*\/8\\Delta \\omega}.\n \\label{eq:t_d}\n\\end{equation}\nOn the other hand, this consideration does not apply to our case due to the spatial dependence of the Rabi frequency $\\Omega_a({\\bf r})$. \nDue to spatial dependence of the intensity variations, we do not really have an optimal pulse duration \n$t_d$, and our choice of $t_d$ is a compromise between on one hand \nhaving sufficiently large transfer of atoms to state $\\ket{b}$\nand on the other hand avoiding population being transferred back from\nstate $\\ket{b}$ to state $\\ket{a}$ in positions where the intensity is large and thus the transfer \ndynamics is faster.\nHence, we have to choose shorter pulse durations than what~\\refeq{eq:t_d} suggests. \nFurthermore, due to the nodal lines in the pump beam, it is never possible to use this arrangement for \na total conversion of atomic population from $\\ket{a}$ to $\\ket{b}$. \nWith these aspects kept in mind, we may still use~\\refeq{eq:t_d} as a rough estimate for the order of magnitude for our choice of $t_d$.\n\n\\section{Numerical Investigation}\n\nIn the previous section, we elaborated on how to inscribe complex knotted vortex lines into matter waves. Whereas the \nsimple argument of~\\refeq{eq:simple_argument} allows us to conclude that our setup should work at least in the perturbative regime for \nshort time-scales and small amounts of atoms in state $\\ket{b}$, we cannot infer what happens beyond this perturbative regime. \nIn this section, we aim to numerically study excitation and decay of our highly excited states into more elementary unknots or \nvanishing of the latter by collision and annihilation of vortex lines. \nWe will do so by considering specific examples. However, these examples are by no means an exhaustive treatment of the\nthe huge potential manifold of realizations possible. To thoroughly understand the dynamics remains \nan elusive goal, which goes beyond the scope of this paper. \n\n\\subsection{Hopf-link}\n\nOne of the simplest torus knots is the so-called Hopf-link, which corresponds to setting $n=2$ in~\\refeq{eq:ansatz_light} and is \ndepicted e.g. by the green isodensity surface in~\\reffig{fig:hopf_projection}(b). Using the method outlined in~\\ref{sec:knotted_light_field} and~\\refeq{eq:determine_alp}, we find the following superposition \ncoefficients of Laguerre-Gaussian polynomials for a Hopf-link:\n\\begin{eqnarray}\n \\frac{\\Omega_a\\Omega_b^*}{8\\omega\\Delta} &= A[(1-2w^2+2w^4)\\mathrm{LG}_{00}+(2w^2-4w^4)\\mathrm{LG}_{01}+2w^4\\mathrm{LG}_{02}\\nonumber\\\\\n &-4\\sqrt{2}w^2\\mathrm{LG}_{20}]\\Theta(t_{d}-t).\n\\end{eqnarray}\nTo this end, let us consider the dynamics for $\\kappa_{ab}=4\\pi a_{ab} N\/a_0=1887$, $\\gamma_z=0.125$. \nWe set the two-photon detuning to $\\delta=0$.\nThis can be achieved by using typical parameters of ${}^{87}$Rb with $\\omega=2\\pi\\times 10 $Hz, $\\omega_z=2.5\\pi $Hz, $N=93000$, \n$a_{ab}=5.5$nm, $\\lambda=780$nm, which amounts to typical \nunits of length scales of $a_0=3.4\\mu$m in the $(x,y)$-plane and $a_0\/\\gamma_z=27.2\\mu m$ in the $z$-direction.\nThomas-Fermi like dynamics, where the nonlinearity is large compared to the broadening due to the Laplacian, \nis preferable to ensure robustness of the vortex cores and avoid the trivial broadening of the latter. \nFor that reason, we have to choose a sufficiently large value for $\\kappa_{ij}$.\nFor the other $a_{ij}$, we assume the ratio $a_{aa}$:$a_{ab}$:$a_{bb}$ to be given by $1.03$:$1$:$0.97$~\\cite{Cornell:PRL:1998}.\nWith respect to our Rabi pulse, we choose \n$A=450$, $w=1.5$, $\\sigma=0.675$ corresponding to $2.3\\mu$m, \nand a pulse duration of $t_{d}=0.002$ corresponding to $31.8\\mu$s.\n\nWe use the common Fourier split-step method~\\cite{Agrawal_Kivshar:2003} and adaptive Runge-Kutta algorithm to fourth order for the time-step for numerical computation. \nParts of the code were written using~\\cite{XMDS}. To find the appropriate state $\\psi_a(t=0)$, we used imaginary time-evolution (e.g.~\\cite{Tosi:PRE:2000}).\nIt is crucial to use a variable time-step, since the dynamics during the time duration of the Rabi-pulse $t_d$ has to be temporally \nfully resolved, and thus the time-step has to be much smaller than $t_d$ within $t_d$. After $t>t_d$, the time-step can be chosen \nmuch larger, since it only needs to resolve the characteristic time-scale of the BEC dynamics. We used time-steps varying between $\\Delta t=\\Range{d-8}{5d-4}$ and\na spatial resolution of $\\Delta x=0.05-0.06$ with $220^3$ points.\n\n\\reffig{fig:hopf_projection}(a-d) depict an isodensity surface of the condensate wave function $\\psi_b$ with small value, \nwhich serves as an illustration of the knotted vortex core, and its phase profile in different ways, shortly after the imprinting occurred.\nHere, the initial wave function $\\psi_a(t=0)$ was computed as the ground state of the trap and $\\psi_b(t=0)=0$. \n\nA basic qualitative anticipation of the dynamics can be found by \nprojecting the knot orthogonally to the beam- or $z$-axis and considering the momentum associated with \nthe (unit-)areas, as elaborated in~\\cite{Ricca:Proc:2013}. Consider the projection in the $(x,y)$-plane, as \nshown in~\\reffig{fig:hopf_projection}(c). \nThe index $I_j$ assigned to each of the areas $R_j$ enclosed by the vortex lines $\\gamma$ can be found by evaluating the sum~\\cite{Ricca:Proc:2013}\n\\begin{equation}\n I_j=\\sum_{{\\bf \\rho}\\cap {\\bf \\gamma}} \\mathrm{sign} \\left( {\\bf e_z} {\\bm \\rho} \\times {\\bf t} \\right),\n \\label{eq:I_j}\n\\end{equation}\nwhich runs over all intersections of the chosen vector ${\\bf\\rho}$ with the projected vortex line $\\gamma$ for a given region $R_j$. \nHere, ${\\bm \\rho}$ denotes an arbitrary vector pointing from the inside to the outside of the enclosed area \nand $\\mathrm{sign}$ is the usual sign-function taking the values $\\mathrm{sign}(\\ast)=\\pm 1$. \nFurthermore, the vector $\\bf{t}$ is a tangent vector to the vortex line curve (denoted as $\\gamma$) at the point where ${\\bm \\rho}$ crosses $\\gamma$. \nThe direction of $\\bf t$ is given by the orientation of the arcs, which is determined by the phase~[see~\\reffig{fig:hopf_projection}(c)].\nThe values assigned to the regions in~\\reffig{fig:hopf_projection}(c) correspond to the values computed by~\\refeq{eq:I_j}.\nThe associated momenta of a region $R_j$ can be found by~\\cite{Ricca:Proc:2013}\n\\begin{equation}\n \\left({\\bf p}_z\\right)_j=\\oint_{R_j} {\\bm \\omega} d^2r I_j,\n\\end{equation}\nwhere $\\bm \\omega$ denotes the vorticity. \n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig2.eps}\n\\caption{\n(color online) (a) Illustration of the phase of $\\psi_b$ in the $(x,y)$-plane after short time evolution. (b) Additional to the phase in the $(x,y)$-plane from Fig. (a), \nthe green surface represents a low-value isodensity surface of $|\\psi_b|^2$ around the \nvortex line imprinted on the BEC. (c) Projection of Fig (b) into the $(x,y)$-plane and associated indices according to~\\refeq{eq:I_j}. Expected dynamics should thus be, that \nthe central region moves faster in the $z$-direction relative to its neighbouring regions. \n(d) shows again the isosurface from Fig. (b), however, the coloring of the isosurface illustrates the value of the phase at each position of the isosurface. \n\\label{fig:hopf_projection}}\n\\end{center}\\end{figure}\n\nWhat one can deduce from these arguments is a movement in the positive $z$-direction with the central region traveling fastest [\\reffig{fig:hopf_projection}(c)]. \nLet us now look at the numerically computed dynamics. \nSnapshots of the latter are shown in~\\reffig{fig:hopf_to_unknot} (see also movies \\href{hopf_1.mp4}{\\textit{hopf\\_1.mp4}} and \\href{hopf_2.mp4}{\\textit{hopf\\_2.mp4}}).\nClearly, there is an overall center-of-mass mass motion towards the positive $z$-direction, as expected. \nHowever, reconnection of vortex lines as well as finite size effects of the condensate leading to sharp gradients in the density lead to a very involved dynamics, \nwhich we were not able to predict or understand in simple terms.\nSurprisingly, the formation decays into two unknots that propagate into the positive and negative $y$-direction, respectively. \n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig3.eps}\n\\caption{\n(color online) Dynamics of the condensate wavefunction $\\psi_b$, whose vortex lines form a Hopf-link. Both upper and lower row show the dynamics in timesteps of \n$\\Delta t=0.1$ (corresponding to $\\Delta t=1.6$ms) starting from $t=0.05$ (corresponding to $t=0.8$ms). \nBoth rows use the same isosurface for $|\\psi_b|^2$, however, in the lower row we cropped the box at a smaller value, so that the ``boundary'' of\nthe condensate does not obfuscate the dynamics of the vortex lines. Additional to that, we used the phase to color the isosurface in the lower row. \nWe can see an overall drift the positive $z$-direction, which can be understood \nfrom~\\reffig{fig:hopf_projection}. \nSee also the movies \\href{hopf_1.mp4}{\\textit{hopf\\_1.mp4}} and \\href{hopf_2.mp4}{\\textit{hopf\\_2.mp4}} for the full propagation dynamics.\n\\label{fig:hopf_to_unknot}}\n\\end{center}\\end{figure}\n\n\\subsection{Trefoil}\\label{sec:trefoil}\n\nIn this section we will study the dynamics of a trefoil imprinted on a BEC. The dynamics is generic, quite different initial conditions lead to similar results,\nthe basics of which can be understood in the simple terms described before. \n\nIn this case, let $w=1.2$. Instead of using the prefactors $a_{l,p}(w)$ from~\\refeq{eq:determine_alp}, \nwe use the optimized prefactors given in~\\cite{King:thesis,Dennis:Nature:2010} of the Laguerre-Gaussian modes for the light field. \n\\begin{equation}\n \\fl \\frac{\\Omega_a\\Omega_b^*}{8\\omega\\Delta} = A( 1.51\\mathrm{LG}_{00} - 5.06\\mathrm{LG}_{10} + 7.23\\mathrm{LG}_{20}- 2.03\\mathrm{LG}_{30} - 3.97\\mathrm{LG}_{03} )\\Theta(t_{d}-t).\n\\end{equation}\nTo this end, let us use $\\sigma=0.78$ corresponding to $2.65\\mu$m and $A=700$. \nFurthermore, let us choose $\\kappa = 3753$, $\\gamma_z = 0.05$. \nThis can be realized using again $\\omega=2\\pi\\times 10 $Hz, $\\omega_z=1\\pi $Hz,\n$N=1.85\\times10^5$, $a_{ab}=5.5$nm and $\\lambda=780$nm. \nWith respect to our Rabi pulse, we use\na pulse duration of $t_{d}=0.004$ corresponding to $63.7\\mu$s.\n\n\\reffig{fig:trefoil_projection}(a) illustrates the phase of $\\psi_b$ and (b) its vortex core shortly after imprinting. \nProjecting onto the $(x,y)$-plane and assigning values according to~\\refeq{eq:I_j} leads to~\\reffig{fig:trefoil_projection}(c).\nAgain, we can expect that the overall knot will propagate in the positive $z$-direction, and the central part carries the largest momentum.\n\\reffig{fig:trefoil_projection}(d) illustrates the expected reconnection dynamics according to the rules found in e.g.~\\cite{Koplik:PRL:1993}.\nThe arrows indicate the direction of ${\\bf t}$, which is again determined by the phase. The inset (grey dashed box) illustrates how \nthe vortex lines should reconnect. If we apply this simple rule to all three crossings, we can expect that the green vortex line evolves into what is depicted by the solid \nblack line, i.e. a decay into two unkots.\n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig4.eps}\n\\caption{\n(color online) (a) Illustration of the phase of $\\psi_b$ in the $(x,y)$-plane after short time evolution. (b) Additional to the phase in the $(x,y)$-plane from Fig. (a), \nthe green line represents a low-value isodensity surface of $|\\psi_b|^2$ around the \nvortex line imprinted on the BEC. (c) Projection of Fig (b) into the $(x,y)$-plane and associated indices. \nSince the indices reflect the momenta of the regions, we expect that \nthe central region moves faster in the $z$-direction relative to its neighboring regions. \n(d) shows the expected breakup of the green vortex lines \ninto the solid black lines. The black arrows illustrate the orientation, which is determined by the phase. The grey dashed inset illustrates \nthe expected product of collision and reconnection of the vortex lines according to, e.g.~\\cite{Koplik:PRL:1993}.\n\\label{fig:trefoil_projection}}\n\\end{center}\\end{figure}\n\nLet us now consider the dynamics, which is shown in~\\reffig{fig:vortex_expelled} (see also the movie \n\\href{trefoil_1.mp4}{\\textit{trefoil\\_1.mp4}} and \\href{trefoil_2.mp4}{\\textit{trefoil\\_2.mp4}} for full dynamics).\nWe see that the vortex lines of this specific trefoil knot first reconnect [see~\\reffig{fig:vortex_expelled}], and the \ncentral regions travels fastest into the positive $z$-direction according to our expectation~\\reffig{fig:trefoil_projection}(b--c).\nAfter that, the central region expels an unknot or vortex ring which decouples from the rest of the knot, as shown in~\\reffig{fig:vortex_expelled}. \nThe single unknot propagates to the positive $z$-direction faster than the remaining part of the knot. \nInterestingly, our dynamics differs from the usual dynamics \nof a clear breakup into two unknots as expected from [\\reffig{fig:trefoil_projection}(c), solid black line] \nand what has been found in~\\cite{Barenghi:PRE:2012,Irvine:arXiv:2015}. \nDifferences to previous observations are due to finite size effects of the cloud, that in our case \nthe knot has a large aspect ratio, which with our rescaling basically means that the dynamics in the $z$-direction is much slower, smallness of the knot, and finally that we \nactually consider two coupled fields (dynamics of $\\psi_a$ not shown). \n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\textwidth]{fig5.eps}\n\\caption{\n(color online) Dynamics of the condensate wavefunction $\\psi_b$, whose vortex lines form a trefoil knot. Both upper and lower row show the same dynamics in timesteps of \n$\\Delta t=0.1$ (corresponding to $\\Delta t=0.8$ms) starting from $t=0.05$ (corresponding to $t=0.8$ms). \nBoth rows use the same isosurface for $|\\psi_b|^2$, however, in the lower row we cropped the box at a smaller value, so that the ``boundary'' of\nthe condensate does not obfuscate the dynamics of the vortex lines. Additional to that, we used the phase to color the isosurface in the lower row. \nUpon evolution, vortex cores reconnect, which leads to a decay into a vortex ring being expelled from \nthe central region, which can be understood from~\\reffig{fig:trefoil_projection}).\nSee also the movie \\href{trefoil_1.mp4}{\\textit{trefoil\\_1.mp4}} and \\href{trefoil_2.mp4}{\\textit{trefoil\\_2.mp4}} for the full propagation dynamics.\n\\label{fig:vortex_expelled}}\n\\end{center}\\end{figure}\n\n\\section{Conclusions}\nRecently there has been a growing interest in the dynamics of knotted vortex lines in BECs. \nHowever, thus far there has been no actual proposal on how to excite such matter waves in a controlled fashion. \nIn this work, we have presented a setup with physically realistic parameters to create such knotted vortex lines in ultracold matter waves. \nWe combined recent theoretical and experimental results from complex light shaping which allowed us to create knotted vortex lines embedded in light fields. We \nuse the latter as a probe field for three-level atoms in a $\\Lambda$-type setup to inscribe the nodal lines to BECs. \nThe setup is quite generic, and allows investigation of a large variety of potential dynamics. \nThe finite time span to populate state $\\psi_b$ reduces production of soundwaves. \nThe finiteness of the condensate, the smallness of the knot as well as the reconnections of the vortex lines give\nrise to a very involved dynamics. \nThe velocity in the $z$-direction of the vortex knot should not be large. \nThis velocity can be controlled by the ratio between the trapping frequencies $\\gamma_z=\\omega_z\/\\omega$ relative to the condensate. \nThe probe field which determines $\\Omega_a$ should not be too spatially broad, otherwise the value of $\\gamma_z$ required to fully embed the knotted vortex line \nbecomes impractical. We have assumed that the light field can be treated within the paraxial approximation, although we note that, in principle, \nthis condition can be relaxed to non-paraxial knotted light fields~\\cite{Dennis:OL:11}. \nDue to the tight focusing of the optical beam, only moderate powers of the light fields that determine the Rabi-frequencies $\\Omega_i$ are required. \nThus, the setup should work generically for a large range of parameters. \nWe have shown two illustrative examples of a trefoil knot and a Hopf-link, and discussed the dynamics using already well-established techniques. \nThe data presented in this paper is available online at~\\cite{Maucher:data}.\n\n\\section{Acknowledgements} \nThis work was funded by the Leverhulme Trust Research Programme Grant RP2013-K-009, SPOCK: Scientific Properties Of Complex Knots. \nWe would like to thank P. M. Sutcliffe, J. L. Helm, T. P. Billam, D. Sugic and M. Dennis for stimulating discussions. \n\n\\section*{References}\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDetailed understanding of non-contact friction and energy transfer processes in nanostructures is of great importance, both \nfrom the conceptual and practical viewpoints.\nExisting theoretical studies, starting with the seminal paper by Pendry \\cite{Pendry1}, mostly consist of calculations of friction coefficients, i.e. friction force between two parallel dielectric plates (e.g. supported graphenes) in uniform relative motion which is experimentally not easily measured (e.g. current drag in one graphene caused by current flow in another \none) \\cite{Theory1,Theory2,Theory3,Theory4,Theory5,Theory6,Theory7}.\n\nWhile the experiments with two slabs in parallel relative motion with constant velocity are difficult to perform, we suggest here \nthat for the same systems experiments with slabs in relative oscillatory motion with fixed or variable frequency might be easier to \nperform, and could lead to new and interesting observations. \nRecently a similar approach has been realized experimentally \\cite{QF_exp_osc1,QF_exp_theory_osc,QF_exp_osc2,QF_exp_osc3}.\nIn these experiments the system (usually an AFM tip above the surface) oscillates at some characteristic \nfrequency. These oscillations are then, because of various dissipation mechanisms (which includes quantum friction), damped. \nOur model is based on a slightly different concept; one of the slabs, e.g. the AFM tip, is driven with variable frequency. This means that the friction can \nbe deduced from the energy dissipated in one oscillating cycle. \nIn this paper we provide a general theoretical description of such processes, expecting that this method might become a useful tool to study dynamical properties of \nlow-dimensional systems \\cite{Munez}.\n\n\nThe main objective of this paper is therefore a theoretical description of these phenomena in systems consisting \nof two non-touching polarizable media, specifically conservative (van der Waals or Casimir) and \ndissipative forces (quantum friction) between two quasi-twodimensional (q2D crystals) in \nrelative parallel and oscillatory motion. While the case of slabs in parallel uniform motion has been extensively studied \\cite{Pendry1,Persson1,Volokitin1,\nVolokitin2,Theory4,Pendry2,Philbin}, here we develop an analogous theory describing interaction of atomically thick slabs (q2D crystals) in oscillatory motion. \n\n\n \nIn Sec.\\ref{Sec2} the expressions for van der Waals and dissipative energies and forces are derived for such a q2D system in a very general case,\nfor variable slab temperatures and dynamical properties characterized by their surface response functions $D_1$ and \n$D_2$, and for variable oscillating frequencies and amplitudes. \nWe assume 2D translational invariance and neglect retardation for the slab distances in consideration. \nFor the sake of clarity and comparison, in Appendix\\ref{AppA} we derive analogous results for the \ncase of parallel uniform motion, recovering but also generalizing some earlier results \\cite{trenjePRB,PFK}. \n \n \n\nIn Sec.\\ref{DerofD} we derive general expressions for surface response functions $D_i$ for multilayer slabs, later to \nbe specified for monolayers of a substance like graphene or silicene adsorbed on dielectric substrates. Surface response functions \n$D_1$ and $D_2$ will be the key ingredients in the expressions describing dissipative and \nreactive processes in Sec.\\ref{Sec2} and Sec.\\ref{DerofD}. \nIn Sec.\\ref{DerofD} we also show how to calculate surface response functions $D_i$ for a specific case of q2D crystals on a dielectric substrate\nThe expression for the surface excitation propagator of a system of \ntwo coupled slabs is also derived.\n\n \nIn Sec.\\ref{nabnak} we present the models used to describe the q2D crystal and substrate dynamical response. \nWe study the specific case of a graphene monolayer on a dielectric substrate, which is chosen to be ionic \ncrystal SiO$_2$. \nThe substrate is considered as a homogenous semiinfinite ionic crystal SiO$_2$ with the appropriate dielectric function in \nthe longwavelength limit. Graphene monolayer dynamical response is determined from \nfirst principles. Also some computational details are specified. \n\n\nIn Sec.\\ref{resuuult} general expressions of previous sections are applied to the system of \ntwo slabs, where each slab represents a graphene($E_{Fi}$)\/SiO$_2$ system, and where graphene doping is characterized \nby Fermi energy $E_F$ relative to the Dirac point. \n\n \nIn Sec.\\ref{cmspe} we demonstrate how the spectra of electronic excitations in one slab \nand in two coupled slabs depend on graphene doping $E_F$.\n \n\nThe form of these coupled ecitations is responsible for the behaviour of the atractive forces and dissipation.\nWe first discuss in Sec.\\ref{weakvdW} the modification of van der Waals force for oscillating in comparison \nwith the static slabs. Van der Waals energies depend on two factors. They increase with the increased \ngraphene doping, but are reduced for the asymmetric doping when excitations in two slabs are off-resonance. \nDynamical vdW energy shows unusual behavior: it starts as plateau, and then decreases.\nThis is, because the fast Dirac plasmon in one slab for low driving frequencies $\\omega_0<\\omega_p$, still perfectly follows Doppler shifted charge density fluctuations in another slab. \nFor larger driving frequencies this is not the case and vdW energies decrease. \nFinally, for small or zero doping the $\\pi\\rightarrow\\pi^*$ and $\\pi\\rightarrow\\sigma$ excitations \ncause linear weakening of the dynamical vdW energy. \n \nIn Sec.\\ref{DISsub} we calculate and discuss how dissipated power depends on various parameters: \ndriving amplitude $\\rho_0$ and frequency $\\omega_0$, on the separations between slabs $a$ and on the \nsubstrate. We find simple $\\rho^2_0$ dependence, while the $\\omega_0$ dependence is determined \nby the intensity of resonant coupling between hybridized Dirac plasmons and substrate TO phonons.\nWe found that in realistic grahenes (in comparison with Drude model when excitation of undamped Dirac plasmons provides \nunrealistically strong $2\\omega_p$ peak in the dissipated power) the dissipation power peak is strongly reduced and red shifted.\nWe also explain why the substrate substantially reduces dissipated power peak. \nFor larger separations $a$ additional peaks appear in dissipated power originating from the excitations of hybridized substrate phonons. \n\nIn Sec.\\ref{DISdop} we explore how the dissipated power depends on graphene dopings.\nWe show that if one graphene is pristine ($E_F=0$) it causes the disappearance of strong $2\\omega_p$ peak \nin the dissipated power. Moreover, for larger separations the doping causes shifts, appearance and disappearance of many peaks originating \nfrom resonant coupling between hybridized substrate phonons and Dirac plasmons.\n\n \nIn Sec.\\ref{Sec5} we present the conclusions.\n\n\n\n\n\n\\section{General theory: Oscillating slabs}\n\\label{Sec2}\n\n\n\\subsection{Van der Waals energy and force}\nIn Appendix \\ref{vdWenergyforce} we have derived van der Waals energy and force between two slabs \nin uniform relative motion in some detail because it will help us to treat a \nsimilar problem of two oscillating slabs. \n\nWe shall later assume that the slabs consist of graphene monolayers with variable doping, deposited on dielectric slabs of thickness $\\Delta$ \ndescribed by local dielectric functions $\\epsilon(\\omega)$, as shown in Fig.\\ref{Fig1}. The left slab mechanically oscillates with frequency $\\omega_0$ and \namplitude ${\\hbox{\\boldmath $\\rho$}}_0$ relative to the right slab. Again we calculate the diagram in Fig.\\ref{FigA1} as in the \\ref{vdWenergyforce}, but now the slab parallel coordinates change in time \nas \n\\begin{equation}\n{\\hbox{\\boldmath $\\rho$}}-{\\hbox{\\boldmath $\\rho$}}_1\\rightarrow{\\hbox{\\boldmath $\\rho$}}-{\\hbox{\\boldmath $\\rho$}}_1-{\\hbox{\\boldmath $\\rho$}}_0(\\sin\\omega_0t-\\sin\\omega_0t_1)\n\\label{kukuli}\n\\end{equation}\nso that instead of (\\ref{nabij3}) we have\n\\begin{equation}\n\\begin{array}{c}\nE_{c}=\\int^{\\infty}_{-\\infty}dt_1\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\ne^{-i{\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0(\\sin\\omega_0t-\\sin\\omega_0t_1)}\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty} dzdz_1dz_2dz_3 \nS_1({\\bf Q},z,z_1,t-t_1)V({\\bf Q},z,z_3)\n\\nonumber\\\\\n\\nonumber\\\\\nD_2({\\bf Q},z_3,z_2,t-t_1)V({\\bf Q},z_2,z_1). \n\\end{array}\n\\label{bijem3}\n\\end{equation}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.9\\columnwidth]{Fig1.pdf}\n\\caption{Geometry of the system.}\n\\label{Fig1}\n\\end{figure} \nIf we use \n\\[\ne^{iz\\sin\\phi}=\\sum^{\\infty}_{m=-\\infty}J_m(z)e^{im\\phi} \n\\]\nwhere $J_m$ are Bessel functions, after Fourier transformation in $\\omega$ space, using expressions (\\ref{Def1}--\\ref{Def3}), (\\ref{impsv}) and \nintegration over $z$ coordinates we obtain \n\\begin{equation}\n\\begin{array}{c}\nE_{c}=\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}\n\\sum^{\\infty}_{m,m'=-\\infty}J_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)J_{m'}({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\n\\left[2n_1(\\omega)+1\\right]e^{i(m-m')\\omega_0t}\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\nImD_1({\\bf Q},\\omega)ReD_2({\\bf Q},\\omega+m\\omega_0).\\hspace{3cm}\n\\end{array}\n\\end{equation}\nHere we have also used the fact that $ImD_2({\\bf Q},\\omega)$ is an antisymmetric function of \n$\\omega$ and does not contribute to integration. \nWe see that the energy oscillates in time with frequencies $(m-m')\\omega_0$. \nIf we assume to measure energies on a time scale $\\Delta t>T$, where $T=\\frac{2\\pi}{\\omega_0}$ is the maximal \nduration of one cycle, then we can average over $T$\n\\begin{equation}\n\\frac{1}{T}\\int^{T}_0dte^{i(m-m')\\omega_0t}=\\delta_{mm'},\n\\label{Timeaverage}\n\\end{equation}\nand find the result independent of time: \n\\[\n\\begin{array}{c}\nE_{c}=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}\n\\sum^{\\infty}_{m=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\\ \\left\\{[2n_1(\\omega)+1]\nImD_1({\\bf Q},\\omega)ReD_2({\\bf Q},\\omega+m\\omega_0)+\\right.\n\\nonumber\\\\\n\\nonumber\\\\\n\\left.[2n_2(\\omega)+1]ImD_2({\\bf Q},\\omega)ReD_1({\\bf Q},\\omega+m\\omega_0)\\right\\},\n\\end{array}\n\\]\nwhere the expression in curly brackets is fully analogous to the one in (\\ref{jura66}), but now $\\omega'\\rightarrow\\omega_m=\\omega+m\\omega_0$.\nInclusion of higher order processes follows the same procedure as for the parallel motion in \\ref{vdWenergyforce}. After \nintegration over the coupling constant, we obtain the result analogous to (\\ref{jujujuju7}) \n\\begin{eqnarray}\nE_{c}=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\sum^{\\infty}_{m=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\times\n\\label{vdwFIN}\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}A({\\bf Q},\\omega,\\omega_m)\\hspace{3cm}\n\\nonumber\n\\end{eqnarray}\nwhere $A$ is given by (\\ref{jalko}) and (\\ref{jujucka}), with $\\omega_m=\\omega+m\\omega_0$.\n\nAgain, the limiting cases can be obtained from Sec.\\ref{vdWenergyforce}.\nFor $\\omega_0=0\\ (\\omega'=\\omega)$ and ${\\hbox{\\boldmath $\\rho$}}_0=0$ we find the well known result for van der \nWaals interaction when the slabs are at rest \\cite{vdW2007,Pedro}: \n\\begin{eqnarray}\nE_{c}(a)=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\int^{\\infty}_{0}\\frac{d\\omega}{2\\pi}\\ sgn\\omega\\times\\hspace{2cm} \n\\nonumber\\\\\n\\nonumber\\\\\n\\hspace{3cm}Im\\ln\\left[1-e^{-2Qa}D_1({\\bf Q},\\omega)D_2({\\bf Q},\\omega)\\right]\n\\nonumber\n\\end{eqnarray}\nFor finite frequency $\\omega_0$ and $D_1=D_2=D$ we find: \n\\begin{eqnarray}\nE_{c}(a)=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\n\\sum^{\\infty}_{m=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}sgn\\omega\\ \nIm \\ln\\left[1-e^{-2Qa}D({\\bf Q},\\omega)D({\\bf Q},\\omega_m)\\right].\n\\nonumber\n\\end{eqnarray}\nWe notice that the frequency integrals are the same as \nin (\\ref{jujujuju7}--\\ref{sinko}). Also, the attractive van der Waals force \nbetween two oscillating slabs is given by \n\\begin{eqnarray}\nF_{\\perp}(a)=-\\frac{dE_{c}(a)}{da}=\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\n\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}Qe^{-2Qa}\n\\sum^{\\infty}_{n=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\times\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}B({\\bf Q},\\omega,\\omega_m) \n\\end{eqnarray}\nwhere the function $B$ is given by (\\ref{prcko}) and (\\ref{prdf99}). \nThe same holds for the $\\omega_0\\rightarrow 0$ or $D_1=D_2=D$ limits when the \nexpressions for $B$ become (\\ref{jurec}) or (\\ref{kurec}), respectively.\n\\subsection{Dissipated power}\nWe can perform the calculation of the dissipated power for two slabs oscillating parallel to each \nother with amplitude ${\\hbox{\\boldmath $\\rho$}}_0$ and frquency $\\omega_0$ in analogy with the previous treatment of \ntwo slabs in uniform relative motion in Sec.\\ref{jurniga}. Again, we have to transform the parallel \ncoordinates in the left slabs as in (\\ref{kukuli}).\nThen (\\ref{losse5}), after integration over $t_1$ becomes \n\\begin{equation}\n\\begin{array}{c}\nP_{12}(t)=-i\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\int\\frac{d\\omega}{2\\pi}\\sum^{\\infty}_{m,m'=-\\infty}\n\\\\\n\\\\\n(-1)^{m+m'}e^{i(m'-m)\\omega_0t}(m'\\omega_0-\\omega)\\ \nJ_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)J_m'({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\\\\n\\\\\nS_1({\\bf Q},|\\omega|,z,z_1)\\otimes V({\\bf Q},z,z_3)\\otimes \n\\\\\n\\\\\nD_2({\\bf Q},m'\\omega_0-\\omega,z_3,z_2)\n\\otimes V({\\bf Q},z_2,z_1) \n\\end{array}\n\\end{equation}\nWe see that the energy transfer rate is time dependent and oscillates \nwith frequency $(m'-m)\\omega_0$. Again, from (\\ref{Timeaverage}) we see \nthat for time intervals large with respect to the oscillation period $T$ the \nterms $m\\ne m'$ do not contribute and the energy transfer rate is\n\\begin{equation}\n\\begin{array}{c}\nP_{12}=\n-i\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\int\\frac{d\\omega}{2\\pi}\\sum^{\\infty}_{m=-\\infty}(m\\omega_0-\\omega)\\ \nJ^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\\\\n\\\\\nS_1({\\bf Q},|\\omega|,z,z_1)\\otimes V({\\bf Q},z,z_3)\\otimes \n\\\\\n\\\\\nD_2({\\bf Q},m\\omega_0-\\omega,z_3,z_2)\n\\otimes V({\\bf Q},z_2,z_1) \n\\end{array}\n\\label{loss11}\n\\end{equation} \nIf we now use (\\ref{Def1}), the definitions (\\ref{Def2}) and (\\ref{Def3}) \nof the surface correlation function and the surface excitation propagator, respectively, and the connection (\\ref{impsv}) between the surface \ncorrelation function and the imaginary part of \nsurface excitation propagator, equation (\\ref{loss11}) can be written as\n\\begin{eqnarray}\nP_{12}=-\\frac{\\hbar}{\\pi}\\sum^{\\infty}_{m=-\\infty}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\hspace{3cm} \n\\nonumber\\\\\n\\label{loss12}\\\\\n\\int\\frac{d\\omega}{2\\pi}\\ \\omega_m\\ [2n_1(\\omega)+1]\\ ImD_1({\\bf Q},\\omega) ImD_2({\\bf Q},\\omega_m).\\hspace{1cm} \n\\nonumber\n\\end{eqnarray} \nEvaluating (\\ref{loss12}) we have used the fact that the real part of the function under summation and integration is \nodd and the imaginary part is an even function of $n$ and $\\omega$.\n$P_{12}$ is the energy transferred from the left to the right slab.\nNow we have to repeat the discussion in Sec.\\ref{jurniga} and substract the part of this energy which \nwill be reversibly returned to the left slab. The same arguments, leading to (\\ref{losse17}), will give \nthis energy to be\n\\begin{eqnarray}\nP'_{12}=\\hbar\\sum^{\\infty}_{n=-\\infty}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_n({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\hspace{2cm} \n\\nonumber\\\\\n\\label{loss13}\\\\\n\\int\\frac{d\\omega}{2\\pi}\\ \\omega\\ [2n_1(\\omega)+1]\\ Im D_1({\\bf Q},\\omega) Im D_2({\\bf Q},\\omega_n). \n\\nonumber\n\\end{eqnarray} \nExpression (\\ref{loss13}) represents the energy transferred from the \nleft to right but which will be reversibly returned, as shown in Fig.\\ref{FigA3}b. Therefore the energy \nwhich is irreversibly transferred from the left to the right, i.e. the dissipated power, is \n\\begin{equation}\n\\begin{array}{c}\nP_{1}=P_{12}-P'_{12}=\n2\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0) \n\\\\\n\\\\\n\\int\\frac{d\\omega}{2\\pi}\\ [2n_1(\\omega)+1] ImD_1({\\bf Q},\\omega) ImD_2({\\bf Q},\\omega_m).\n\\end{array}\n\\label{diss1}\n\\end{equation}\nAnalogous calculation would give the energy dissipated in the process where the charge \nfluctuation in the right slab induces fluctuations in the left slab. We have to \nexchange $1$ and $2$ in (\\ref{diss1}) and replace $m\\rightarrow-m$. \nRepeating the steps in (\\ref{jutro}) the final result becomes: \n\n\n\\begin{eqnarray}\nP=P_1+P_2=\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\n4\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\\hspace{2cm}\n\\nonumber\\\\\n\\label{njunja}\\\\\n\\left[n_1(\\omega)-n_2(\\omega_m)\\right]ImD_1({\\bf Q},\\omega)ImD_2({\\bf Q},\\omega_m).\n\\nonumber\n\\end{eqnarray}\nThis expression is analogous to (\\ref{jutro}). For $T=0$ $2n(\\omega)+1\\rightarrow sgn\\omega$ and \n(\\ref{njunja}) can be written as \n\\begin{eqnarray}\nP=\\hspace{5cm}\n\\label{njunja1}\\\\\n\\nonumber\\\\\n4\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\int^{m\\omega_0}_{0}\\frac{d\\omega}{2\\pi}\\hspace{2cm}\n\\nonumber\\\\\n\\nonumber\\\\\nImD_1({\\bf Q},\\omega)ImD_2({\\bf Q},m\\omega_0-\\omega).\n\\nonumber\n\\end{eqnarray}\nAdding higher order terms (\\ref{Eka1},\\ref{Eka2}) we obtain the energy \ndissipated per unit time: \n\\begin{eqnarray}\nP=2\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\times\n\\nonumber\\\\\n\\label{losshop}\n\\\\\n\\hspace{3cm}\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}C({\\bf Q},\\omega,\\omega_m)\n\\nonumber\n\\end{eqnarray}\nwhere $C$ is given by (\\ref{kaka}). Limiting casses \nare also obtained from (\\ref{losshop}). For $\\omega_0=0$ and\/or for $\\rho_0=0$ \nobviously $P=0$.\n\\section{Derivation of the slab surface excitation propagators $D_{1,2}({\\bf Q},\\omega)$}\n\\label{DerofD}\nThe main quantities which appear in the formula for van der Waals interaction $E_c$ or \ndissipated power $P$ are the surface excitation propagators $D_{1}({\\bf Q},\\omega)$ and $D_{2}({\\bf Q},\\omega)$ of the left (first) and right (second) slab, respectively.\nThe derivation of $D_1$ and $D_2$ is analogous for both slabs, so here we shall derive just one surface \nexcitation propagator $D$. The structure of the monolayer-substrate composite (e.g. graphene on \nSiO$_2$) is shown in Fig.\\ref{Fig2}. The slab consists of the graphene monolayer adsorbed at some small distance $h$ (e.g. $h=0.4$nm) above the substrate of macroscopic thickness $\\Delta$. The \ndielectric, e.g. the SiO$_2$ slab is placed in the region $-\\Delta-h\\le z\\le-h$ and the graphene \nlayer occupies $z=0$ plane. The same model system is used in Refs.\\cite{Ivan1,Ivan2} where \nthe authors explore plasmon-phonon hybridization, stopping power and wake effect produced \nby the proton moving parallel to the composite. The unit cell for such huge nanostructure would \nconsist of hundreds of atoms, so it is impossible to perform full {\\em ab initio} ground state and structure optimization calculation. Moreover, an {\\em ab initio} calculation of the response function would be even more demanding so we need an approximation for the response function calculation. The easiest (and probably the best) approximation is to treat \nthe SiO$_2$ slab as a homogeneous dielectric described by some local dielectric function $\\epsilon_S(\\omega)$ and to consider graphene as a purely 2D system described \nby the response function $R({\\bf Q},\\omega)$, as sketched in Fig.\\ref{Fig2}. \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=4.0cm,height=7cm]{Fig2.pdf}\n\\caption{(color online) Simplified model where the SiO$_2$ substrate is shown as a homogenous dielectric slab described by the local dielectric function $\\epsilon_S(\\omega)$ and graphene is described by 2D response function $R({\\bf Q},\\omega)$. $D({\\bf Q},\\omega)$ is the surface excitation propagator of the substrate\/graphene composite.}\n\\label{Fig2}\n\\end{figure}\nIn order to derive the surface excitation \npropagator $D({\\bf Q},\\omega)$ we start from its definition:\n\\begin{eqnarray}\nD({\\bf Q},\\omega)=v_Q\\int^0_{-\\infty}\\ dzdz' e^{-Q(z+z')}R({\\bf Q},\\omega,z,z')= \n\\nonumber\\\\\n\\label{defD}\\\\\n\\frac{1}{v_Q}\\left\\{W({\\bf Q},\\omega,z=0,z'=0)-v_Q\\right\\};\\ i=1,2.\n\\nonumber\n\\end{eqnarray}\nwhich connects the surface excitation propagator with the screened Coulomb interaction $W({\\bf Q},\\omega,z=0,z'=0)$ at $z=z'=0$ surface.\nHere $R({\\bf Q},\\omega,z,z')$ represents the nonlocal dielectric function of graphene\/dielectric composite which we assume occupies the region $z,z'\\le 0$. \n\nIt is well known \\cite{gr2D1,gr2D2,gr2D3,gr2D4} that physical properties of a graphene \nmonolayer in the low ($Q,\\omega$) region can be described to a very good approximation \nassuming the monolayer to be strictly twodimensional, so that the nonlocal independent \nelectron response function can be written as\n\\begin{equation}\nR^{0}({\\bf Q},\\omega,z,z')=R^{0}({\\bf Q},\\omega)\\delta(z)\\delta(z') \n\\end{equation}\nwhere we assume that the graphene lies in the $z=0$ plane and the response function \n$R^{0}({\\bf Q},\\omega)$ can be derived from first \nprinciples, as decribed in Sec.\\ref{nabnak}. \nDynamically screened response function $R({\\bf Q},\\omega)$ in RPA is given as \na series of terms\n\\begin{equation}\nR({\\bf Q},\\omega)=R^{0}+R^{0}v_QR^{0}+...=\\frac{R^0({\\bf Q},\\omega)}{1-v_QR^0({\\bf Q},\\omega)}.\n\\label{RPARR}\n\\end{equation} \nIf we assume for the moment that there is no dielectric in the system (e.g. $\\epsilon_S(\\omega)=1$) then the screened Coulomb \ninteraction is simply given by \n\\begin{equation}\nW({\\bf Q},\\omega,z=0,z'=0)=v_Q+v_QR({\\bf Q},\\omega)v_Q.\n\\label{scrCint}\n\\end{equation}\nUsing the definition (\\ref{defD}) the surface excitation propagator becomes\n\\begin{equation}\nD({\\bf Q},\\omega)=v_QR({\\bf Q},\\omega).\n\\end{equation}\nWhen the dielectric slab is introduced, the external charges and charge density fluctuations in the graphene layer do not interact via the bare Coulomb \ninteraction $v_Q$ but via the Columob interaction modified by the presence of \nthe dielectric slab \\cite{Laplace} \n\\begin{equation}\nv_Q\\rightarrow\\tilde{v}_Q(\\omega)=v_Q\\left[1+D_S({\\bf Q},\\omega)\\right],\n\\label{tildeV11}\n\\end{equation}\nwhere the substrate surface excitation propagator is\n\\begin{equation}\nD_S({\\bf Q},\\omega)=D_S(\\omega)\\frac{1-e^{-2Q\\Delta}}{1-D^2_S(\\omega)e^{-2Q\\Delta}}e^{-2Qh}\n\\label{labina}\n\\end{equation}\nand\n\\begin{equation}\nD_S(\\omega)=\\frac{1-\\epsilon_S(\\omega)}{1+\\epsilon_S(\\omega)}\n\\end{equation} \nrepresents the surface excitation propagator of a semiinfinite \n($\\Delta\\rightarrow\\infty$, $h=0$) dielectric.\nThis causes that the screened Colulomb interaction (\\ref{scrCint}) becomes the function \nof $\\tilde{v}_Q(\\omega)$\n\\begin{equation}\nW\\rightarrow\\tilde{W}=\\tilde{v}_Q(\\omega)+\\tilde{v}_Q(\\omega)\\tilde{R}({\\bf Q},\\omega)\\tilde{v}_Q(\\omega),\n\\label{modifiedW}\n\\end{equation}\nwhere, because charge density fluctuations inside graphene also interact via $\\tilde{v}_Q(\\omega)$, the screened response function is modified as\n\\begin{equation}\n\\tilde{R}({\\bf Q},\\omega)=\\frac{R^0({\\bf Q},\\omega)}{1-\\tilde{v}_Q(\\omega)R^0({\\bf Q},\\omega)}.\n\\label{curoni}\n\\end{equation}\nFinally, after inserting (\\ref{modifiedW}) into (\\ref{defD}) we obtain the surface excitation propagator in the presence of \nthe dielectric\n\\begin{eqnarray}\nD({\\bf Q},\\omega)=\\frac{1}{v_Q}\\left\\{\\tilde{v}_Q(\\omega)\\tilde{R}_i({\\bf Q},\\omega)\\tilde{v}_Q(\\omega)+\\right.\n\\label{newD}\\\\\n\\hspace{3cm}\\left.\\tilde{v}_Q(\\omega)-v_Q\\right\\}.\n\\nonumber\n\\end{eqnarray}\nwhich can be rewritten in a more transparent form as\n\\begin{eqnarray}\nD({\\bf Q},\\omega)=\\hspace{5cm}\n\\nonumber\\\\\n\\nonumber\\\\\n\\frac{D_S({\\bf Q},\\omega)+v_QR({\\bf Q},\\omega)+2v_QR({\\bf Q},\\omega)D_S({\\bf Q},\\omega)}{1-v_QR({\\bf Q},\\omega)D_S({\\bf Q},\\omega)}.\n\\end{eqnarray}\nThe spectrum of coupled excitations in a single slab can be calculated from\n\\begin{equation}\nS({\\bf Q},\\omega)=-\\frac{1}{\\pi}ImD({\\bf Q},\\omega).\n\\end{equation}\nFor the coupled slabs described by their surface excitations propagators \n$D_1$ and $D_2$, separated by the distance $a$, in a similar way we can derive \nthe propagator $\\tilde{D}$ for the coupled system \n\\begin{equation}\n\\tilde{D}({\\bf Q},\\omega)=\n\\frac{D_1({\\bf Q},\\omega)+D_2({\\bf Q},\\omega)+2D_1({\\bf Q},\\omega)D_2({\\bf Q},\\omega)}{1-e^{-2Qa}D_1({\\bf Q},\\omega)D_2({\\bf Q},\\omega)}\n\\end{equation}\nand the excitation spectrum of this system is\n\\begin{equation}\n\\tilde{S}({\\bf Q},\\omega)=-\\frac{1}{\\pi}Im\\tilde{D}({\\bf Q},\\omega).\n\\end{equation}\n\\section{Description of substrate and graphene dynamical response}\n\\label{nabnak}\nThe results in Sec.\\ref{DerofD} are quite general and can be applied to a monolayer \nof any material on any dielectric substrate.\nNow we shall specify the dielectric substrate to be the homogenous \nlayer of ionic crystal SiO$_2$.\n\nDielectric properties (or dynamical response) of bulk ionic crystals in the long-wavelength limit can \nbe described in terms of their optical phonons at the $\\Gamma$ point. More complex polar crystals such as SiO$_2$ possess a multitude of different optical \nphonons of different symmetries and polarizations. However, here we suppose that \nSiO$_2$ posses two well-defined, non-dispersing transverse\noptical (TO) phonon modes at the frequencies $\\omega_{TO1}$\nand $\\omega_{TO2}$ with the corresponding damping rates\n$\\gamma_{TO1}$ and $\\gamma_{TO2}$, giving rise to a \ndielectric function of the form \\cite{Ivan1,Ivan2}\n\\begin{eqnarray}\n\\epsilon_S(\\omega)=\\epsilon_{\\infty}+(\\epsilon_i-\\epsilon_{\\infty})\\frac{\\omega^2_{TO2}}{\\omega^2_{TO2}-\\omega^2-i\\omega\\gamma_{TO2}}+\n\\nonumber\\\\\n(\\epsilon_0-\\epsilon_i)\\frac{\\omega^2_{TO1}}{\\omega^2_{TO1}-\\omega^2-i\\omega\\gamma_{TO1}} ,\n\\label{dielectric}\n\\end{eqnarray}\nwhere $\\epsilon_0$, $\\epsilon_i$, and $\\epsilon_{\\infty}$ represent \nthe dielectric constant for SiO$_2$ at the zero, intermediate, and very large\nfrequencies. This dielectric function will be inserted in the expression (\\ref{labina}) for the substrate surface excitation propagator $D_S({\\bf Q},\\omega)$. \n\nThe graphene response function $R({\\bf Q},\\omega)$ is given by (\\ref{curoni}) in terms of the noninteracting response function\n\\begin{equation}\nR^{0}({\\bf Q},\\omega)=L\\ R^{0}_{{\\bf G}=0{\\bf G}'=0}({\\bf Q},\\omega)\n\\label{Chi02D}\n\\end{equation}\nwhere the 3D Fourier transform of independent electron response function is given \nby \\cite{PRB13} \n\\begin{eqnarray}\nR^{0}_{{\\bf G}{\\bf G}'}({\\bf Q},\\omega)=\\hspace{5cm}\n\\nonumber\\\\\n\\frac{2}{\\Omega}\\sum_{{\\bf K}\\in S.B.Z.}\\sum_{n,m}\\ \\frac{f_n({\\bf K})-f_m({\\bf K}+{\\bf Q})}\n{\\hbar\\omega+i\\eta+E_n({\\bf K})-E_m({\\bf K}+{\\bf Q})}\\times\n\\label{Resfun0}\\\\\n\\rho_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G})\\ \\rho^*_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G'}),\n\\nonumber\n\\end{eqnarray} \nwhere $f_{n{\\bf K}}=[e^{(E_{n{\\bf K}}-E_F)\/kT}+1]^{-1}$ is the Fermi-Dirac distribution at \ntemperature $T$. The charge vertices in (\\ref{Resfun0}) have the form \n\\begin{equation}\n\\rho_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G})=\n\\int_\\Omega\\ d{\\bf r}e^{-i({\\bf Q}+{\\bf G}){\\bf r}}\\ \\phi^*_{n{\\bf K}}({\\bf r})\\phi_{n{\\bf K}+{\\bf Q}}({\\bf r})\n\\label{Matrel}\n\\end{equation}\nwhere ${\\bf Q}$ is the momentum transfer vector parallel to the $x-y$ plane, ${\\bf G}=({\\bf G}_\\parallel,G_z)$ are $3D$ reciprocal lattice vectors and \n${\\bf r}=({\\hbox{\\boldmath $\\rho$}},z)$ is a $3D$ position vector. Integration in (\\ref{Matrel}) is performed over the normalization volume $\\Omega=S\\times L$, where $S$ is the \nnormalization surface and $L$ is the superlattice constant in $z$ direction (separation between graphene layers is superlattice arrangement). \nPlane wave expansion of the wave function has the form \n\\[\n\\phi_{n{\\bf K}}({\\hbox{\\boldmath $\\rho$}},z)=\\frac{1}{\\sqrt{\\Omega}}e^{i{\\bf K}{\\hbox{\\boldmath $\\rho$}}}\\ \\sum_{\\bf G}C_{n{\\bf K}}({\\bf G})e^{i{\\bf G}{\\bf r}},\n\\]\nwhere the coefficients $C_{n{\\bf K}}$ are obtained by solving the Local Density Approximation-Kohn Sham (LDA-KS) equations selfconsistently as will be discussed below.\nHowever, this straightforward calculation of graphene response functions $R({\\bf Q},\\omega)$ is not \nsufficient if we want to investigate the hybridization between the Dirac plasmon and Fuchs-Kliewer (FK) phonons at dielectric surfaces. Namely, due to \nthe very low energy of FK phonons ($\\sim 50$meV) the crossing of their dispersion relations with Dirac plasmon occurs for very small wave vectors ($Q<0.001$a.u.). \nOn the other hand even for very dense $K$-point mesh sampling, as for example $601\\times 601\\times 1$ used in this calculation, \nthe minimum transfer wave vector $Q$ which can be reached (e.g. $Q=0.0026$a.u.$^{-1}$ in this calculation) is still bigger than FK phonon-Dirac \nplasmon crossing wave vector. Therefore we have to find the way how to calculate $R({\\bf Q},\\omega)$ for a denser Q-point mesh \nin the optical $Q\\approx 0$ limit. One possible way is that instead of calculating response function $R^{0}({\\bf Q},\\omega)$ we calculate the optical \n($Q=0$) conductivity $\\sigma(\\omega)$. The optical conductivity in graphene can be written as \\cite{gr2D3} \n\\begin{equation}\n\\sigma(\\omega)=\\sigma^{\\mathrm{intra}}(\\omega)+\\sigma^{\\mathrm{inter}}(\\omega), \n\\label{curren1}\n\\end{equation}\nwhere \n\\begin{equation}\n\\sigma^{\\mathrm{intra}}(\\omega)=\\frac{i\\rho_0}{\\omega+i\\eta_{\\mathrm{intra}}}\n\\label{curren2}\n\\end{equation}\nis intraband or Drude conductivity and where \n\\begin{equation}\n\\rho_0=-\\frac{2}{\\Omega}\\sum_{{\\bf K},n}\\frac{\\partial f^i_n({\\bf K})}{\\partial E_n({\\bf K})}\n|j^{x}_{n{\\bf K},n{\\bf K}}({\\bf G}=0)|^2\n\\label{curren3}\n\\end{equation}\nrepresents the effective number of charge carriers. \nThe interband conductivity is \n\\begin{eqnarray}\n\\sigma^{\\mathrm{inter}}(\\omega)=\n\\frac{-2i}{\\omega\\Omega}\\sum_{{\\bf K},n\\neq m}\\ \\frac{\\hbar\\omega}{E_n({\\bf K})-E_m({\\bf K})}\\times\n\\nonumber\\\\\n\\frac{f^i_n({\\bf K})-f^i_m({\\bf K})}\n{\\hbar\\omega+i\\eta_{\\mathrm{inter}}+E_n({\\bf K})-E_m({\\bf K})}\\times\n\\label{curren4}\\\\\nj^{x}_{n{\\bf K},m{\\bf K}}({\\bf G}=0)\\ [j^{x}_{n{\\bf K},m{\\bf K}}({\\bf G}'=0)]^*\n\\nonumber\n\\end{eqnarray}\nwhere the current vertices are given by\n\\begin{equation}\nj^{\\mu}_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G})=\n\\int_\\Omega\\ d{\\bf r}e^{-i({\\bf Q}+{\\bf G}){\\bf r}}\\ \nj^{\\mu}_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf r}),\n\\label{curren5}\n\\end{equation}\nand \n\\begin{eqnarray}\nj^{\\mu}_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf r})=\n\\frac{\\hbar e}{2im}\n\\left\\{\\phi_{n{\\bf K}}^*({\\bf r})\\partial_\\mu\\phi_{m{\\bf K}+{\\bf Q}}({\\bf r})\\right.\\hspace{2cm}\n\\label{curren6}\\\\\n\\hspace{2cm}-\\left.[\\partial_\\mu\\phi_{n{\\bf K}}^*({\\bf r})]\\phi_{m{\\bf K}+{\\bf Q}}({\\bf r})\\right\\}.\n\\nonumber\n\\end{eqnarray}\nIn the optical $Q\\approx 0$ limit the independent electron response function can be written in terms of optical \nconductivities (\\ref{curren1}) as \\cite{Zoran}\n\\begin{equation}\nR^0({\\bf Q}\\approx 0,\\omega)=L\\ \\frac{Q^2}{i\\omega}\\sigma(\\omega).\n\\label{chi-sigma}\n\\end{equation}\nFinally, the RPA or screened response function $R({\\bf Q},\\omega)$ can be obtained from (\\ref{chi-sigma}) \nusing (\\ref{RPARR}). \n\nIn the calculation of Sec.\\ref{resuuult} we shall assume the \ngraphene response to be isotropic in the small $({\\bf Q},\\omega)$ limit. This means \nthat the graphene response functions and the corresponding surface excitation functions are functions of \n$Q$ and not of ${\\bf Q}$. \n\n\\subsection{Computational details}\n\\label{Comp}\nThe first part of the calculation consists of determining the KS ground state of the single layer graphene and the corresponding wave functions \n$\\phi_{n{\\bf K}}({\\hbox{\\boldmath $\\rho$}},z)$ and energies $E_n({\\bf K})$. For graphene unit cell constant we use the experimental value of $a=4.651\\ \\mathrm{a.u.}$ \\cite{lattice}, and superlattice unit cell constant (separation of graphene layers) is $L=5a$. For calculating KS wave functions and energies we use a plane-wave self-consistent field DFT code (PWSCF) within the QUANTUM ESPRESSO (QE) package \\cite{QE}. The core-electron interaction was approximated by the norm-conserving pseudopotentials \\cite{normcon}, and the exchange correlation (XC) potential by the Perdew-Zunger local density approximation (LDA) \\cite{lda1}. To calculate the ground state electronic density we use $21\\times21\\times1$ Monkhorst-Pack K-point \nmesh \\cite{MPmesh} of the first Brillouin zone (BZ) and for the plane-wave cut-off energy we choose 50 Ry. \nThe second part of calculation consists of determining the independent electron response function (\\ref{Resfun0}) and conductivity \n(\\ref{curren1}--\\ref{curren4}). In order to achieve better resolution in the long wavelength ($Q\\approx 0$) and low energy ($\\omega\\approx 0$) \nlimit the response function (\\ref{Resfun0},\\ref{Matrel}) and conductivity (\\ref{curren1}--\\ref{curren6}) are evaluated from the wave functions $\\phi_{n{\\bf K}}({\\bf r})$ and energies $E_n({\\bf K})$ calculated for the $601\\times601\\times1$ Monkhorst-Pack K-point mesh which coresponds to 361801 K-points in the first Brillouin zone (1BZ). Band summations ($n,m$) in (\\ref{Resfun0}), (\\ref{curren3}) and (\\ref{curren4}) are performed over 30 bands. In the calculation we use two kinds of damping parameters: $\\eta_{\\mathrm{intra}}=10$meV for transitions within the same bands ($n\\leftrightarrow n$), and $\\eta_{\\mathrm{inter}}=50$meV for transitions between different bands ($n\\leftrightarrow m$). \nFor bulk SiO$_2$ dielectric function given by (\\ref{dielectric}) we use the following \nparameters: $\\epsilon_0=3.9$, $\\epsilon_i=3.05$, $\\epsilon_{\\infty}=2.5$, \n$\\omega_{TO1}=55.6$ meV, $\\omega_{TO2}=138.1$ meV, $\\gamma_{TO1}=5.368$ meV and $\\gamma_{TO2}=8.947$ meV taken from Ref.\\cite{Dielectricpar}. \nFor the gap between graphene and the SiO$_2$ surface, we take $h=4\\AA[7.55$ a.u.$]$ \\cite{height}.\n\n\n\\section{Results for graphene monolayers on SiO$_2$ substrates}\n\\label{resuuult}\nTheoretical expressions derived in Sec.\\ref{Sec2} (and in Appendix \\ref{AppA}) are quite \ngeneral, i.e. are valid for any pair of crystal slabs described by their response \nfunctions, while the corresponding surface excitation functions derived in \nSec.\\ref{DerofD} are valid for any 2D adsorbed monolayer on any dielectric \nsubstrate. In this section we shall apply these results to calculate reactive and \ndissipative response of various combinations of slabs consisting of graphene monolayers with variable doping on SiO$_2$ substrate, using the dynamical surface response functions of \nthese materials given in Sec.\\ref{DerofD}. \n\nBefore proceeding with detailed calculations a few general comments are \nin order. Though the derived expressions for van der Waals and dissipated \npower (\\ref{vdwFIN}) and (\\ref{losshop}), respectively, include temperature \ndependence, in the systems studied here inclusion of finite temperature leads to practically \nno effects, therefore all results will be reported for $T=0$. \nThe dependence of these two physical properties on the two parameters, the distance \nbetween the slabs $a$ and the oscillation amplitude $\\rho_0$, can be analyzed if \nwe recognize in the expressions (\\ref{vdwFIN}) and (\\ref{losshop}) the function \n\\begin{equation}\nf_m(x)=\\int^{2\\pi}_0\\frac{d\\phi}{2\\pi}J^2_m(x\\cos\\phi),\n\\label{fm}\n\\end{equation}\nwhich is possible because of the assumed isotropy of graphene response. The function \n$f_m(x)$ is shown in Fig.\\ref{Fig3} for first four $m$'s, where $x=Q\\rho_0$. \n\\begin{figure}[t]\n\\includegraphics[width=6cm,height=5cm]{Fig3.pdf}\n\\caption{Function $f_m(x)$ for m=0 (blue solid line), $m=1$ (black solid line), $m=2$ (black dashed line) and $m=3$ (black dashed-dotted line). Vertical dashed line denotes the maximum argument $x_{cut}$ defined by parameters ($a$ and $\\rho_0$) used in the calculation.}\n\\label{Fig3}\n\\end{figure} \nAnother important factor in (\\ref{vdwFIN}) and (\\ref{losshop}) is $e^{-2Qa}$ which defines the \ncutoff wave vector $Q_c$, depending on the slab separation $a$. \nThe separations we shall consider in this calculation are $a=10-50$nm which defines the cutoff wave \nvector $Q_c\\approx 0.05a.u.$. On the other hand, the ampitudes which will be considered are \n$\\rho_0\\approx 0.1-1$nm. This finally provides the maximum argument $Q$ of the functions (\\ref{fm}) which is $x_{cut}\\approx 1$. From Fig.\\ref{Fig3} is obvious that up to $x_{cut}$ only the $m=0$ \nand $m=1$ terms will contribute. Moreover, for $x1$ and therefore \n\\begin{equation}\nf_0\\approx 1-\\frac{x^2}{4};\\ \\ \\ f_1(x)\\approx \\frac{x^2}{8}. \n\\label{approx}\n\\end{equation}\nIn Fig.\\ref{Fig3} we see that approximation (\\ref{approx}) is valid almost up \nto $x_{cut}$.\n\n\\subsection{Spectra of coupled modes}\n\\label{cmspe}\n \nIn this section we shall first discuss the spectra of coupled plasmon\/phonon excitations \nin one and two graphene\/SiO$_2$ slabs separated by distance $a$ in order to understand \nthe dominant dissipation mechanisms.\n\nFig.\\ref{Fig4}(a) shows the spectrum of surface excitations $S(Q,\\omega)=-Im D(Q,\\omega)$ \nin graphene(200meV)\/SiO$_2$ slab (as shown in Fig.\\ref{Fig2}) and Fig.\\ref{Fig4}(b) in the system \nwhich consists of two graphene\/SiO$_2$ slabs (as shown in Fig.\\ref{Fig1}) separated by distance $a=5$nm. In the lonwavelength limit the SiO$_2$ surface suports two surface polar (FK) TO phonons with \nflat dispersions and the doped graphene contains a Dirac plasmon with square root dispersion. \nCoupling between these modes results in three branches, as shown in Fig.\\ref{Fig4}(a). For larger \n$Q$ the first and second flat branches are phononlike, i.e. their induced \nelectrical fields mostly come from polarization modes on the dielectric\nsurface. On the other hand, the third square root branch is \nplasmon-like, i.e. its induced electrical field mostly comes from charge density \noscillations localised in the graphene layer. However, in the $Q\\rightarrow 0$ limit \nthe strong hybridization (avoided crossings) between these modes occur and they \npossess mixed plasmon-phonon character. When another slab is brought in the vicinity the \nthree modes in each slab interact which results in the mode splitting and formation of \nsix coupled modes as shown in Fig.\\ref{Fig4}(b). \nFigure \\ref{Fig4}(c) shows the spectrum of surface excitations in the \ngraphene($0$meV)\/SiO$_2$ slab. Because the pristine graphene does not support Dirac plasmon \nthe spectrum consist just of two weak phonon branches $\\omega_{TO1}$ and $\\omega_{TO2}$ damped \nby $\\pi\\rightarrow\\pi^*$ excitations. The spectrum of surface excitations in two equal \ngraphene(0meV)\/SiO$_2$ slabs separated by 5nm (not shown here) is very similar to the one \nshown in Fig.\\ref{Fig4}(c) which indicates weak interaction between phonons \nin the two slabs. This could be the consequence of strong screening of FK phonons \nby graphene adlayers which reduces the range of their induced electrical \nfield. Figure \\ref{Fig4}(d) shows the spectrum in the system which consists of two different \nslabs, graphene($0$meV)\/SiO$_2$ and graphene($200$meV)\/SiO$_2$, separated by $5$nm. \nOne can notice interesting hybridization between the Dirac plasmon and two phonons in one slab and two phonons in another slab giving five branches. \n\\begin{figure*}[tt]\n\\includegraphics[width=1.0\\columnwidth]{Fig4a.jpg}\n\\includegraphics[width=1.0\\columnwidth]{Fig4c.jpg}\n\\includegraphics[width=1.0\\columnwidth]{Fig4b.jpg}\n\\includegraphics[width=1.0\\columnwidth]{Fig4d.jpg}\n\\caption{(Color online) The spectra of surface excitations in (a) graphene(200meV)\/SiO$_2$ single \nslab (as shown in Fig.\\ref{Fig2}), (b) in the system consisting of two equal graphene(200meV)\/SiO$_2$ slabs, (as shown in Fig.\\ref{Fig1}) separated by distance 5nm, (c) single \ngraphene(0meV)\/SiO$_2$ slab and (d) in the system consisting of two unequal slabs, \ngraphene(200meV)\/SiO$_2$ and graphene(0meV)\/SiO$_2$, separated by distance 5nm.} \n\\label{Fig4}\n\\end{figure*}\nIn the next section we shall explore how particular plasmon-phonon modes contribute to the \ndissipated power in two oscillating slabs. \n\n\\subsection{Modification of van der Waals force}\n\\label{weakvdW}\nVan der Waals energy and attractive force are usually calculated and measured for static \nobjects. Here we show how their relative oscillating motion can reduce this attraction, \nwhich can be relevant not only from the theoretical standpoint but \nalso in some experimental situations and applications. This phenomenon is present \nalso in the case of parallel motion, as shown in the Appendix \\ref{AppA}, but \nthis situation would be more difficult to realize in practice. \n\nMaking use of the approximation (\\ref{approx}) for the lowest order \nterms of the functions $f_0$ and $f_1$ given by (\\ref{fm}) we can \nrewrite the expression (\\ref{vdwFIN}) for the van der Waals energy as \n\\begin{eqnarray}\nE_{c}(a)=\\frac{\\hbar}{2}\\int\\frac{QdQ}{2\\pi}\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\\hspace{3cm}\n\\label{java}\\\\\n\\nonumber\\\\\n\\left\\{ \\left[1-\\frac{Q^2\\rho_0^2}{4}\\right]A(Q,\\omega,\\omega)+\\frac{1}{4}Q^2\\rho_0^2\\ A(Q,\\omega,\\omega-\\omega_0)\\right\\}\n\\nonumber\n\\end{eqnarray}\nwhere $A$ is given by (\\ref{jalko}) and (\\ref{jujucka}). \nIn the $T\\rightarrow 0$ limit and neglecting higher order terms $A$ reduces to\n\\begin{eqnarray}\nA(Q,\\omega,\\omega')=\\hspace{5cm}\n\\nonumber\\\\\ne^{-2Qa}sgn\\omega\\left\\{ImD_1(Q,\\omega)ReD_2(Q,\\omega')+(1\\leftrightarrow 2)\\right\\}\n\\nonumber\n\\end{eqnarray}\nWe see that for $\\rho_0\\rightarrow 0$ the van der Waals energy reduces \nto the standard result for the static case, and for $\\rho_0\\ne 0$ and \n$\\omega_0\\ne 0$ the lowest order corrections scale with $\\rho^2_0$. \nFrom (\\ref{java}), and also from (\\ref{njunja2}), we see that the slab separation $a$ (because of exponential \nfactor $e^{-2Qa}$) reduces the wave vector range to $Q<1\/2a$.\n\nFig.\\ref{Fig5} shows van der Waals energies $E_c$ of two variously doped, unsupported full conductivity \n(\\ref{curren1}--\\ref{curren4}) graphenes as functions of the driving frequency $\\omega_0$. \nThe driving amplitude is $\\rho_0=20$nm and separation between slabs is $a=10$nm. \nFor the case of two heavily and equally doped graphenes $1-1$eV (thick black solid line)\nthe 'static' ($\\omega_0=0$) van der Waals energy is the largest in comparison with other doping combinations. This is reasonable considering that \nthen except of $\\pi$ and $\\pi+\\sigma$ plasmons (and corresponding electron-hole \nexcitations) the graphenes support strong Dirac plasmons which are all in resonance. \nTherefore, the charge density fluctuation in one slab $ImD_1(\\omega)$ resonantly induces electrical field in another slab \n$ReD_2(\\omega)$ to which it couples, and \nvice versa. \nAs the driving frequency $\\omega_0$ increases the fluctuation and the induced field do not match any more, i.e. $ImD_1(\\omega)$ and $ReD_2(\\omega+n\\omega_0)$ become \nDoppler shifted and vdW energy is expected to decrease. \nHowever, the vdW energy first exhibits a wide plateau until $\\omega_0<50$THz. \nWe performed a separate vdW energy calculation for two unsupported \nDrude (\\ref{curren1},\\ref{curren2}) graphenes (not shown here) and noticed that it shows the same \nfeatures as presented in Fig.\\ref{Fig5}. This suggests that Dirac plasmons are responsible for all characteristic \nfeatures in vdW energy (for larger dopings).\nTherefore, the plateau arises probably because the Dirac plasmon fluctuation in one slab, e.g. at $\\omega_p$, can be efficiently screened by induced \nplasmon field in another slab which is not necessarily at the same frequency $\\omega_p$. \nMoreover, graphene, regardless of doping, exhibits perfect screening $ReD(Q\\approx 0,\\omega\\approx 0)\\approx -1$ \\cite{Duncan2012} causing that \nthe static point charge feels image potential. This causes that $E_c$ shows almost identical plateau \nfor the case of differently doped graphenes $1-0.2$eV (black solid line) and $1-0eV$ \n(thin black solid line). \nAs the doping difference increases plateau energy decreases which is \nreasonable because of plasmon resonance breakdown. \nFor larger $\\omega_0>50$THz the Dirac plasmon in one slab does not match any more the \nperfect screening regime in another one, resulting in a rapid decrease or weakening of vdW energy.\nIn the case of weakly doped graphenes, such as the combinations $0.2-0.2$eV (red dashed line) and $0.2-0$eV (thin red dashed \nlines), the 'static' $\\omega_0\\approx 0$ van der Waals energy reduces in comparison with the heavy\ndoping (combinations with $1$eV) cases. This is reasonable considering that Dirac plasmon spectral weight decreases with \ndoping. Additionally, it can be noted that for lower doping the vdW plateau shifts to $\\omega_0<25$THz. \nThis is because the perfect screening frequency region can be roughly estimated as \n$ReD(\\omega<\\omega_p)\\approx-1$, so, as the plasmon energy decreases the frequency interval whithin which fluctuations are perfectly screened becomes narrower. \nIt is interesting to notice that for some frequencies (e.g. $\\omega_0>100$THz) the resonant but low doping vdW energy (e.g. $0.2-0.2$eV case) \novercomes the heavily doped but off resonance vdW energy (such as the cases $1-0.2$eV and $1-0$eV). \nThe static $\\omega_0=0$ vdW energy of pristine graphenes $0-0$eV (blue dashed dotted line) is the weakest and shows smooth decreasing, almost \nlinear behaviour. In this case there are no Dirac plasmons in the graphenes spectra. Therefore, only resonant coupling between $\\pi\\rightarrow\\pi^*$ electron-hole \nexcitations, $\\pi$ and $\\pi+\\sigma$ plasmons contribute to the vdW energy. As the frequency $\\omega_0$ increases the overlap between these \nelectronic excitations decreases causing smooth and linear vdW energy weakening. \nThe same linear behaviour (for $\\omega_0>50$THz) can be noticed for doping combinations $0.2-0.2$eV and $0.2-0$eV \nwhich proves that for lower dopings the dominant vdW energy weakening mechanism becomes off-resonant coupling \nbetween $\\pi\\rightarrow\\pi^*$ electron-hole excitations, $\\pi$ and $\\pi+\\sigma$ plasmons. \n\nIt should be noted here that such designed (graphene based) slabs might enable modification of attraction between slabs, e.g. controlled 'sticking' \nand 'un-sticking' of two slabs. For example, two heavily doped graphenes ($1-1$eV case in Fig.\\ref{Fig5}) are strongly bound, however \nbinding energy between pristine graphenes ($0-0$eV case achieved, e.g. simply by electrostatic gating) is reduced more than \ntwice. Moreover, for larger $\\omega_0$ (and fixed doping) the dynamical binding energy is substantially reduced, leading to 'un-sticking' \nof two slabs, and vice versa, their 're-sticking' by reducing the driving frequency. \n\n\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig5.pdf}\n\\caption{(Color online). Van der Waals energies $E_c$ of two variously doped, unsupported full conductivity \n(\\ref{curren1}--\\ref{curren4}) graphenes as functions of driving frequency $\\omega_0$.\nThe left-right graphene dopings are $1-1$eV (thick black solid), $1-0.2$eV (black solid), 1-1eV (thin black \nsolid), $0.2-0.2$eV (red dashed), $0.2-0$eV (thin red dashed), $0-0$eV (blue dashed-dotted), as also denoted in the figure.\nSeparation between graphenes is $a=10$nm and oscillating amplitude is $\\rho_0=20$nm. } \n\\label{Fig5}\n\\end{figure}\n\n\n\\subsection{Dissipated power - substrate dependence}\n\\label{DISsub}\nIn this section we shall explore how the dissipation power in two oscillating \nslabs depends on the conductivity model we use to describe graphene and how \nsubstrate influences the dissipation power. \n\n\nIn order to facilitate the analysis of the results we shall again use the \napproximation (\\ref{approx}). The lowest order term \nwhich contributes in (\\ref{losshop}) is $f_1$, and from Fig.\\ref{Fig3} it is \nobvious that, for $x0.5$ epochs) and a sufficient number of labels ($>20\\%$ of the data is labeled). The latter finding is surprising since the addition of an unsupervised learning algorithm depends on the presence of labels in order to deliver marginal benefits over gradient descent.\n\nThe underlying form of the learned rule that makes \\textbf{HAT} successful is still a mystery; we find that while the meta-learner may learn a useful update rule during training, the meta-learner does not converge to this useful rule in the long run and instead devolves into a linear function \\textbf{Converged-Rule}. This converged function preserves fully-converged weights by reinforcing incoming weights for neurons with high activations.\n\n\\subsection{Future Work}\n\nThe discovery that \\textbf{HAT} does not stably converge to a function makes analysis quite difficult. However, there is potential for future work to do more subtle analyses.\n\nImagine a time $t$ during training in which the meta-learner $M$ has converged to a useful function, but the learner $L$ has not yet finished training. A follow-up to this thesis might be to discover whether there such a time $t$ exists, what the structure of $M$ at time $t$ is, and how $M$ changes the weights of $L$ at time $t$. One potential methodology might be to observe the function $f$ not as a 3-dimensional function in $(v_i, w_{ij}, v_j)$ but rather as a 4-dimensional function in $(v_i, w_{ij}, v_j, t)$. Observing the function along the $t$-axis and checking for phase changes would shed light on whether a single useful update rule is learned during training or whether \\textbf{HAT}'s learning is truly transient and continuous. If this follow-up were to succeed, then we could have an a priori rule to apply without having to metalearn update rules.\n\nExtracting the local rules from multiple domains could either find that \\textbf{HAT} learns a universal rule or that functional distance between two rules describes the ``difference'' between their originating domains.\n\\vspace*{-1mm}\n\\begin{itemize}\n \\itemsep-0.4em \n \\item Suppose we always metalearn the same rule, regardless of problem domain. \\textbf{Optimal-Hebb} is then a universal learning rule.\n \\item Suppose \\textbf{Optimal-Hebb} is not universal for all problems. For local rules $R_A, R_B$ on problems $A,B$, integrating $\\int_{\\mathbb{R}^3} (R_A-R_B)\\cdot dF(v_i, w_{ij}, v_j)$ for input distribution $F$ gives an explicit measure for how similar $A$ and $B$ are. This provides a systematic way to identify pairs of learning problems that are good candidates for transfer learning.\n\\end{itemize}\n\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIt is a property of fundamental importance within statistical physics that generic and realistic thermodynamic systems exhibit one particular state -- thermal equilibrium -- which is always approached, irrespective of the initial condition. Yet the important question of which\nmicroscopic conditions are necessary or sufficient for the thermalization of a closed quantum many-body system is still largely unanswered~\\cite{Polkovnikov11}. This is of particular importance, especially because there exists a specific class of isolated quantum systems, termed integrable, for which relaxation to thermal states is prevented due to the presence of an extensive number of (quasi-)local conservation laws~\\cite{Rigol06,Rigol07,Kollar11}.\nSuch particular systems often represent isolated points in the parameter space of physical many-body systems and demand a precise tuning of the microscopic parameters. Nevertheless, these models are very valuable because they often represent fixed points of renormalization group theories and as such contain the low-temperature equilibrium properties of a much wider class of systems. This directly leads to an apparent dilemma in quantum many-body theory which has attracted a lot of interest recently. In particular, beyond equilibrium these integrable models become nongeneric as they fail to thermalize. Instead, they are trapped in extended prethermal states described by nonthermal generalized Gibbs ensembles~\\cite{Polkovnikov11,Rigol06,Rigol07,Rigol08,Kollar11,cazalilla06,cardycalabrese06,Barthel08}. Resolving this dilemma is one of the major challenges for the understanding of the coherent dynamics of quantum many-body systems.\n\nIn this work we address this question for a paradigmatic low-energy model: the Luttinger liquid~\\cite{haldane81,haldane81b,giamarchi04}, representing the fixed point theory of systems of interacting fermionic particles in one dimension at low temperatures. The Luttinger liquid is an integrable theory failing to thermalize but rather exhibiting a description in terms of a generalized Gibbs ensemble~\\cite{cazalilla06,Iucci09,Iucci10}. Here, we will be interested in the nonequilibrium dynamics in the presence of a weak fermionic band curvature, which represents a generic perturbation, irrelevant in the low-energy equilibrium limit, but relevant on intermediate to long time scales in order to drive the crossover towards thermalization. \n\nThe increasing number of cold atom experiments performed under out of equilibrium conditions~\\cite{Greiner2002ux,kinoshita,Hofferberth06,trupke13,schmiedmayer12,nagerl13,meinert14,preiss15,hild14,cheneau12} has driven significant interest in the theoretical understanding of the non-equilibrium dynamics in quantum many-body systems. Importantly, these experiments share a remarkable isolation from the environment, thereby probing the purely coherent unitary time evolution on the experimentally relevant time scales. This has paved the way to experimentally study the constrained relaxational dynamics of quantum systems close to integrability~\\cite{kinoshita,Langen14,agarwal14,schmiedmayernphys12}, showing unconventional properties due to the anticipated (quasi-)local conservation laws. Although the inherent integrability breaking terms, resulting from, e.g., imperfections in the particle-particle interactions or higher orbital modes, are considered to be weak, they are believed to eventually cause relaxation to thermal states on long-time scales. Yet a full understanding of this process has not been achieved so far.\nWithin the current understanding, however, the thermalization dynamics of quantum many-body systems with weak integrability-breaking perturbations is expected to occur via a two-stage process. Initially, the dynamics of local observables at transient and intermediate time scales are controlled by the corresponding integrable theory\nserving as a metastable attractor for the non-integrable dynamics~\\cite{Moeckel,Kollar11,Stark13}. \nThis trapping in a metastable state has been termed prethermalization~\\cite{berges_pretherm,Moeckel} and is expected to exist for several non-integrable models and models close to integrability~\\cite{Moeckel,Eckstein09,Essler14,Kollar11,Rosch08,Marcuzzi13,Nessi15,Fagotti14,Fagotti15,Babadi2015,Bertini15}. \nIn the quasi-particle picture, prethermalization is associated with the initial formation of well-defined excitations \\cite{Moeckel} which leads to a dephasing of all terms that are not diagonal in quasi-particle modes, i.e. to a projection of the initial density matrix onto the diagonal ensemble in the quasi-particle basis. After this intermediate quasi-particle formation, the dynamics eventually crosses over to the thermalization regime, where weak quasi-particle scattering leads to a slow redistribution of energy and establishes detailed balance between the different modes. This causes asymptotic thermalization on long time scales compatible with the Eigenstate-Thermalization-Hypothesis~\\cite{Srednicki94,Deutsch91,Rigol08,Gibbs,Biroli10}.\n\nIn equilibrium, the fermionic band curvature in the Luttinger liquid, because irrelevant in the renormalization group sense, does not modify static correlation functions, which are well described by the quadratic Luttinger theory. Importantly, however, the curvature has a strong impact on frequency-resolved fermionic quantities. This has been observed in Coulomb drag experiments \\cite{Debray01,Debray02}, which could not be explained in terms of a quadratic Luttinger theory. In a hydrodynamic representation, the band curvature describes resonant scattering processes between the elementary phononic excitations of the system, such that perturbation theory is plagued by divergences due to the resonant nature of the interactions. Important first approaches to the interacting Luttinger liquid applied a self-consistent Born approximation in order to determine the phonon self-energy on the mass-shell \\cite{andreev80,samokhin98,zwerger06}. However, these works were unable to explain the frequency-dependence of the self-energy, which appeared to be non-negligible for dynamic observables. Using a combination of bosonization and subsequent refermionization a general theory has been developed which has been very successful in determining spectral equilibrium properties such as the dynamic structure factor and the fermionic spectral function in thermal equilibrium \\cite{pustilnik03,pustilnik07,imambekov09,imambekov09a}. Importantly for the scope of the present work, however, it has not yet been possible to generalize this methodology to systems out of equilibrium. Only recently, these equilibrium results have been recovered by a quantum hydrodynamic approach \\cite{lamacraft15,lamacraft15a}, showing that hydrodynamics is also capable of controlling the resonant phonon interactions.\n\nThe theoretical finding of these works is that the elementary excitations are no longer described in terms of bosonic quasi-particles with exact energy-momentum relation $\\omega=u|q|$ but dissolve into a continuum of excitations. This continuum, however, is energetically confined between two well-defined excitation branches $\\epsilon^-_q<\\omega<\\epsilon^+_q$ (with $\\epsilon^{\\pm}_q\\rightarrow 0$ as $q\\rightarrow0$) at which the spectral weight of the bosonic excitations features algebraic divergences, reflected in corresponding divergences of the dynamical structure factor. This fine structure in the bosonic spectral weight, and equivalently self-energy, makes the development of a general kinetic theory for \\emph{frequency-resolved} observables a very demanding task, which has not yet found a satisfactory solution. However, as will be shown in this work, static properties and their time evolution are nevertheless accessible.\n\nThe goal of this work is to study \\changed{the escape out of the prethermalization regime and the crossover towards thermalization} in Luttinger liquids with \\changed{quadratic} fermionic dispersion on the basis of a hydrodynamic description. Specifically, we aim at formulating a kinetic theory for the momentum distribution of the phononic degrees of freedom \\changed{taking into account the leading nonlinear corrections due to the quadratic dispersion. While in this way we are able to describe the escape out of the prethermalization regime in a controlled way, the final asymptotic thermalization of the system might be modified by the more subleading, off-resonant contributions which we do not consider here.}\nThe kinetic equation describes the time-evolution of the phonon momentum distribution and is\nsuitable in the long-wavelength limit and for weak quenches but still goes beyond the regime of linear response. In turn this kinetic theory gives a valid description for the fermionic occupation distribution in the vicinity of the Fermi points where the anticipated fine-structure of the bosonic spectral weight only gives subleading contributions.\nThis \"semi-static\" -- and as a consequence tractable -- description, covers the forward time evolution of any static, i.e. frequency independent, observable.\nWe show that the dynamics of precisely these frequency independent observables depend only on the time evolution of the momentum distribution of excitations $n_q$ and can be captured within a kinetic theory. \nThe justification for this approach is the subleading width of the excitation spectrum $|\\epsilon^+_q-\\epsilon^-_q|\\ll u|q|$ compared to the phonon energy for all relevant $q$ (below the Luttinger liquid cutoff), which is equivalent to the statement that even in the presence of the non-linearity the continuum of excitations in the hydrodynamic description is tightly bound to the mass-shell. This condition replaces the common quasi-particle criterion \\cite{kamenevbook} and enables a thorough kinetic description.\n\n\\mh{The applicability of the kinetic equation requires the preformation of well-defined quasi-particles out of the bare particles which occurs during the process of prethermalization before the quasi-particle scattering sets in. We, however, find that close to the integrable point of vanishing fermionic interactions quasi-particle formation becomes very slow shifting the applicability of the theory for weakly interacting fermions to long time scales and far distances. We give quantitative estimates of the corresponding spatio-temporal scales of the breakdown of the kinetic theory. Not too close to the noninteracting point, however, the kinetic equation is well justified and allows us to study the escape out of the prethermalization regime towards thermalization.}\nIn the regime of applicability, the kinetic equation leads in the asymptotic long-time limit to a linearized quantum Boltzmann equation whose attractor is the desired thermal Gibbs state.\nWe find that the thermalization dynamics out of the prethermal state is triggered by short wavelength modes and afterwards progressing algebraically slowly towards longer wavelengths. \nWhether this is a generic feature of weakly-perturbed integrable theories, is an important and interesting question for future work. \n\nThe main result of this work is a spatio-temporal decomposition of correlations in the studied nonlinear Luttinger Liquid, which is illustrated in Fig.~\\ref{fig:QuenchDiag}. \nBy analyzing the equal-time fermionic Green's function $G\\sub{t,x}^{<}$, the Fourier transform of the fermionic occupation distribution, we find three regimes which we term prequench, prethermal, and thermal and which are separated by two crossover scales $x\\sub{th}(t)$ and $x\\sub{pt}(t)$ obeying $x\\sub{th}(t) < x\\sub{pt}(t)$. The crossover scale $x\\sub{pt}(t)=2ut$ sets the light cone~\\cite{cardycalabrese06} with $u$ the sound velocity of the elementary bosonic excitations of the integrable theory. Causality implies that for distances $x \\gg x\\sub{pt}(t)$ the system's properties are not yet influenced by the nonequilibrium protocol, but are rather given by the initial state yielding the notion of the prequench regime. Inside the light cone for distances $x2ut, x>x\\sub{th}$, the Green's function is determined by the quasi-particles of the initial state and feature algebraic decay in real-space corresponding to the pre-quench state of the system, modulated by a amplitude decaying as a stretched exponential in time. In the intermediate regime $2ut0 \\end{array}\\right.\n}\nand both the quadratic Hamiltonian as well as the non-linearity are modified by this interaction change.\nThe eigenbasis of $H\\sub{LL}$, which is expressed in terms of the physically more transparent phononic creation and annihilation operators $\\creo{a}{q}, \\anno{a}{q}$ according to the canonical Bogoliubov transformation \n\\eq{Bog1}{\n\\theta_x &=& \\theta_0 +\\frac{i}{2}\\int_q\\left(\\frac{2\\pi}{|q|K}\\right)^{1\/2} e^{-iqx-\\frac{|q|}{\\Lambda}}\\left(\\creo{a}{q}-\\anno{a}{-q}\\right),\\\\\n\\phi_x&=&\\phi_0-\\frac{i}{2}\\int_{q}\\left(\\frac{2\\pi K}{|q|}\\right)^{1\/2}\\mbox{sgn}(q)\\ e^{-iqx-\\frac{|q|}{\\Lambda}}\\left(\\creo{a}{q}+\\anno{a}{-q}\\right)\\label{Bog2},} is therefore obviously transformed by the quench. This transformation depends on the interaction via the Luttinger parameter $K$. \n\nThe state of the system before the quench corresponds in general no longer to an equilibrium state after the quench, and the system will consequently undergo a nontrivial time evolution according to the new Hamiltonian. The occupations of bosonic modes after the quench can be computed via the above Bogoliubov transformation. Before the quench, the interacting system is in equilibrium at zero temperature, such that $G^K_{q,t=0}=\\langle \\{\\anno{a}{q},\\creo{a}{q}\\}\\rangle=1$ in the prequench basis. This yields the postquench occupations\n\\begin{eqnarray}\nn_{t=0,q}&=&\\langle \\creo{a}{q}\\anno{a}{q}\\rangle_{t=0}=\\frac{1}{2}\\left[\\frac{\\lambda^2+1}{\\lambda}n_{i,q}+\\frac{\\left(\\lambda-1\\right)^2}{2\\lambda}\\right],\\nonumber\\\\\nm_{t=0,q}&=&\\langle\\creo{a}{q}\\creo{a}{-q}\\rangle_{t=0}=\\frac{1-\\lambda^2}{4\\lambda}\\left(2n_{i,q}+1\\right), \\mbox{ with } \\lambda=\\frac{K\\sub{f}}{K\\sub{i}}.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\label{Occ}\n\\end{eqnarray}\nHere, $n_{i,q}$ is the initial occupation of the bosonic modes and $\\lambda=\\frac{K\\sub{f}}{K\\sub{i}}$ the ratio between the final $K\\sub{f}=\\sqrt{1+\\frac{g\\sub{f}}{\\pi v\\sub{F}}}$ and the initial $K\\sub{i}=\\sqrt{1+\\frac{g\\sub{i}}{\\pi v\\sub{F}}}$ Luttinger parameter. \nIn this work, we focus on a zero temperature initial state, $n_{i,q}=0$ for all $q$. The phonon density after the quench $n_{t,q}> 0$ is always larger than the density before the quench, resulting in a nonzero excitation energy $\\Delta E=\\langle H\\sub{f}\\rangle-\\langle H\\sub{i}\\rangle>0$ generated by the quench. Non-zero off-diagonal occupations $m_{t,q}\\neq0$ indicate that the correlations are not diagonal in the post-quench quasi-particle basis and in order to relax to an equilibrium state, $m_{t,q}$ must decay to zero. In the present setting, we choose $m_{t,q}=e^{-2iu|q|t}\\langle\\creo{a}{q}\\creo{a}{-q}\\rangle_{t}$, such that the off-diagonal occupations remain always real, being either positive or negative, depending on the quench.\n\nIn the phonon basis, \n\\eq{Eq5a}{\nH\\hspace{-1mm}=\\hspace{-1.2mm}\\int_q \\hspace{-1.2mm} u|q|\\creo{a}{q}\\anno{a}{q}\\hspace{-0.5mm}+\\hspace{-1mm}\\int_{q,k}\\hspace{-3mm}\\sqrt{|qk(k+q)|}\\ v(k,q) \\left(\\creo{a}{q+k}\\anno{a}{q}\\anno{a}{k}+\\mbox{h.c.}\\right),}\nwith the vertex function $v(k,q)=v\\left(\\frac{q}{|q|},\\frac{k}{|k|},\\frac{k+q}{|k+q|}\\right)$, which depends on the signs of the in- and outgoing momenta. In the interaction representation the phonon scattering Hamiltonian is\n\\eq{Eq5b}{\nH\\hspace{-1mm}_I(t)=\\hspace{-1mm}\\int_{q,k}\\hspace{-3mm}\\sqrt{|qk(k+q)|}\\ v(k,q)\\left(\\creo{a}{q+k}\\anno{a}{q}\\anno{a}{k}e^{iut(|q+k|-|q|-|k|)}+\\mbox{h.c.}\\right).\n}\nInstead of solving the full problem, we aim at extracting the dominant contributions of the nonlinearity which are relevant for intermediate and large times and which drive the crossover towards thermalization.\nIn view of Eq.~\\eqref{Eq5b}, off-resonant processes, for which $|q|+|k| \\not= |k+q|$, will dephase and as a consequence become negligible for the intermediate and long-time evolution of the system~\\cite{zwerger06}. Resonant processes on the other hand, here set by $|q|+|k| = |k+q|$, will at intermediate and long times become relevant in the renormalization group sense, as discussed in Ref.~\\cite{Heyl2015nd}. \n\\changed{The off-resonant processes can be eliminated perturbatively \\cite{Heyl2015nd}, yielding subleading corrections for intermediate and large times, which we will neglect in the following. For the asymptotic thermalization process, these subleading corrections will yield non-universal corrections (i.e. observable in microscopic constants and prefactors). For instance, the presence of off-resonant scattering events will eventually lower the asymptotic temperature compared to a system with purely resonant scattering events. The influence of off-resonant interactions on the decay rate of the bosonic and fermionic quasi-particles has been investigated in Ref.~\\cite{Proto14a}. The decay rate extracted from this computation is orders of magnitude lower than the rate due to purely resonant scattering processes. Furthermore, it has a subleading scaling behavior $\\sim Tq^4$ compared to $\\sim \\sqrt{q^3T}$ for resonant scattering processes at small momenta $q$ \\cite{zwerger06, andreev80}. Consequently, it is thus no influence on the leading order long time behavior. This allows us for the present purpose to restrict the phonon scattering to the resonant processes alone:}\n\\eq{Eq5}{\nH\\hspace{-1mm}=\\hspace{-1.2mm}\\int_q \\hspace{-1.2mm} u|q|\\creo{a}{q}\\anno{a}{q}\\hspace{-0.5mm}+\\hspace{-1mm}v_0\\int_{q,k}'\\hspace{-3mm}\\sqrt{|qk(k+q)|} \\left(\\creo{a}{q+k}\\anno{a}{q}\\anno{a}{k}+\\mbox{h.c.}\\right),}\nwhere the integral $\\int_{q,k}'$ is performed for momenta $|q+k|=|q|+|k|$ and $v_0=v(1,1)=\\frac{3}{m}\\sqrt{\\frac{\\pi}{K}}$ is the strength of the nonlinearity at resonance \\cite{buchholdmethod,zwerger06}.\n\nAs we are interested in fermionic correlation functions, we switch from an operator based formalism to a field theoretical formulation on the Keldysh contour, which is explained in the appendix \\ref{appendix2}, see also Ref.~\\cite{buchholdmethod}. This allows us to treat both spatial and temporal forward time correlations on an equal footing. We will focus our analysis on the so-called fermionic lesser Green's function \n\\eq{Green}{G^{<}_{t,x}=-i\\langle\\cre{\\psi}{t,x}\\anno{\\psi}{t,0}\\rangle} at equal forward times $t$ from which all fermionic equal time correlations can be deduced. Especially, in terms of a physical interpretation it is the Fourier transform of the fermionic momentum distribution \n\\eq{Mom}{\nn^{\\mbox{\\tiny F}}_{t,q}=i\\int_x e^{iqx}G^<_{t,x}.\n}\nIn the field theory representation, the bosonized fermionic lesser Green's function at equal times is\n\\eq{GF1}{\nG^{<}_{\\eta,t,x}=-i\\langle\\cre{\\psi}{\\eta,-,t,x}\\anno{\\psi}{\\eta,+,t,0}\\rangle=-i\\Lambda\\frac{e^{-i\\eta k\\sub{F}x}}{2\\pi}e^{-\\frac{i}{2}\\mathcal{G}^<_{\\eta,t,x}}.\n}\nHere, $\\cre{\\psi}{\\nu},\\ann{\\psi}{\\nu}$ label Grassmann fields with the index $\\nu=(\\eta,\\gamma,t,x)$ representing right and left movers ($\\eta=\\pm$), the contour variables on the Keldysh plus and minus contour ($\\gamma=\\pm$), the forward time coordinate $t$ and the relative spatial distance $x$. The corresponding lesser exponent $\\mathcal{G}^<$ is defined as \n\\eq{GF2}{\n\\mathcal{G}^<_{\\eta, t,x}=2i\\log\\left\\langle e^{i\\left(\\eta\\phi_{+,t,0}-\\theta_{+,t,0}-\\eta\\phi_{-,t,x}+\\theta_{-,t,x}\\right)} \\right\\rangle.\n}\nThe extra index $(\\pm)$ of the Luttinger fields labels position on the plus-minus contour, see appendix. Combining Eq.~\\eqref{GF2} and the Bogoliubov transformation above, one finds that $\\mathcal{G}^{<}_{-\\eta,t,x}=\\mathcal{G}^{<}_{\\eta,t,-x}$. The Green's function of the left movers is the spatially mirrored Green's function of the right movers, and it is sufficient to consider only the Green's function of the right movers \n\\eq{GF3}{\nG^{<}_{t,\\eta x}\\equiv G^{<}_{+,t,\\eta x}=G^{<}_{\\eta,t,x}\n}\nand equivalently for the exponent $\\mathcal{G}^<$. According to the linked cluster theorem, the logarithm in Eq.~\\eqref{GF2} is defined as the sum of all connected diagrams in an expansion of the exponent. As a consequence, it can be expressed to leading order in terms of the full Green's functions, with the next non-vanishing correction being proportional to the equal-time one-particle irreducible four-point vertex, which is zero in the microscopic theory. Its effective correction remains negligibly small. In particular, the four-point vertex will only contribute to $\\mathcal{O}[(um)^{-4}]$ which is two orders of magnitude smaller than the desired accuracy and its contribution can be safely neglected. The static one-particle irreducible four-point vertex represents a negligible correction for any equilibrium problem since it can only be generated via multiple concatenation of subleading three-point vertices. Especially it is not responsible for the modifications of the dynamic structure factor reported in Refs.~\\cite{pustilnik07,imambekov09,lamacraft15}, since at zero temperature vertex corrections vanish exactly due to causality \\cite{buchholdmethod,forsternelson76}. Consequently, the modifications of the dynamic structure factor happen entirely on the basis of the irreducible two-point vertex, i.e. the phonon self-energy. In the present case, the four-point vertex is exactly zero before the quench since this state corresponds to a zero temperature state as well as immediately after the quench, since a flat quasi-particle distribution in Eq.~\\eqref{Occ} leads to a vanishing vertex correction. \nIn terms of the Luttinger fields and apart from four-point vertex corrections, the exponent for the fermionic Green's function is\n\\eq{GF4}{\n\\mathcal{G}^<_{t,x}=\\sum_{\\alpha,\\beta=\\theta,\\phi}\\left(2\\delta_{\\alpha\\beta}\\hspace{-0.1cm}-\\hspace{-0.1cm}1\\right)\\left[G^K_{\\alpha\\beta,t,0}\\hspace{-0.1cm}\n-\\hspace{-0.1cm}G^K_{\\alpha\\beta,t,x}\\hspace{-0.1cm}+\\hspace{-0.1cm}G^A_{\\alpha\\beta,t,x}\\hspace{-0.1cm}-\\hspace{-0.1cm}\nG^R_{\\alpha\\beta,t,x}\\right]\n,}\nwhere $G^{R\/A}_{\\alpha\\beta}$ is the retarded, advanced Green's function for $\\alpha,\\beta=\\theta,\\phi$ and $G^K_{\\alpha\\beta}$ is the corresponding Keldysh Green's function, i.e. $G^R_{\\alpha\\beta,t,x}=-i\\langle \\alpha_{q,x,t}\\beta_{c,0,t}\\rangle$.\nApplying the Bogoliubov transformation to the phonon basis, the equal time exponent becomes\n\\begin{widetext}\n\\eq{GF5}{\n\\mathcal{G}^<_{t,x}=i\\int_q\\left[\\mbox{$\\frac{\\pi e^{-\\frac{|q|}{\\Lambda}}}{|q|}$}(\\cos(qx)-1)\\left[\\mbox{$\\frac{K^2+1}{K}$}(2n_{t,q}+1)+2\\mbox{$\\frac{K^2-1}{K}$}\\cos(2u|q|t)m_{t,q}\\right]\\right]+2\\arctan(\\Lambda x)+4i\\int_q\\left[\\frac{\\pi e^{-\\frac{|q|}{\\Lambda}}}{|q|}\\sin(|q|x)\\sin(2u|q|t)m_{t,q}\\right].\n}\\end{widetext}\n Here, $n_{t,q}=\\langle \\creo{a}{t,q}\\ann{a}{t,q}\\rangle$ and $m_{t,q}=|\\langle\\ann{a}{t,-q}\\ann{a}{t,q}\\rangle|$ are the equal time normal and anomalous phonon densities, which evolve in time due to phonon scattering. The absence of the quasi-particle self-energy in this expression is caused by the equal time properties of the Green's function and underlines the fact that time-local, i.e. static, observables, even if explicitly forward-time dependent, are not modified by the frequency resolved fine structure of the self-energies once the time dependent distribution $n_{t,q}$ is known. In the remainder of this paper, we will analyze the time evolution of the exponent \\eqref{GF5} after the interaction quench and its implications for the fermionic Green's function \\eqref{GF1}.\n\nConcerning the relevance of the interacting Luttinger model, before closing the section, we would like to mention that only recently pioneering experiments in ultra-cold gases both in and out of equilibrium explored the transient and prethermalization dynamics of systems~\\cite{Hofferberth06,trupke13,schmiedmayer12,Langen14,nagerl13,meinert14,preiss15,hild14,cheneau12,agarwal14,schmiedmayernphys12,schmiedmayernjp13,bloch13,Guan2013} effectively described by a quadratic Luttinger model, the bosonic theory of the Hamiltonian in Eq.~\\eqref{Eq2}. In particular, in Refs.~\\cite{Hofferberth06,trupke13,schmiedmayer12,Langen14} prethermal states in the relative phase of a suddenly split condensate have been identified that have been stable on the experimentally accessible time scales. For the latter experiments, the cubic nonlinearity studied in the present work constitutes the leading order correction to the quadratic theory in a gradient expansion. Therefore, the framework developed in the subsequent sections to describe the relaxation dynamics in the system, is of direct experimental relevance once the time scales are experimentally accessible to study the escape out of the prethermalization plateau. It is, however, important to note that the concrete experimental setup of the suddenly split condensate requires a further but straightforward extension of the considered model system to include two species of coupled bosonic fields. Moreover, let us emphasize that these experimental systems do not simulate the Luttinger liquid of interacting fermions -- our initial motivation -- but directly the effective bosonic low-energy theory. In this way, it might be possible to obtain experimental access to the dynamics of the bosonic occupation distributions, governed by the kinetic theory formulated below, via time-of-flight imaging.\n\n\\section{Summary of main results}\n\\label{sec:summaryofresults}\n\nBefore formulating and solving the kinetic theory for the interacting Luttinger liquid in detail, we briefly summarize the main results reported in this work. In the subsequent sections, we will then present the detailed calculations. Specifically, the known results on the purely integrable system are reformulated within the present framework in Sec.~\\ref{sec:PT}. The kinetic equation, used to address the presence of the nonlinear phonon scattering, is derived in Sec.~\\ref{sec:kinetic_equation}. This kinetic equation is then solved in Sec.~\\ref{sec:thermalization_dynamics}.\n\nIt is the aim of this work to study the thermalization dynamics of the fermionic equal time Green's function \\eqref{Green}, which is the Fourier transform of the fermionic momentum distribution \\eqref{Mom} and contains the information on quadratic equal time fermion observables.\nWithout loss of generality, we focus on the distribution of the right-movers, i.e., $\\eta=+$. In the presence of phonon scattering, we determine the time-evolution of $G^<_{t,x}$ via a set of kinetic equations derived later in Sec.~\\ref{sec:kinetic_equation}.\n\nWe find that $G^<_{t,x}$ features two distinct spatio-temporal crossover scales $x\\sub{th}(t)$ and $x\\sub{pt}(t)$, separating three regimes with distinct scaling behavior:\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{l l c }\n\t1. & prequench: $\\quad$\t&\t$x\\sub{pt}(t) \\ll |x|$, \\\\\n\t2. & prethermal:\t&\t$x\\sub{th}(t) \\ll |x| \\ll x\\sub{pt}(t)$, \\\\\n\t3. & thermal:\t\t&\t$|x| \\ll x\\sub{th}(t)$. \\\\\n\\end{tabular}\n\\end{center}\n\\renewcommand{\\arraystretch}{1}\nWe find for the associated crossover scales $x_\\mathrm{pt}(t)$ and $x_\\mathrm{th}(t)$:\n\\eq{eq:crossoverScales}{\nx_\\mathrm{pt}(t) = 2ut, \\qquad \\qquad x_\\mathrm{th}(t) =\\frac{x_{\\lambda}}{\\Lambda} \\left( v_0 \\Lambda^2 t \\right)^{\\alpha_\\lambda}.\n}\nThe first crossover at $x\\sub{pt}(t)$ determines the light cone~\\cite{cardycalabrese06} set by the sound velocity $u$ of the phononic elementary excitations and is known from the non-interacting Luttinger model. Two space points a distance $x \\gg x\\sub{pt} (t)$ apart from each other have not been able to exchange information after the quench due to causality. Therefore, the properties at such distances are solely given by the initial condition before the quench such that we term this regime ``prequench''. \\changed{For distances $xx_{\\text{th}}$. In the intermediate regime $x_c(t)1\/2$ generally) is not surprising. The non-linearity in the Luttinger model introduces a microscopic energy scale $v_0\\Lambda^2$ which represents the characteristic time scale of the dynamics induced by the non-linearity, i.e. in the present case the thermalization dynamics beyond the quadratic theory. Additionally, the non-linearity breaks the scale invariance of the quadratic model, which is responsible for the fact that all microscopic scales can be eliminated from macroscopic observables in that case. In the absence of scale invariance, however, the microscopic length scale $\\Lambda$ will appear in certain observables, expressing that their explicit value depends on model specific details.\n\nAs we show in our detailed analysis below, we find that this separation into three spatio-temporal regimes -- prequench, prethermal, and thermal -- reflects itself in a remarkable factorization property of the Green's function\n\\eq{eq:factorization2}{\nG^<_{t,x}=G^<_{0,x}Z\\sub{pt}(s\\sub{pt})Z\\sub{th}(s\\sub{th}),}\nwhich holds everywhere except in the vicinity of the crossover scales $x_\\mathrm{th}(t)$ and $x_\\mathrm{pt}(t)$. Here, we have introduced the following short-hand notations:\n\\begin{equation}\ns\\sub{pt}=\\left\\{\\begin{array}{cl} x &\\mbox{ for } xx_\\mathrm{pt}(t)\\end{array}\\right. , \\quad s\\sub{th}=\\left\\{\\begin{array}{cc}x&\\mbox{ for } xx\\sub{th}(t)\\end{array}\\right. .\n\\end{equation}\nWhile the factorization into $G^<_{0,x}$ and $Z\\sub{pt}$ has been already known for the exact solution of the integrable model~\\cite{cazalilla06}, here, we show that the influence of the nonlinearity can be captured by a further factor in terms of $Z\\sub{th}$. The thermal contribution $Z\\sub{th}(s\\sub{th})$ exhibits interesting spatio-temporal dynamics in particular in the long-time regime $ut \\gg x\\sub{th}(t)$. It is defined as\n\\begin{equation}\n Z_\\mathrm{th} (s_\\mathrm{th}) = \\exp\\left(-\\frac{K^2+1}{K} \\frac{\\pi \\tilde T_t|s_\\mathrm{th}|}{u}\\right)\n\\end{equation}\nand features two different spatio-temporal regimes.\n\n\n\\emph{(i) thermalized regime:}\nDeep in the thermalized region $|x|\\ll x\\sub{th}(t)$ where $s\\sub{th}=x$, $Z\\sub{th}=\\exp(-|x|\/\\xi_{\\tilde{T}_t})$ exhibits the conventional exponential decay with distance that the system experiences in thermal states with an associated thermal length\n\\begin{equation}\\label{tlength}\n\t\\xi_{\\tilde T_t} = \\frac{K}{1+K^2} \\frac{u}{\\pi \\tilde T_t}.\n\\end{equation}\nThe effective temperature $\\tilde T_t$, however, entering this equation remains a dynamical quantity with\n\\begin{equation}\n \\tilde T_t=T+u \\Lambda \\Delta_\\lambda(v_0 \\Lambda^2 t)^{-\\mu},\n\\end{equation}\napproaching the temperature $T$ of the final thermal ensemble algebraically slowly. We find that the numerical simulations of the kinetic equation are consistent with an analytical estimate for the exponent $\\mu=2\/3$. Thus, the system in this spatial region appears to be hotter than in the final asymptotic thermal state. The associated excess energy stored at short distances has to be transported to larger distances which, however, is an algebraically slow process since this energy transport in the presence of detailed balance is carried out by dynamical slow modes, emerging as a consequence of exact conservation laws \\cite{Lux13}.\n\n\\emph{(ii) prethermal and prequench regime:} Within the prethermal and prequench region $x\\sub{th}(t) \\ll x$, the amplitude $Z\\sub{th}(s\\sub{th})=Z\\sub{th}(x\\sub{th}(t))$ approaches a space-independent but time-dependent constant quantifying the temporal decay of the prethermal correlations:\n\\begin{equation}\n Z\\sub{th}(x\\sub{th}(t)) = \\exp[-x\\sub{th}(t)\/\\xi_{\\tilde T_t}].\n\\end{equation}\nBecause $x\\sub{th}(t) \\propto (v_0 \\Lambda^2 t)^{\\alpha_\\lambda}$, we have, remarkably, that this amplitude decays in stretched exponential form. This decay is sub-exponential and thus inherently nonperturbative in nature, highlighting the capabilities of our present approach.\n\n\\section{Dynamics in the absence of phonon scattering}\nIn order to systematically understand the effect of phonon scattering on the relaxation dynamics after the interaction quench, we first determine the dynamics of the exponent $\\mathcal{G}^<_{t,x}$ in the absence of scattering, i.e. for $\\frac{1}{m},v_0\\rightarrow 0$. This quench scenario has been extensively discussed in \\cite{Karrasch12,Kennes13,cazalilla06,Iucci09,Iucci10}, and we will only briefly list the known results in the present formalism in order to make contact to the relaxation dynamics in the presence of phonon scattering, which are discussed subsequently.\n\\subsection{Ground state properties}\nFor a system in the ground state, $n_{t,q}=m_{t,q}=0$ and the exponent evaluates to\n\\eq{GF6}{\n\\mathcal{G}^<_{t,x}=-i\\frac{K^2+1}{2K}\\log(1+\\Lambda^2x^2)+2\\arctan(\\Lambda x),}\nwhich leads to a time-independent fermionic Green's function\n\\eq{GF7}{\nG^<_{t,x}=-\\frac{i\\Lambda}{2\\pi}e^{-ik\\sub{F}x-i\\arctan(\\Lambda x)}\\sqrt{1+\\Lambda^2x^2}^{-\\frac{K^2+1}{2K}}\n,}\nwell known from the literature \\cite{haldane81,giamarchi04}. It features an algebraic decay in space $\\sim x^{-\\frac{K^2+1}{2K}}$ and a power law singularity of the fermionic momentum distribution close to the Fermi momentum $n^{\\mbox{\\tiny F}}_q\\sim|q-k\\sub{F}|^{-\\frac{(K-1)^2}{2K}}$ \\cite{giamarchi04}. \n\\subsection{Quench from the ground state}\\label{sec:PT}\nInitializing the fermions in the ground state and performing an interaction quench leads to constant non-zero phonon densities in the post-quench basis, according to Eq.~\\eqref{Occ}. In the absence of scattering, the phonon densities are constants of motion and remain time independent, $n_{t,q}=n_{0,0}\\equiv n$ and $m_{t,q}=m_{0,0}\\equiv m$. In this situation, only dephasing of the off-diagonal Green's functions induces relaxation and the exponent is\n\\begin{widetext}\\begin{eqnarray}\n\\label{GF8}\n\\mathcal{G}^<_{t,x}&=&2\\arctan(\\Lambda x)-i\\mbox{$\\frac{K^2+1}{2K}$}(2n+1)\\log(1+\\Lambda^2x^2)+im\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x-2ut)^2}{1+\\Lambda^2(x+2ut)^2}$}\\right)-i\\mbox{$\\frac{K^2-1}{2K}$}m\\left[\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x-2ut)^2}{1+4u^2t^2\\Lambda^2}$}\\right)+\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x+2ut)^2}{1+4u^2t^2\\Lambda^2}$}\\right)\\right]\\nonumber\\\\\n&=&\\mathcal{G}^<_{0,x}+im\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x-2ut)^2}{1+\\Lambda^2(x+2ut)^2}$}\\right)-i\\mbox{$\\frac{K^2-1}{2K}$}m\\log\\left[\\mbox{$\\frac{(1+\\Lambda^2(x+2ut)^2)(1+\\Lambda^2(x-2ut)^2)}{(1+4u^2t^2\\Lambda^2)^2(1+x^2\\Lambda^2)^2}$}\\right].\\end{eqnarray}\\end{widetext}\nHere, $\\mathcal{G}^<_{0,x}$ is the exponent corresponding to the prequench state, i.e. the ground state of interacting fermions with the prequench Luttinger parameter $K_i$. Consequently the fermion Green's function \\eqref{GF1} factorizes\n\\eq{GF9}{\nG^<_{t,x}=G^<_{0,x}\\tilde{Z}\\sub{pt}(x,t).\n}\nThe factor $\\tilde{Z}\\sub{pt}$ is defined by Eqs.~\\eqref{GF8} and \\eqref{GF1} and describes the time-dependent modification of the initial zero temperature Green's function due to the quench. In view of the following discussion it is useful to investigate this factor on distances away from the light cone $x=2ut$. For distances $|x|\\ll 2ut$, the temporal factors in Eq.~\\eqref{GF8} cancel each other and $\\tilde{Z}\\sub{pt}(t,x)\\overset{|x|\\ll2ut}{\\rightarrow} Z\\sub{pt}(x)$ looses its time dependence. On the other hand, for distances $|x|\\gg2ut$, the spatial dependence drops out and $\\tilde{Z}\\sub{pt}(t,x)\\overset{|x|\\gg2ut}{\\rightarrow} Z\\sub{pt}(2ut)$. This defines the prethermal amplitude\n\\eq{GF10}{\nZ\\sub{pt}(s)=\\left(\\sqrt{1+\\Lambda^2s^2}\\right)^{\\frac{K^2-1}{2K}\\frac{1-\\lambda^2}{4\\lambda}}.\n}\nThe process associated with the crossover of $Z\\sub{pt}(s)$ from a temporal to a spatial dependence as a function of time is the formation of quasi-particles corresponding to the post-quench Hamiltonian. This is the typical prethermalization scenario in the absence of quasi-particle scattering. For short times, the properties of the system are dominated by the initial state of the system, and the fermion Green's function is only modified by a global amplitude but has the same spatial scaling behavior as for the initial state. The effect of the quadratic Hamiltonian in the time evolution is the dephasing of all terms, which are not diagonal in the basis of the post-quench quasi-particles, leading to a diagonal ensemble in the quasi-particles with a non-equilibrium phonon density. This non-equilibrium distribution of phonons induces a scaling behavior of the fermion Green's function in real space, which is different from the zero and finite temperature cases.\n\nIn the absence of phonon scattering, the diagonal phonon densities $n_{t,q}$ are constants of motion and do not relax, the density matrix $\\rho$ therefore does not approach a Gibbs state but is rather described in the asymptotic limit $t\\rightarrow\\infty$ by a generalized Gibbs ensemble (GGE), which respects the constants of motion and maximizes the entropy. It is given by\n\\eq{GF11}{\n\\rho\\sub{GGE}=Z\\sub{GGE}^{-1}e^{-\\int_q \\nu_q \\hat{n}_q},}\nwhere the Lagrange parameters $\\nu_q=2\\log\\left(\\frac{\\lambda+1}{|\\lambda-1|}\\right)$ depend on the quench parameter and $Z\\sub{GGE}$ is the normalization factor.\n\nThe fermion Green's function for the two different regimes is then\n\\eq{GF12}{\nG^<_{t,x}=G^<_{0,x}\\times\\left\\{\\begin{array}{ll} Z\\sub{pt}(2ut) &\\mbox{ for } |x|\\gg 2ut\\\\\nZ\\sub{pt}(x)& \\mbox{ for } |x|\\ll 2ut\\end{array}\\right. ,\n}\nwith the non-equilibrium scaling behavior \n\\eq{GF13}{G^<_{t,x}\\overset{t\\rightarrow\\infty}{\\sim} |x|^{-\\gamma\\sub{Eq}\\gamma\\sub{GGE}},\n}\nwhere $\\gamma\\sub{Eq}=\\frac{K^2+1}{2K}$ is the equilibrium exponent and $\\gamma\\sub{GGE}=\\frac{\\lambda^2+1}{2\\lambda}=2n+1$ (see, Eq.~\\eqref{Occ}) is the non-equilibrium correction resulting from a non-thermal quasi-particle distribution.\n\\section{Phonon scattering and the kinetic equation}\n\\label{sec:kinetic_equation}\n\nIn the previous sections, we have demonstrated that the forward time evolution of the fermionic equal-time Green's function can be determined solely from the momentum dependent excitation distributions $n_{t,q}, m_{t,q}$. All quadratic, equal-time observables on the other hand can be computed from the fermionic equal-time Green's function via a unitary transformation, such that the knowledge of $n_{t,q}$ and $m_{t,q}$ gives access to the forward time evolution of all the frequency independent quadratic fermion observables. Therefore the time-evolution of this specific set of observables can be captured by the time evolution of the frequency independent and well-defined quantities $n_{t,q}, m_{t,q}$, which does not necessitate the frequency resolved fine structure in the fermionic spectrum. \nIn order to determine the time-evolution of the phonon densities, we derive kinetic equations for the excitation distribution function \\cite{kamenevbook} in the limit of well defined excitations, closely following the steps in Ref.~\\cite{buchholdmethod} and briefly discussing the approximations. \n\nBefore we start with the explicit derivation, we review very briefly the known results for nonlinear Luttinger liquids (c.f. \\cite{imambekov12}) and place the present approach into this context. At zero temperature and without band curvature, long wavelength physics of the interacting fermion model can be exactly mapped to the quadratic Luttinger model and therefore has well-defined, sharp phononic excitations, expressed by a spectral function of the phonons $\\mathcal{A}_{q,\\omega}=i(G^R_{q,\\omega}-G^A_{q,\\omega})=2\\pi \\delta(\\omega-u|q|)$. In the presence of band curvature, however, the phonons themselves interact via a resonant three-point scattering vertex, which leads to a broadening of the spectral function around the mass-shell $\\omega=u|q|$. This broadening can be described in terms of two excitations branches at frequencies $\\omega=\\epsilon^{\\pm}_q$, where $\\epsilon^-_qu|q|$ labels a phononic branch (such that $|\\epsilon^{+}_q-\\epsilon^-_q|\/q\\rightarrow 0$ for $q\\rightarrow0$)\\cite{imambekov12,lamacraft15}. The spectral weight of the excitations in the nonlinear Luttinger liquid is distributed continuously between these two branches. Whereas the solitonic branch represents an exact boundary (i.e. no spectral weight is located at frequencies $\\omega<\\epsilon_q^-$), featuring a power law singularity for frequencies above $\\epsilon^-_q$, the phononic branch represents an algebraically sharp boundary (i.e. the spectral weight for frequencies $\\omega>\\epsilon_q^+$ is strongly algebraically suppressed), featuring a power law singularity from both sides \\cite{imambekov12}. While the power-law singularities at the edges of the spectral weight obviously cannot be explained by a frequency independent self-energy, the characteristic width of the spectral weight $\\delta\\omega_q=\\epsilon^+_q-\\epsilon^-_q=\\frac{q^2}{m^*}$ can be captured by an imaginary part of the on-shell value of the self-energy $\\Sigma^R_{q,\\omega=u|q|}$, which determines the renormalized mass $m^*$ \\cite{zwerger06,aristov,lamacraft15,imambekov12}. These results hold for the zero temperature limit of the problem. At finite temperature $T>0$, however, a self-consistent Born-approximation for the on-shell self-energy predicts a scaling of the spectral weight $\\delta\\omega_q\\sim \\sqrt{|q|^3T}$ \\cite{lamacraft13,gangardt13}, which has been also observed in numerical simulations of interacting one-dimensional bosons \\cite{lamacraft13}. For $\\delta\\omega_q\\ll u|q|$, i.e. the width of the spectral weight of the excitations being much smaller than the average excitation energy, the spectral weight is still sharply concentrated at the mass-shell and one can still think (physically) of well defined excitations although the fine structure of the spectral weight is very different from what one is used to for weakly interacting quasi-particles. As a consequence, it is possible to derive a kinetic equation for the excitation densities in this regime, applying the common quasi-particle and local time approximations, and we will implement this approach below. It neglects the specific structure of the spectral weight of nonlinear Luttinger liquids, which is valid for \"static\" variables in the quasi-particle limit $\\delta\\omega_q\\ll u|q|$.\nWe begin by introducing the interaction picture for the Heisenberg operators \n\\eq{Eq21}{\n\\cre{a}{t,q}\\rightarrow \\cre{a}{t,q}e^{-iu|q|t},}\nwhich leaves the Hamiltonian \\eqref{Eq5} unmodified but shifts the spectral weight of diagonal modes to zero frequency and eliminates the phase $\\sim e^{i2u|q|t}$ of off-diagonal correlation functions \\cite{buchholdmethod}. \n The Green's functions in the interaction representation are labeled with a tilde. The Keldysh Green's function in Nambu space is\n\\eq{Eq22}{\ni\\tilde{G}^K_{t,q,\\delta}=\\left(\\hspace{-0.15cm}\\begin{array}{ll}\\langle\\{\\ann{a}{t+\\frac{\\delta}{2},q},\\cre{a}{t-\\frac{\\delta}{2},q}\\}\\rangle & \\langle\\{\\ann{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},-q}\\}\\rangle\\\\\n\\langle\\{\\cre{a}{t+\\frac{\\delta}{2},-q},\\cre{a}{t-\\frac{\\delta}{2},q}\\}\\rangle&\\langle\\{\\cre{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},q}\\}\\rangle\n\\end{array}\\hspace{-0.15cm}\\right),\\ \\ \\ \n}\nwhere $\\{\\cdot,\\cdot\\}$ is the anti-commutator and we introduced an additional relative time shift $\\delta$ associated with spectral properties of the system. The retarded Green's function is\n\\eq{Eq23}{\ni\\tilde{G}^R_{t,q,\\delta}&=&\\theta(\\delta)\\left(\\hspace{-0.15cm}\\begin{array}{ll}\\langle[\\ann{a}{t+\\frac{\\delta}{2},q},\\cre{a}{t-\\frac{\\delta}{2},q}]\\rangle & \\langle[\\ann{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},-q}]\\rangle\\\\\n\\langle[\\cre{a}{t+\\frac{\\delta}{2},-q},\\cre{a}{t-\\frac{\\delta}{2},q}]\\rangle&\\langle[\\cre{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},q}]\\rangle\n\\end{array}\\hspace{-0.15cm}\\right)\\nonumber\\\\\n&=&\\theta(\\delta)\\left(\\hspace{-0.15cm}\\begin{array}{cc}\\langle[\\ann{a}{t+\\frac{\\delta}{2},q},\\cre{a}{t-\\frac{\\delta}{2},q}]\\rangle & 0\\\\\n0&\\langle[\\cre{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},q}]\\rangle\n\\end{array}\\hspace{-0.15cm}\\right).}\nThe off-diagonal retarded and advanced Green's functions are exactly zero. This is a consequence of the Hamiltonian, which does not introduce a coupling between the modes $q$ and $-q$, such that the commutator $[\\ann{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},-q}]=0$ for all times $t,\\delta$. The anti-hermitian Keldysh Green's function is parametrized according to \\cite{kamenevbook,buchholdmethod}\n\\eq{Eq24}{\n\\tilde{G}^K_{t,q,\\delta}=\\left(\\tilde{G}^R\\circ\\sigma_z\\circ F-F\\circ\\sigma_z\\circ\\tilde{G}^A\\right)_{t,q,\\delta}}\nin terms of the time-dependent, hermitian quasi-particle distribution function $F$ and the Pauli matrix $\\sigma_z$, the latter preserving the symplectic structure of bosonic Nambu space. The $\\circ$ represents matrix multiplication with respect to momentum space and convolution with respect to time. Switching to Wigner coordinates by Fourier transforming the Keldysh Green's function with respect to relative time\n\\eq{Eq25}{\n\\tilde{G}^K_{t,q,\\omega}=\\int_{\\delta}\\tilde{G}^K_{t,q,\\delta}\\ \\ e^{i\\omega\\delta}\n}\nand applying the Wigner approximation, which, due to the RG-irrelevant interactions, is justified in the same regime for which the Luttinger description is applicable \\cite{buchholdmethod,Heating}, we find\n\\eq{Eq26}{\n\\tilde{G}^K_{t,q,\\omega}=\\tilde{G}^R_{t,q,\\omega}\\sigma_zF_{t,q,\\omega}-F_{t,q,\\omega}\\sigma_z\\tilde{G}^A_{t,q,\\omega},} \nwhich is diagonal in momentum and frequency space. Inverting Eq.~\\eqref{Eq26} by multiplying it with $\\left(\\tilde{G}^R\\right)^{-1}$ from the left and $\\left(\\tilde{G}^A\\right)^{-1}$ from the right, yields the kinetic equation for the distribution function\n\\eq{Eq27}{\ni\\partial_tF_{t,q,\\omega}\\hspace{-0.1cm}=\\hspace{-0.1cm}\\sigma_z\\Sigma^R_{t,q,\\omega}F_{t,q,\\omega}\\hspace{-0.1cm}-\\hspace{-0.1cm}F_{t,q,\\omega}\\Sigma^A_{t,q,\\omega}\\sigma_z\\hspace{-0.1cm}-\\hspace{-0.1cm}\\sigma_z\\Sigma^K_{t,q,\\omega}\\sigma_z.\\ \\ \\ \\ \\ \\ \n}\nThe retarded, advanced self-energies $\\Sigma^{R\/A}_{t,q,\\omega}$ are diagonal in Nambu space, while the Keldysh self-energy $\\Sigma^K_{t,q,\\omega}$ consists of non-vanishing diagonal and off-diagonal entries due to the initial off-diagonal occupations $m_{0,q}\\neq0$.\n\nThe kinetic equation for the phonon occupations is obtained by multiplying Eq.~\\eqref{Eq27} on both sides with the spectral function $\\tilde{\\mathcal{A}}_{t,q,\\omega}=i\\left(\\tilde{G}^R_{t,q,\\omega}-\\tilde{G}^A_{t,q,\\omega}\\right)$ and integrating over frequency space. For interacting Luttinger Liquids, the spectral function $\\tilde{\\mathcal{A}}_{t,q,\\omega}$ is very narrowly peaked at the mass shell and the kinetic equation is essentially locked onto $\\omega=0$ in this way (in the interaction picture, the mass shell is at $\\omega=0$). As a consequence, one finds kinetic equations for the diagonal densities\n\\eq{Eq28}{\n\\partial_tn_{t,q}=-\\sigma^R_{t,q}(2n_{t,q}+1)+\\sigma^K_{t,q}\n}\nand the off-diagonal densities\n\\eq{Eq29}{\n\\partial_tm_{t,q}=-2\\sigma^R_{t,q}m_{t,q}-\\Gamma^K_{t,q}.\n}\nThey can be expressed in terms of the imaginary part of the retarded on-shell self-energy \\eq{Nun1}{\\sigma^R_{t,q}=\\frac{1}{2}\\int_{\\omega}\\tilde{\\mathcal{A}}_{t,q,\\omega}\\left(\\Sigma^R_{t,q,\\omega}-\\Sigma^A_{t,q,\\omega}\\right)\\approx\\frac{1}{2}\\left(\\Sigma^R_{t,q,\\omega=0}-\\Sigma^A_{t,q,\\omega=0}\\right)} and the Keldysh on-shell self-energies \\eq{Nun2}{\\sigma^K_{t,q}=\\frac{i}{2}\\int_{\\omega}\\tilde{\\mathcal{A}}_{t,q,\\omega}\\left(\\Sigma^K_{t,q,\\omega}\\right)_{11}\\approx\\frac{i}{2}\\left(\\Sigma^K_{t,q,\\omega=0}\\right)_{11}} and \\eq{Nun3}{\\Gamma^K_{t,q}=\\frac{i}{2}\\int_\\omega\\mathcal{A}_{t,q,\\omega}\\left(\\Sigma^K_{t,q,\\omega}\\right)_{12}\\approx\\frac{i}{2}\\left(\\Sigma^K_{t,q,\\omega=0}\\right)_{12}.} The Keldysh self-energy is always anti-hermitian and therefore purely imaginary in frequency and momentum space, such that Eqs.~\\eqref{Eq28}, \\eqref{Eq29} are real. Since the criterion $|\\epsilon^+_q-\\epsilon^-_q|\\ll u|q|$ is equivalent to $\\sigma^R_{t,q}\\ll u|q|$ at zero and finite temperature equilibrium, we also apply the latter criterion for the present out-of-equilibrium situation in order to estimate the validity of our approach.\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=1\\linewidth]{Diag}\n \\caption{Diagrammatic illustration of the Dyson-Schwinger equations up to cubic order. Here, $G$ represents the full Green's function, $S^{(3)}$ the bare three-body vertex and $\\Gamma^{(3)}$ the full three-body vertex. For convenience, this displays only the topology of the diagrams, which has not been extended to Keldysh space.}\n \\label{fig:Diag}\n\\end{figure}\n\nThe phonon scattering terms in Eq.~\\eqref{Eq5} are resonant, i.e. they describe scattering between a continuum of energetically degenerate states, and as a consequence, perturbation theory diverges. In order to determine the self-energies $\\sigma^R_{t,q}, \\sigma^K_{t,q}, \\Gamma^K_{t,q}$, we apply non-perturbative Dyson-Schwinger equations, which are truncated at cubic order. This takes into account renormalization effects of the cubic vertex and yields non-perturbative self-energies. The topology of the corresponding diagrams is shown in Fig.~\\ref{fig:Diag}.\n If we neglect the cubic vertex correction, the Dyson-Schwinger equations reduce to the self-consistent Born approximation \\cite{buchholdmethod}. For an initial state with constant phonon density, as it is the case for the present setup, the vertex correction has been shown to be exactly zero \\cite{buchholdmethod,forsternelson76}, however it obtains a non-zero value in the time-evolution of the system. The kinetic equations \\eqref{Eq28}, \\eqref{Eq29} are solved iteratively, starting at a certain time $t$, the self-energies and vertex correction are computed as functions of the distributions $n_{t,q}, m_{t,q}$. Subsequently $\\partial_tn_{t,q}, \\partial_tm_{t,q}$ are determined, and used in turn to compute the distributions $n_{t+\\Delta,q}, m_{t+\\Delta,q}$ for an infinitesimally later time. This procedure is repeated in order to determine the time-evolution of the phonon densities and self-energies. A more detailed, technical derivation of the iterative solution for the kinetic equation, self-energies and vertex correction can be found in \\cite{buchholdmethod}.\n\n\\section{Thermalization and Prethermalization Dynamics}\n\\label{sec:thermalization_dynamics}\n\nAs one can see from the kinetic equations in Eq.~\\eqref{Eq28} and Eq.~\\eqref{Eq29}, the diagonal and off-diagonal phonon densities are no longer constants of motion in the presence of phonon scattering and energy is redistributed between the different momentum modes. On a general level, when the system thermalizes, as we will show below, the steady state of the dynamics in the presence of a cubic scattering as in Eq.~\\eqref{Eq5}, is solely determined by the associated temperature $T$ and independent of any further details of the initial nonequilibrium state. Specifically, the diagonal modes acquire a Bose-Einstein distribution $n_{\\infty,q} = n_{t\\rightarrow\\infty,q}=\\left(e^{ u|q|\/T}-1\\right)^{-1}$ whereas the off-diagonal distributions $m_{q}=0$ have to vanish.\n\n\nImportantly, in the resonant approximation, the final temperature $T$ ($k\\sub{B}=1$ in the following) can be computed directly from the initial state as will be shown now. In a closed system, the total energy is conserved. Moreover, the conservation of the kinetic energy is an additional exact feature of the derived kinetic equation. As a consequence, also the interaction energy itself is individually conserved. The latter is not an artifact of the kinetic equation but a feature of the resonant nature of the interactions, which, by definition of resonance, commute with the quadratic part of the Hamiltonian \\eqref{Eq5} already on an operator level. \nThis implies that the relaxation dynamics due to the interactions takes place in closed subsets of degenerate eigenstates of the quadratic Hamiltonian, which would in the absence of phonon scattering only acquire a global phase and were not able to thermalize. \nConsequently, the kinetic energy of the initial ($e_0$) and final state ($e_f$) have to be equal, which yields:\n\\eq{Eq20}{\ne_{0}=un_{\\lambda}\\Lambda^2=\\int_q \\hspace{-0.1cm}u|q| n_{0,q}\\overset{!}{=}\\int_q u |q|n_{\\infty,q}=\\frac{T^2_{\\lambda}\\pi^2}{3u}=e_{f}.\\ \\ \\ \n}\nHere, $n_{0,q}$ is the initial momentum distribution, see Eq.~\\eqref{Occ}, and $n_{\\infty,q} = \\left(e^{\\beta u|q|}-1\\right)^{-1}$ is the final, thermal distribution. This gives:\n\\eq{temperature}{T_{\\lambda}=\\frac{u\\Lambda}{\\pi}\\sqrt{3n_{\\lambda}},} which depends on the details of the quench only through the quench parameter $\\lambda$ such that we denote the temperature via $T\\sub{$\\lambda$}$ in the following. Importantly, this temperature yields a criterion for the applicability of the Luttinger theory for the present quench scenario, since Luttinger theory is only well-defined for temperatures lower than the cutoff $T_{\\lambda}1$.\n\n\\begin{figure*}\n \\includegraphics[width=1\\linewidth]{OccFull}\n \\caption{Simulation of the time-evolution of the diagonal phonon density $n_{\\tau,q}$ (left column) and off-diagonal density $m_{\\tau,q}$ (right column) for different quench parameters $\\lambda$. In each row, the individual lines correspond to different times $\\tau=(0,1,2,3,4,5)$. \nLeft column: The total phonon density increases in time (from light to dark green) and the dotted lines represent the corresponding asymptotic density in the limit $\\tau\\rightarrow\\infty$, which is a Bose distribution with the quench dependent temperature $T_{\\lambda}=(0.035, 0.124, 0.24)u\\Lambda$ (from the top to the bottom row). The distribution function is separated into two regimes according to Eq.~\\eqref{Eq37}, with a linear increase in momentum for small momenta and a corresponding thermal distribution for larger momenta. The crossover momentum separating the two regimes is marked with a dot.\nRight column: The off-diagonal phonon density is decreasing in time (from light to dark red), displaying two distinct momentum regimes: For momenta larger than the crossover, $q>q\\sub{th}$, the off-diagonal occupation decreases exponentially in momentum, while it remains close to its initial value $m_{0,q}=m_{\\lambda}$ for momenta smaller than the crossover. While any momentum mode $n_{\\tau,q>0}$ will thermalize at a finite time $\\tau<\\infty$, the zero momentum mode remains pinned to its initial value $n_{\\tau,q=0}=n_{\\tau=0,q=0}$. The latter is not an artifact of the approximation but a consequence of exact fermionic particle number conservation, as outlined in the main text.\n}\n \\label{fig:OccFull}\n\\end{figure*}\n\nThe time evolution of the phonon densities for three different quench parameters $\\lambda$ is shown in Fig.~\\ref{fig:OccFull}. It features two characteristic regimes, which are separated by a time-dependent crossover momentum $q\\sub{th}(\\tau)$, which turns out to be the inverse thermal length scale $x\\sub{th}(\\tau)=1\/q\\sub{th}(\\tau)$. According to the numerical simulations, $q\\sub{th}(\\tau)$ can be parametrized as $q\\sub{th}(\\tau)=Q_{\\lambda}\\tau^{\\alpha_{\\lambda}}$, where the exponent $\\alpha_{\\lambda}$ and the amplitude $Q_{\\lambda}$ are monotonic functions of the quench parameter (for $\\lambda>1$). According to Fig.~\\ref{fig:OccFull}, away from the crossover, the phonon distribution can be written as\n\\eq{Eq37}{\nn_{\\tau,q}=\\left\\{\\begin{array}{cl}n_{\\lambda}+c_{\\tau,\\lambda}|q|& \\mbox{ for } |q|q\\sub{th}(\\tau)\\end{array}\\right. .\n}\nFor small momenta $|q|q\\sub{th}$ fast quasi-particle scattering events have established a local equilibrium and the phonon density is well described by a Bose distribution function $n\\sub{B}(u|q|,\\tilde{T}_{\\tau,\\lambda})=\\left(e^{u|q|\/\\tilde{T}_{\\tau,\\lambda}}-1\\right)^{-1}$, which can be approximated by a classical Rayleigh-Jeans distribution, as in Eq.~\\eqref{Eq37}, for intermediate momenta $q\\sub{th}||$\\eqref{Eq39b}$|$, which leads to the condition on the distance $xq\\sub{th}$ are described by a single, well defined temperature $\\tilde{T}_{t,\\lambda}$ such that $n_{t,q}=n_{\\mbox{\\tiny B}}(u|q|,T)\\approx T\/(u|q|)$. For momenta $qq\\sub{th}$, the modes are described by the same temperature, indicating the presence of local detailed balance in the momentum regime larger than the crossover. In this regime, the temperature decays algebraically, revealing energy transport from the thermalized to the non-thermalized region, carried by dynamical slow modes. The inset shows the decay of the effective temperature for large times, allowing for numerical estimate $\\mu=2\/3$, which corresponds to the red, dotted line.\n}\n \\label{fig:MomTherm}\n\\end{figure*}\n\nIn order to determine the asymptotic dynamics in the thermalized regime, we define a momentum and time dependent temperature by inverting the on-shell Bose distribution function \n\\eq{Eq41}{\n\\tilde{T}_{t,\\lambda,q}=\\frac{u|q|}{\\log\\left(\\frac{n_{t,q}+1}{n_{t,q}}\\right).}\n}\nThe time evolution of $\\tilde{T}_{t,\\lambda,q}$ is shown in Fig.~\\ref{fig:MomTherm}. For momenta $qq\\sub{th}$, $\\tilde{T}_{t,\\lambda,q}$ becomes momentum independent and a global property of the high momentum modes. The decay of $\\tilde{T}_{t,\\lambda}=\\tilde{T}_{t,q>q\\sub{th},\\lambda}$ follows a power law in time, which can be expressed\n\\eq{Eq42}{\n\\tilde{T}_{t,\\lambda}=T_{\\lambda}+u\\Lambda\\Delta_{\\lambda}\\left(v_0\\Lambda^2t\\right)^{-\\mu},\n}\nwhere $\\mu$ is the relaxation exponent associated with the dynamical slow modes. For a one-dimensional system with energy and momentum conserving dynamics $\\mu=2\/3$, since this behavior corresponds to the Kardar-Parisi-Zhang (KPZ) universality class \\cite{KPZ,spohn04,lamacraft13,vanBeijeren,spohn15}. Performing a single parameter fit from the numerical simulations, we find that for large times $\\mu=2\/3$ agrees very well with the numerical data for various different quench scenarios. However, for intermediate times, we find scaling behavior with $\\mu>2\/3$ for some quenches, which might be traced back to the presence of subleading correction terms due to couplings to other diffusive modes \\cite{Lux13,narayan02,Mukerjee}. Numerically a distinction of these possible scaling contributions is only possible for simulation times of multiple decades, such that we cannot exclude a different exponent $\\mu<2\/3$ at the largest times \\cite{Lux13}, which is however not observed in our simulations.\n\n\nWhile the establishment of a local detailed balance, leading to effective thermalization and thermal-like fermionic correlation functions is an effect of local quasi-particle scattering, the asymptotic thermalization dynamics describing energy transport over large distances in momentum space is determined by macroscopic diffusive modes in the system. This is observable by an algebraically decaying temperature towards the final temperature of the system $T_{\\lambda}$. The discussion on the dynamical slow modes remains valid even in the presence of off-resonant scattering processes and therefore the universal properties of the asymptotic thermalization process remain unmodified. However, non-universal properties, such as the final temperature as well as the relaxation rate will be modified by the off-resonant processes. Their precise computation would be a task for numerical simulations.\n\n\\section{Conclusion}\nIn this work, we have analyzed the relaxation dynamics of interacting Luttinger liquids, microscopically represented by one-dimensional interacting fermions with band curvature, after a sudden quench in the fermionic interaction. The theoretical analysis is based on quantum kinetic equations for the phonon distribution function and non-perturbative Dyson-Schwinger equations, which are both well suited to determine the time-evolution of static observables for interacting Luttinger liquids with resonant, cubic interactions, and applicable in a broad parameter regime within the Luttinger framework. The central result is a two-step thermalization procedure including a spatio-temporal prethermalized regime for intermediate distances and times, which leads to fermionic correlation functions described by a generalized Gibbs state on these distances, and corresponds to fast quasi-particle formation after the quench. On smaller distances, a thermalized regime occurs due to the scattering and associated redistribution of energy between the quasi-particle modes. This regime is described by thermal correlation functions with a characteristic thermal correlation length and a thermal quasi-particle distribution with an effective temperature that decays algebraically in time towards its asymptotic value.\n\n This work shows in which way thermalization and prethermalization occur and spread in space for RG-irrelevant, and in this sense weak, integrability breaking interactions. In this setup both thermalization and prethermalization occur locally in space. While the prethermalized region spreads ballistically in space, the thermalized region spreads sub-ballistically due to the subleading, RG-irrelevant nature of the interactions. This allows for a well-defined prethermal regime in time and space, which would not be possible for a constant, momentum independent scattering vertex, for which thermalization would occur immediately on all different length scales.\nThis underpins the statement that typical candidates for clearly observable prethermalized regimes within generic thermalization dynamics are quasi-particle theories with RG irrelevant interactions.\n\n\n\n\\begin{acknowledgments}\nWe acknowledge valuable discussions with Alessio Recati. This research was supported by the\nGerman Research Foundation (DFG) through the Institutional\nStrategy of the University of Cologne within the German\nExcellence Initiative (ZUK 81) and the European Research\nCouncil (ERC) under the European Unions Horizon\n2020 research and innovation programme (grant agreement\nNo 647434)\nas well as the\nDeutsche Akademie der Naturforscher Leopoldina under grant numbers LPDS 2013-07 and LPDR 2015-01.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec:Introduction}}\n\nWe develop an empirical framework to identify and estimate the heterogeneous\neffects of treatments on outcomes of interest, where the treatments\nare the result of agents' interaction (e.g., bargaining,\noligopolistic entry, decisions in the presence of peer effects or\nstrategic effects). Treatments are determined as an equilibrium of\na game and these strategic decisions of players endogenously affect\ncommon or player-specific outcomes. For example, one may be interested\nin the effects of entry of newspapers on local political behavior,\nentry of carbon-emitting companies on local air pollution and health\noutcomes, the presence of potential entrants in nearby markets on\npricing or investment decisions of incumbents, the exit decisions\nof large supermarkets on local health outcomes, or the provision of\nlimited resources when individuals make participation decisions under\npeer effects and their own gains from the treatment.\\footnote{The entry and pollution is our leading example introduced in Section\n\\ref{sec:stylized_ex}; the other examples are discussed in detail\nin Appendix \\ref{sec:Examples}.} As reflected in some of these examples, our framework allows us to\nstudy the \\textit{externalities of strategic decisions}, such as societal\noutcomes resulting from firm behavior. Ignoring strategic interaction\nin the treatment selection process may lead to biased, or at least\nless informative, conclusions about the effects of interest.\n\nWe consider a model in which agents play a discrete game of complete\ninformation, whose equilibrium actions (i.e., a profile of binary\nendogenous treatments) determine a post-game outcome in a nonseparable\nmodel with endogeneity. We are interested in the various treatment\neffects of this model. In recovering these parameters, the setting\nof this study poses several challenges. First, the first-stage game\nposits a structure in which binary dependent variables are simultaneously\ndetermined in threshold crossing models, thereby, making the model,\nas a whole, \\textit{incomplete}. This is related to the problem of\nmultiple equilibria in the game. Second, due to this simultaneity,\nthe selection process for each treatment in the profile does not exhibit\nthe conventional monotonic property \\`a la \\citet{imbens1994identification}.\nFurthermore, we want to remain flexible with other components of the\nmodel. That is, we make no assumptions on the joint distributions\nof the unobservables nor parametric restrictions on the player's payoff\nfunction and how treatments affect the outcome. In addition, we do\nnot impose any arbitrary equilibrium selection mechanism to deal with\nthe multiplicity of equilibria, nor require that players be symmetric.\nIn nonparametric models with multiplicity and\/or endogeneity, identification\nmay be achieved using excluded instruments with large support. Although\nsuch a strong requirement can be met in practice, estimation and inference\ncan still be problematic (\\citet{andrews1998semiparametric}, \\citet{khan2010irregular}).\nThus, we avoid such assumptions for instruments and other exogenous\nvariables.\n\nThe first contribution of this study is to establish that under strategic\nsubstitutability, regions that predict the equilibria of the treatment\nselection process in the first-stage game can present a monotonic\npattern in terms of the number of treatments selected.\\footnote{To estimate payoff parameters, \\citet{berry1992estimation} partly\ncharacterizes equilibrium regions. To calculate the bounds on these\nparameters, \\citet{CT09} simulate their moment inequalities model\nthat are implied by the shape of these regions, especially the regions\nfor multiple equilibria. While their approaches are sufficient for\ntheir analyses, full analytical results are critical for the identification\nanalysis in this current study.} The second contribution of this study is to show, after restoring\nthe \\textit{generalized monotonicity} in the selection process, how\nthe model structure and the data can provide information about treatment\nparameters, such as the average treatment effects (ATEs). We first\nestablish the bounds on the ATE and other related parameters with\npossibly discrete instruments. We also show that tighter bounds on\nthe ATE can be obtained by introducing (possibly discrete) exogenous\nvariables excluded from the first-stage game. This is especially motivated\nwhen the outcome variable is affected by externalities generated by\nthe players. We can derive sharp bounds as long as the outcome variable\nis binary. To deal with the multiple equilibria problem in our analysis,\nwe assume that instruments vary sufficently to offset the effect of\nstrategic substitutability. We provide a simple testable implication\nfor the existence of such instrument variation in the case of mutually\nindependent payoff unobservables. This requirement of variation is\nqualitatively different and substantially weaker than a typical large\nsupport assumption. A marked feature of our analyses is that for the\nsharp bounds on the ATE, player-specific instruments are not necessary.\n\nOur bound analysis\nbuilds on \\citet{VY07} and \\citet{SV11}, which consider point and partial identification in single-agent nonparametric triangular models\nwith binary endogenous variables. Unlike them, however, we allow for multi-agent strategic interaction\nas a key component of the model. Some studies have extended a single-treatment\nmodel to a multiple-treatment setting (e.g., \\citet{heckman2006understanding},\n\\citet{jun2011tighter}), but their models maintain monotonicity in\nthe selection process and none of them allow simultaneity among the\nmultiple treatments resulting from agents' interaction, as we do in\nthis study.\n\nIn interesting recent work, \\citet{heckman2015unordered}, and \\citet{lee2016identifying}\nextend the monotonicity of the selection process in multi-valued treatments\nsettings. \\citet{heckman2015unordered} introduce unordered monotonicity,\nwhich is a different type of treatment selection mechanisms than ours.\n\\citet{lee2016identifying} consider more general non-monotonicity\nand do mention entry games as one example of the treatment selection\nprocesses they allow. However, they assume known payoffs and bypass\nthe multiplicity of equilibria by assuming a threshold-crossing equilibrium\nselection mechanism, both of which we do not assume in this study.\nIn addition, \\citet{lee2016identifying}'s focus is on the identification\nof marginal treatment effects with continuous instruments. In another\nrelated work, \\citet{chesher2014generalized} consider a class\nof generalized instrumental variable models in which our model may\nfall and propose a systematic method of characterizing sharp identified\nsets for admissible structures. This present study's characterization\nof the identified sets is analytical, which helps investigate how the\nidentification is related to exogenous variation in the model and\nto the equilibrium characterization in the treatment selection. Also,\ncalculating the bounds on the treatment parameters using their approach\ninvolves projections of identified sets that may require parametric\nrestrictions. Lastly, \\citet{Han18,Han19c} consider identification\nof dynamic treatment effects and optimal treatment regimes in a nonparametric\ndynamic model, in which the dynamic relationship causes non-monotonicity\nin the determination of each period's outcome and treatment.\n\nWithout triangular structures, \\citet{manski1997monotone}, \\citet{MP00}\nand \\citet{Man13} also propose bounds on the ATE with multiple treatments\nunder various monotonicity assumptions, including an assumption on\nthe sign of the treatment response. We take an alternative approach\nthat is more explicit about treatments interaction while remaining\nagnostic about the direction of the treatment response. Our results\nsuggest that provided there exist exogenous variation excluded from\nthe selection process, the bounds calculated from this approach can\nbe more informative than those from their approach.\n\nIdentification in models for binary games with complete information\nhas been studied in \\citet{Tam03}, \\citet{CT09}, and \\citet{bajari2010identification},\namong others.\\footnote{See also \\citet{galichon2011set} and \\citet{beresteanu2011sharp}\nfor a more general setup that includes complete information games\nas an example.} This present study contributes to this literature by considering post-game outcomes that are often not of players' direct concern.\nAs related work that considers post-game outcomes, \\citet{ciliberto2015market}\nintroduce a model in which firms make simultaneous decisions of entry\nand pricing upon entry. Consequently, their model can be seen as a\nmulti-agent extension of a sample selection model. On the other hand,\nthe model considered in this study is a multi-agent extension of a\nmodel for endogenous treatments. At a more general level, our approach is an attempt to bridge the treatment effect literature and the industrial organization (IO) literature. We are interested in the evaluation of treatments that are the result of agents' strategic interaction, an aspect that is key in the IO literature. To conduct the counterfactual analysis, however, we closely follow the treatment effect literature, instead of the structural approach of the IO literature. For example, \\citet{CT09} and \\citet{ciliberto2015market}\nimpose economic structure and parametric assumptions to recover model primitives for policy analyses. In contrast, our parameters of interest are\ntreatment effects as functionals of the primitives (but excluding\nthe game parameters), and thus, allow our model to remain nonparametric. In addition, as the goal is different, we employ a different approach to partial identification\nunder the multiplicity of equilibria than theirs.\\footnote{Even if we are willing to\nassume a known distribution for the unobserved payoff types, their approach to multiplicity is not applicable\nto the particular setting of this study.}\n\n\n\nTo demonstrate the applicability of our method, we take the bounds\nwe propose to data on airline market structure and air pollution in\ncities in the U.S. Aircrafts and airports land operations are a major\nsource of emissions, and thus, quantifying the causal effect of air\ntransport on pollution is of importance to policy makers.\nWe explicitly allow market structure to be determined endogenously\nas the outcome of an entry game in which airlines behave strategically\nto maximize their profits and where the resulting pollution in this\nmarket is not internalized by the firms. Additionally, we do not impose\nany structure on how airline competition affects pollution and allow\nfor heterogenous effects across firms. In other words, not only do\nwe allow the effect of a different number of firms in the market on\npollution to be nonlinear and not restricted, but also\ndistinguish the identity of the firms. The latter is important if we believe that behavior post-entry differs across\nairlines. For example, different airlines might operate the market with a higher frequency\nor with different types of airplanes, hence affecting pollution in a different\nway.\nTo implement our application, we combine data from two sources. The\nfirst contains airline information from the Department of Transportation,\nwhich we use to construct a dataset of airlines' presence in each\nmarket. We then merge it with air pollution data in each airport from\nair monitoring stations compiled by the Environmental Protection Agency.\nIn our preferred specification, our outcome variable is a binary measure\nof the level of particulate matter in the air.\n\nWe consider three sets of ATE exercises to investigate different aspects\nof the relationship between market structure and pollution in equilibrium.\nThe first simply quantifies the effects of each airline operating\nas a monopolist compared to a situation in which the market is not\nserved by any airline. We find that the effect of each airline on\npollution is positive and statistically significant. We also find\nevidence of heterogeneity in the effects across different airlines.\nThe second set of exercises examines the ATEs of all potential market\nstructures on pollution. We find that the probability of high pollution\nis increasing with the number of airlines in the market, but at a\ndecreasing rate. Finally, the third set of exercises quantifies the\nATE of a single airline under all potential configurations of the\nmarket in terms of its rivals. We observe that in all cases, Delta\nentering a market has a positive effect on pollution and this effect\nis decreasing with the number of rivals. The results from the last\ntwo set of exercises are consistent with the results of a Cournot-competition\noligopolistic model in which incumbents \\emph{accommodate} new entrants\nby reducing the quantity they produce.\n\nThis paper is organized as follows. Section \\ref{sec:stylized_ex}\nsummarizes the analysis of this study using a stylized example. Section\n\\ref{sec:General-Theory} presents a general theory: Section \\ref{subsec:Model}\nintroduces the model and the parameters of interest; Section \\ref{subsec:Geometry}\npresents the generalized monotonicity for equilibrium regions for\nmany players; and Section \\ref{subsec:Partial-Identification} delivers\nthe partial identification results of this study. Section \\ref{sec:Monte-Carlo-Studies}\npresents a numerical illustration and Section \\ref{sec:Empirical-Application}\nthe empirical application on airlines and pollution. In the Appendix,\nSection \\ref{sec:Examples} provides more examples to which our setup\ncan be applied. Section \\ref{sec:Extensions} contains four extensions\nof our main results. Finally, Section \\ref{sec:Proofs} collects the\nproofs of theorems and lemmas.\n\n\\section{A Stylized Example\\label{sec:stylized_ex}}\n\nWe first illustrate the main results of this study with a stylized\nexample. Suppose we are interested in the effects of airline competition\non local air quality (or health). Let $Y_{i}$ denote the binary indicator\nof air pollution in market $i$. For illustration, we assume there\nare two potential airlines. In the next section, we present a general\ntheory with more than two players. Let $D_{1,i}$ and $D_{2,i}$ be\nbinary variables that indicate the decisions to enter market $i$\nby Delta and United, respectively. We allow the decisions $D_{1,i}$\nand $D_{2,i}$ to be correlated with some unobserved characteristics\nof the local market that affect $Y_{i}$. Moreover, since $D_{1,i}$\nand $D_{2,i}$ are equilibrium outcomes of the entry game, we allow\nthem to be outcomes from multiple equilibria. The endogeneity and\nthe presence of multiple equilibria are our key challenges in this\nstudy.\n\nLet $Y_{i}(d_{1},d_{2})$ be the potential air quality had Delta and\nUnited's decisions been $(D_{1},D_{2})=(d_{1},d_{2})$; for example,\n$Y_{i}(1,1)$ is the potential air quality from duopoly, $Y_{i}(1,0)$\nis with Delta being a monopolist, and so on. Let $X_{i}$ be a vector\nof market characteristics that affect $Y_{i}$. Our parameter of interest\nis the ATE, $E[Y_{i}(d_{1},d_{2})-Y_{i}(d_{1}',d_{2}')|X_{i}=x]$,\nwhich captures the effect of market structure on pollution. One interesting\nATE is $E[Y_{i}(1,d_{2})-Y_{i}(0,d_{2})|X_{i}=x]$ for each $d_{2}$,\nwhere we can learn the interaction effects of treatments, e.g., how\nmuch the average effect of Delta's entry is affected by United's entry:\n$E\\left[Y_{i}(1,1)-Y_{i}(0,1)\\right]-E\\left[Y_{i}(1,0)-Y_{i}(0,0)\\right]$\n(suppressing $X_{i}$). In our empirical application (Section \\ref{sec:Empirical-Application}),\nwe consider this and other related parameters in a more realistic\nmodel, where there are more than two airlines.\n\nWe show how we overcome the problems of endogeneity and multiple equilibria\nand how to construct bounds on the ATE using the excluded instruments\nand other exogenous variables. Let $Z_{1,i}$ and $Z_{2,i}$ be cost\nshifters for Delta and United, respectively, which serve as instruments.\nAs a benchmark, we first consider naive bounds analogous to \\citet{manski1990nonparametric}\nusing excluded instruments which satisfy \n\\begin{align}\nY_{i}(d_{1},d_{2}) & \\perp(Z_{1,i},Z_{2,i})|X_{i}\\label{as:manski_IV}\n\\end{align}\nfor all $(d_{1},d_{2})$. To simplify notation, we suppress the index\n$i$ henceforth, let $\\boldsymbol{D}\\equiv(D_{1},D_{2})$ and $\\boldsymbol{Z}\\equiv(Z_{1},Z_{2})$,\nand write $E[\\cdot|w]\\equiv E[\\cdot|W=w]$ for a generic r.v. $W$.\nAs an illustration, we focus on calculating bounds on $E[Y(1,1)|X=x]$.\nNote that \n\\begin{align}\nE[Y(1,1)|x]=E[Y(1,1)|\\boldsymbol{z},x] & =E[Y|\\boldsymbol{D}=(1,1),\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z},x]\\nonumber \\\\\n & +\\sum_{\\boldsymbol{d}^{\\prime}\\neq(1,1)}E[Y(1,1)|\\boldsymbol{D}=\\boldsymbol{d}^{\\prime},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{\\prime}|\\boldsymbol{z},x],\\label{eq:Manski_expand-1}\n\\end{align}\nwhere the first equality is by \\eqref{as:manski_IV}. Manski-type\nbounds can be obtained by observing that the counterfactual term $E[Y(1,1)|\\boldsymbol{D}=\\boldsymbol{d}^{\\prime},\\boldsymbol{z},x]=\\Pr[Y(1,1)=1|\\boldsymbol{D}=\\boldsymbol{d}^{\\prime},\\boldsymbol{z},x]$\nis bounded above by one and below by zero. By further using the variation\nin $\\boldsymbol{Z}$, which is excluded from $Y(1,1)$, the lower\nand upper bounds on $E[Y(1,1)\\vert x]$ can be written as \n\\begin{align*}\nL_{Manski}(x) & \\equiv\\sup_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x],\\\\\nU_{Manski}(x) & \\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z}]\\right\\} .\n\\end{align*}\nThe goal of our analysis is to derive tighter bounds than $L_{Manski}(x)$\nand $U_{Manski}(x)$ by introducing further assumptions motivated\nby economic theory.\n\nTo illustrate, we introduce the following semi-triangular model with\nlinear indices. In the next section, we generalize this model by\nintroducing fully nonparametric models that allow continuous $Y$. All the assumptions and results illustrated in the current section are formally stated and proved in the next section. Consider\n\\begin{align}\nY & =1[\\mu_{1}D_{1}+\\mu_{2}D_{2}+\\beta X\\ge\\epsilon],\\label{eq:model_ex1}\\\\\nD_{1} & =1[\\delta_{2}D_{2}+\\gamma_{1}Z_{1}\\ge U_{1}],\\label{eq:model_ex2}\\\\\nD_{2} & =1[\\delta_{1}D_{1}+\\gamma_{2}Z_{2}\\ge U_{2}],\\label{eq:model_ex3}\n\\end{align}\nwhere $(\\epsilon,U_{1},U_{2})$ are continuously distributed unobservables\nthat can be arbitrarily correlated, $(U_{1},U_{2})$ are uniform,\nand assume \n\\begin{align}\n & (\\epsilon,U_{1},U_{2})\\perp(Z_{1},Z_{2})|X,\\label{eq:my_IV}\\\\\n & \\delta_{1}<0\\text{ and }\\delta_{2}<0,\\label{eq:strategic_sub}\\\\\n & sgn(\\mu_{1})=sgn(\\mu_{2}).\\label{eq:mono}\n\\end{align}\nNote that \\eqref{eq:my_IV} replaces \\eqref{as:manski_IV}, \\eqref{eq:strategic_sub}\nassumes strategic substitutability, and \\eqref{eq:mono} is plausible\nin the current example of air quality and entry. Owing to the first\nstage simultaneity,\nthe model \\eqref{eq:model_ex1}--\\eqref{eq:model_ex3} is \\textit{incomplete}, i.e., the model primitives and the\ncovariates do not uniquely predict $(Y,\\boldsymbol{D})$. In this\nmodel, we are \\textit{not} interested in the players' payoff parameters\n$(\\delta_{-s},\\gamma_{s})$ for $s=1,2$, individual parameters $(\\mu_{1},\\mu_{2},\\beta)$\nthat generate the outcome, nor distributional parameters. Instead,\nwe are interested in the ATE as a function of $(\\mu_{1},\\mu_{2},\\beta)$.\nThis is in contrast to \\citet{ciliberto2015market}, where payoff\nand pricing parameters are direct parameters of interest, and thus,\nour identification question and strategy (especially how we deal with\nmultiple equilibria) are different from theirs.\n\nTypically, a standard approach that utilizes instrumental variables\ncompares the reduced-form relationship between the outcome and treatment\nwith the reduced-form relationship between the treatment and instrument.\nWe apply the same idea here by changing the values of $Z_{1}$ and\n$Z_{2}$ and measure the change in $Y$ relative to the change in\n$D_{1}$ and $D_{2}$. To this end, for two realizations $\\boldsymbol{z},\\boldsymbol{z}'$\nof $\\boldsymbol{Z}$, say low and high entry cost for both airlines,\nwe introduce reduced-form objects directly recovered from the data:\n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv\\Pr[Y=1|\\boldsymbol{z},x]-\\Pr[Y=1|\\boldsymbol{z}',x],\\label{eq:h(zzx)-1}\\\\\nh_{\\boldsymbol{d}}(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv\\Pr[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}',x]\\label{eq:hj-1}\n\\end{align}\nfor $d\\in\\{(0,0),(1,0),(0,1),(1,1)\\}\\equiv\\mathcal{D}$. We show that\n\\eqref{eq:h(zzx)-1}--\\eqref{eq:hj-1} deliver useful information\nabout the outcome index function ($\\mu_{1}D_{1}+\\mu_{2}D_{2}+\\beta X$),\nwhich in turn is helpful in constructing bounds on the ATE. Note that\n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}',x) & =h_{11}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{10}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{01}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{00}(\\boldsymbol{z},\\boldsymbol{z}',x)\\nonumber \\\\\n & =\\Pr[Y=1,\\boldsymbol{D}=(1,1)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(1,1)|\\boldsymbol{z}',x]\\nonumber \\\\\n & +\\Pr[Y=1,\\boldsymbol{D}=(1,0)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(1,0)|\\boldsymbol{z}',x]\\nonumber \\\\\n & +\\Pr[Y=1,\\boldsymbol{D}=(0,1)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(0,1)|\\boldsymbol{z}',x]\\nonumber \\\\\n & +\\Pr[Y=1,\\boldsymbol{D}=(0,0)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(0,0)|\\boldsymbol{z}',x],\\label{eq:h_derive0}\n\\end{align}\nwhere $\\boldsymbol{D}=(1,0)$ and $(0,1)$ are the airlines' decisions\nthat may arise as multiple equilibria. The increase in cost (from\n$\\boldsymbol{z}$ to $\\boldsymbol{z}'$) will make the operation of\nthese airlines less profitable in some markets, depending on the values\nof the unobservables $\\boldsymbol{U}=(U_{1},U_{2})$. This will result\nin a change in the market structure in those markets. Specifically,\nmarkets ``on the margin'' may experience one of the following changes\nin structure as cost increases: (a) from duopoly to Delta-monopoly;\n(b) from duopoly to United-monopoly; (c) from Delta-monopoly to no\nentrant; (d) from United-monopoly to no entrant; and (e) from duopoly\nto no entrant. These changes are depicted in Figure \\ref{fig:As_EQ-1},\nwhere each $R_{d_{1},d_{2}}(\\boldsymbol{z})$ denotes the maximal\nregion that predicts $(d_{1},d_{2})$, given $\\boldsymbol{Z}=\\boldsymbol{z}$.\\footnote{See Section \\ref{subsec:notation} in the Appendix for a formal definition.\nThe figure is drawn in a way that $\\gamma_{1}$ and $\\gamma_{2}$\nare negative.} \n\\begin{figure*}[t]\n\\centering \\begin{subfigure}[t]{0.35\\textwidth} \\centering \\begin{tikzpicture}[scale=0.33]\n\\draw[step=1cm,gray,very thin] (-3,-3) rectangle (5,5); \\draw (-3,-3) node[anchor=north east] {0}; \\draw (5,-3) node[anchor=north] {1}; \\draw (-3,5) node[anchor=east] {1};\n\n\\path [draw=none, fill=gray, opacity=0.3] (3.5,1) rectangle (-3,5); \\path [draw=none, fill=gray, opacity=0.3] (1.7,-3) rectangle (5,3);\n\n\\draw[thick,->] (1.7,3) -- (5,3); \\draw[thick,->] (1.7,3) -- (1.7,-3); \\draw[thick,->] (3.5,1) -- (-3,1); \\draw[thick,->] (3.5,1) -- (3.5,5); \n\n\\node [below right, black] at (-3.2,5) {$R_{10}(\\boldsymbol{z})$};\n\\node [below right, black] at (1.4,-0.5) {$R_{01}(\\boldsymbol{z})$};\n\\node [below right, black] at (3.2,5) {$R_{00}(\\boldsymbol{z})$}; \\node [below right, black] at (-3.2,-0.5) {$R_{11}(\\boldsymbol{z})$};\n\n\\draw (-4,6) node[anchor=east] {$U_2$};\n\\draw (6.3,-4) node[anchor=north] {$U_1$};\n\\draw (-3,1) node[anchor=east] {\\small{$\\delta_1+\\gamma_2 z_2$}};\n\\draw (1.7,-3) node[anchor=north] {\\small{$\\delta_2+\\gamma_1 z_1$}};\n\\draw (5,6) node[anchor=east] {\\small{$\\gamma_1 z_1$}};\n\\draw (6.5,3.5) node[anchor=north] {\\small{$\\gamma_2 z_2$}};\n\n\\end{tikzpicture} \\caption{When $\\boldsymbol{Z}=\\boldsymbol{z}$}\n\\end{subfigure\n~ \\begin{subfigure}[t]{0.35\\textwidth} \\centering \\begin{tikzpicture}[scale=0.33]\n\\draw[step=1cm,gray,very thin] (-3,-3) rectangle (5,5); \\draw (-3,-3) node[anchor=north east] {0}; \\draw (5,-3) node[anchor=north] {1}; \\draw (-3,5) node[anchor=east] {1};\n\\path [draw=none, fill=red, opacity=0.3] (1,-1) rectangle (-3,5); \\path [draw=none, fill=red, opacity=0.3] (-1,-3) rectangle (5,0);\n\n\\draw[thick,->] (1.7,3) -- (5,3); \\draw[thick,->] (1.7,3) -- (1.7,-3); \\draw[thick,->] (3.5,1) -- (-3,1); \\draw[thick,->] (3.5,1) -- (3.5,5); \n\n\\draw[thick,red,dashed,->] (-1,0) -- (5,0); \n\\draw[thick,red,dashed,->] (-1,0) -- (-1,-3); \n\\draw[thick,red,dashed,->] (1,-1) -- (-3,-1); \n\\draw[thick,red,dashed,->] (1,-1) -- (1,5); \n\n\\node [below right, red] at (-3.2,3) {$R_{10}(\\boldsymbol{z}')$};\n\\node [below right, red] at (0.3,-1) {$R_{01}(\\boldsymbol{z}')$};\n\\node [below right, red] at (1.1,3) {$R_{00}(\\boldsymbol{z}')$}; \\node [below right, red] at (-4,-1) {$R_{11}(\\boldsymbol{z}')$};\n\n\\draw (-4,6) node[anchor=east] {$U_2$};\n\\draw (6.3,-4) node[anchor=north] {$U_1$};\n\\draw[red] (-3,-0.6) node[anchor=east] {\\small{$\\delta_1+\\gamma_2 z'_2$}};\n\\draw[red] (-0.5,-3) node[anchor=north] {\\small{$\\delta_2+\\gamma_1 z'_1$}};\n\\draw[red] (2.3,6) node[anchor=east] {\\small{$\\gamma_1 z'_1$}};\n\\draw[red] (6.5,1) node[anchor=north] {\\small{$\\gamma_2 z'_2$}};\n\n\\end{tikzpicture}\n\n\\caption{When $\\boldsymbol{Z}=\\boldsymbol{z}'$}\n\\end{subfigure}\\caption{Change in Equilibrium Regions with Compensating Strategic Substitutability.\\label{fig:As_EQ-1}}\n\\end{figure*}\n\nThese changes (a)--(e) are a consequence of the monotonic pattern\nof equilibrium regions, which we formally establish in a general setting\nof more than two players in Theorem \\ref{thm:mono_pattern} of Section\n\\ref{subsec:Geometry}.\n\nIn general, besides these five scenarios, there may be markets that\nused to be Delta-monopoly but become United-monopoly and vice versa,\ni.e., markets that exhibit \\textit{non-monotonic} behaviors; see Remark\n\\ref{rem:nonmonotone} below for details. Owing to possible multiple\nequilibria, we are agnostic about these latter types of changes except\nin extreme cases, where one equilibrium is selected with probability\none. We generally do not know the equilibrium selection mechanism\nin play, much less about how such mechanism changes as cost $\\boldsymbol{Z}$\nchanges. The key idea in this study is to overcome the non-monotonicity\nby shifting the cost sufficiently so that there is no market that\nswitches from one monopoly to another. We show that the shift in cost\nthat compensates the strategic substitutability does just that, as\nis depicted in Figure \\ref{fig:As_EQ-1}. In this figure, we assume\n$\\delta_{2}+\\gamma_{1}z_{1}>\\gamma_{1}z_{1}'$ and $\\delta_{1}+\\gamma_{2}z_{2}>\\gamma_{2}z_{2}'$.\nIn other words, we assume \n\\begin{align}\n\\left|\\gamma_{s}(z_{s}'-z_{s})\\right| & \\ge\\left|\\delta_{-s}\\right|\\text{ for }s=1,2.\\label{eq:EQ_S2}\n\\end{align}\nImportantly, we do not require infinite variation in $\\boldsymbol{Z}$.\\footnote{Of course, changing each $Z_{s}$ from $-\\infty$ to $\\infty$ will\ntrivially achieve our requirement of having no market that switches\nfrom one monopoly to another.} In fact, we show that the compensating strategic substitutability\n\\eqref{eq:EQ_S2} is implied by the following condition, which can\nbe tested using the data: there exist $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$\nsuch that \n\\begin{align}\n\\Pr[\\boldsymbol{D}=(0,0)|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z}'] & >2-\\sqrt 2.\\label{eq:asy2-1}\n\\end{align}\n\nSuppose $\\boldsymbol{z},\\boldsymbol{z}'$ satisfy \\eqref{eq:EQ_S2}.\nThen, by \\eqref{eq:my_IV}, we can derive from \\eqref{eq:h_derive0}\nthat (suppressing $X=x$ for simplicity) \n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}') & =\\Pr[\\epsilon\\leq\\mu_{1}+\\mu_{2},\\boldsymbol{U}\\in\\Delta_{a}\\cup\\Delta_{b}\\cup\\Delta_{e}]\\nonumber \\\\\n & -\\Pr[\\epsilon\\leq\\mu_{1},\\boldsymbol{U}\\in\\Delta_{a}]+\\Pr[\\epsilon\\leq\\mu_{1},\\boldsymbol{U}\\in\\Delta_{c}]\\nonumber \\\\\n & -\\Pr[\\epsilon\\leq\\mu_{2},\\boldsymbol{U}\\in\\Delta_{b}]+\\Pr[\\epsilon\\leq\\mu_{2},\\boldsymbol{U}\\in\\Delta_{d}]\\nonumber \\\\\n & -\\Pr[\\epsilon\\leq0,\\boldsymbol{U}\\in\\Delta_{c}\\cup\\Delta_{d}\\cup\\Delta_{e}],\\label{eq:h_derive}\n\\end{align}\nwhere $\\Delta_{i}$ ($i\\in\\{a,...,e\\}$) are disjoint and each $\\Delta_{i}$\ncharacterizes those markets on the margin described above: $\\Delta_{a}$\ncorresponds to the set of $\\boldsymbol{U}$'s that experience (a),\n$\\Delta_{b}$ corresponds to (b), and so on. Once \\eqref{eq:h_derive}\nis derived, it is easy to see that \n\\begin{align}\nsgn\\{h(\\boldsymbol{z},\\boldsymbol{z}',x)\\} & =sgn(\\mu_{1})=sgn(\\mu_{2}),\\label{eq:sign_match}\n\\end{align}\nwhich is formally shown in Lemma \\ref{lem:asy_to_asy_star}(i). See\nSection \\ref{subsec:proof_S=00003D00003D2} in the Appendix for a\nproof in this specific two-player case, which simplifies the argument\nin the general proof. The result \\eqref{eq:sign_match} is helpful\nfor our bound analysis. Again, focus on $E[Y(1,1)|x]$ and suppose\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)>0$. Then, $\\mu_{1}>0$ and $\\mu_{2}>0$,\nand thus, we can derive the lower bound on, e.g., $E[Y(1,1)|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]$\nin \\eqref{eq:Manski_expand-1} as \n\\begin{align}\nE[Y(1,1)|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x] & =\\Pr[\\epsilon\\le\\mu_{1}+\\mu_{2}+\\beta x|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]\\nonumber \\\\\n & \\ge\\Pr[\\epsilon\\le\\mu_{1}+\\beta x|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]\\label{eq:ex_tigher_bound-1-1}\\\\\n & =E[Y|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x],\\nonumber \n\\end{align}\nwhich is larger than zero, the previous naive lower bound. Similarly,\nwe can calculate the lower bounds on all $E[Y(1,1)|\\boldsymbol{D}=\\boldsymbol{d},\\boldsymbol{z},x]$\nfor $\\boldsymbol{d}\\neq(1,1)$. Consequently, by \\eqref{eq:Manski_expand-1},\nwe have \n\\begin{align*}\nE[Y(1,1)|x] & \\ge\\Pr[Y=1|\\boldsymbol{z},x],\n\\end{align*}\ni.e., the lower bound on $E[Y(1,1)|x]$ is \n\\begin{align*}\n\\tilde{L}(x) & \\equiv\\sup_{\\boldsymbol{z}}\\Pr[Y=1|\\boldsymbol{z},x].\n\\end{align*}\nNote that $\\tilde{L}(x)\\ge L_{Manski}(x)$. In this case, $\\tilde{U}(x)=U_{Manski}(x)$.\nIn Section \\ref{subsec:Partial-Identification}, we show that $\\tilde{L}(x)$\nand $\\tilde{U}(x)$ are sharp under \\eqref{eq:model_ex1}--\\eqref{eq:mono}.\n\nWe can further tighten the bounds if we have exogenous variables that\nare excluded from the entry decisions, i.e., from the $D_{1}$ and\n$D_{2}$ equations. The existence of such variables is not necessary\nbut helpful in tightening the bounds, and can be motivated by the\nnotion of externalities. That is, there can exist factors that affect\n$Y$ but do not enter the players' first-stage payoff functions. Modify\n\\eqref{eq:my_IV} and assume \n\\begin{align}\n(\\epsilon,U_{1},U_{2}) & \\perp(Z_{1},Z_{2},X),\\label{eq:my_IV2}\n\\end{align}\nwhere conditioning on other (possibly endogenous) covariates is suppressed.\nHere, $X$ can be the characteristics of the local market that\ndirectly affect pollution or health levels, such as weather shocks\nor the share of pollution-related industries in the local economy.\nWe assume that conditional on other covariates, these factors affect\nthe outcome but do not enter the payoff functions, since the airlines\ndo not take them into account in their decisions.\n\nTo exploit the variation in $X$ (in addition to the variation in\n$\\boldsymbol{Z}$), let $(x,\\tilde{x},\\tilde{\\tilde{x}})$ be (possibly\ndifferent) realizations of $X$, and define \n\\begin{equation}\n\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x,\\tilde{x},\\tilde{\\tilde{x}})\\equiv h_{00}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{10}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{x})+h_{01}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{x})+h_{11}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\tilde{x}}).\\label{eq:h_tilde-1}\n\\end{equation}\nUnder \\eqref{eq:my_IV2} and analogous to \\eqref{eq:sign_match},\nwe can show that if \n\\begin{align}\nsgn\\{\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)\\} & =sgn(-\\mu_{1})=sgn(-\\mu_{2})\\label{eq:sign_match_x}\n\\end{align}\nis positive (negative), then $sgn\\{\\mu_{1}+\\beta(x-x')\\}=sgn\\{\\mu_{2}+\\beta(x-x')\\}$\nis positive (negative). This is formally shown in Lemma \\ref{lem:asy_to_asy_star}(ii).\n\nAs before, suppose $h(\\boldsymbol{z},\\boldsymbol{z}',x)>0$, and thus,\n$\\mu_{1}>0$ and $\\mu_{2}>0$ by \\eqref{eq:sign_match}. Now, if $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)<0$,\nthen $\\mu_{1}+\\beta x<\\beta x'$ and $\\mu_{2}+\\beta x<\\beta x'$.\nTherefore, we can derive \n\\begin{align*}\nE[Y(1,1)|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]= & \\Pr[\\epsilon\\le\\mu_{1}+\\mu_{2}+\\beta x|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]\\\\\n\\le & \\Pr[\\epsilon\\le\\mu_{1}+\\beta x'|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x']\\\\\n= & \\Pr[Y=1|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x'],\n\\end{align*}\nwhere the second equality also uses \\eqref{eq:my_IV2} and \\eqref{eq:model_ex2}--\\eqref{eq:model_ex3}.\nSimilarly, we have $E[Y(1,1)|\\boldsymbol{D}=(0,1),\\boldsymbol{z},x]\\le\\Pr[Y=1|\\boldsymbol{D}=(0,1),\\boldsymbol{z},x']$,\nand consequently, the upper bound on $E[Y(1,1)\\vert x]$ becomes \n\\[\nU(x)\\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+\\Pr[Y=1,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},x']+\\Pr[\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]\\right\\} \n\\]\nby \\eqref{eq:Manski_expand-1}, and the lower bound is $L(x)=\\tilde{L}(x)$.\nNote that we can further take infimum over $x'$ such that $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)<0$.\n\nTo summarize our illustration, by using two values of $\\boldsymbol{Z}$\nthat satisfy \\eqref{eq:EQ_S2} and $h(\\boldsymbol{z},\\boldsymbol{z}',x)>0$\nand two values of $X$ that satisfy $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)<0$,\nour lower and upper bounds, $L(x)$ and $U(x)$, on $E[Y(1,1)\\vert x]$\nachieve \n\\begin{align*}\nL(x) & =\\tilde{L}(x)\\ge L_{Manski}(x),\\\\\nU(x) & \\ge\\tilde{U}(x)=U_{Manski}(x),\n\\end{align*}\nwhere the inequalities are strict if $\\sum_{\\boldsymbol{d}\\neq(1,1)}\\Pr[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z},x]>0$\nand $\\Pr[Y=0,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},x']>0$.\nWe discuss the sharpness of $L(x)$ and $U(x)$ in the next section.\nSimilarly, we can derive lower and upper bounds on other $E[Y(\\boldsymbol{d})\\vert x]$'s\nfor $\\boldsymbol{d}\\neq(1,1)$, and eventually construct bounds on\nany ATE. The gain from our approach is also exhibited in Figure \\ref{fig:sim1}\nin Section \\ref{sec:Monte-Carlo-Studies}, where we use the same data\ngenerating process as in this section and calculate different bounds\non the ATE, $E[Y(1,1)\\vert x]-E[Y(0,0)\\vert x]$.\n\n\\begin{remark}[Point Identification of the ATE]\\label{rem:Identification-under-Full}\nWhen there exist player-specific excluded instruments with large support,\nwe point identify the ATEs. To invoke an identification-at-infinity\nargument, the following assumptions are instead needed to hold: \n\\begin{align}\n & \\gamma_{1}\\text{ and }\\gamma_{2}\\text{ are nonzero},\\label{eq:infinity1}\\\\\n & Z_{1}|(X,Z_{2})\\text{ and }Z_{2}|(X,Z_{1})\\text{ has an everywhere positive Lebesgue density}.\\label{eq:infinity2}\n\\end{align}\nThese assumptions impose a player-specific exclusion restriction and\nlarge support. Under \\eqref{eq:infinity1}--\\eqref{eq:infinity2},\nwe can easily show that the ATE in \\eqref{eq:ATE} is point identified.\nIn this case, the structure we impose, especially on the outcome function\n(such as the threshold-crossing structure, or more generally Assumption\nM in Section \\ref{subsec:Assumptions} below) is not needed.\n\nThe identification strategy is to exploit the large variation of player\nspecific instruments based on \\eqref{eq:infinity1}--\\eqref{eq:infinity2},\nwhich simultaneously solves the multiple equilibria and the endogeneity\nproblems. For example, to identify $E[Y(1,1)|x]$, consider \n\\begin{align*}\n & E[Y|\\boldsymbol{D}=(1,1),\\boldsymbol{z},x]=E[Y(1,1)|\\boldsymbol{D}=(1,1),\\boldsymbol{z},x]\\\\\n & =E[Y(1,1)|\\delta_{2}+\\gamma_{1}z_{1}\\geq U_{1},\\delta_{1}+\\gamma_{2}z_{2}\\geq U_{2},x]\\rightarrow E[Y(1,1)|x],\n\\end{align*}\nwhere the second equation is by \\eqref{eq:my_IV} and $Y(1,1)=\\mu_{1}+\\mu_{2}+\\beta X$,\nand the convergence is by \\eqref{eq:infinity1}--\\eqref{eq:infinity2}\nwith $z_{1}\\rightarrow\\infty$ and $z_{2}\\rightarrow\\infty$. The\nidentification of $E[Y(0,0)|x]$, $E[Y(1,0)|x]$ and $E[Y(0,1)|x]$\ncan be achieved by similar reasoning. Note that $\\boldsymbol{D}=(1,0)$\nor $\\boldsymbol{D}=(0,1)$ can be predicted as an outcome of multiple\nequilibria. However, when either $(z_{1},z_{2})\\rightarrow(\\infty,-\\infty)$\nor $(z_{1},z_{2})\\rightarrow(-\\infty,\\infty)$ occurs, a unique equilibrium\nis guaranteed as a dominant strategy, i.e., $\\boldsymbol{D}=(1,0)$\nor $\\boldsymbol{D}=(0,1)$, respectively.\\end{remark}\n\n\\begin{remark}[Non-Monotonicity of Treatment Selection]\\label{rem:nonmonotone}In\nthe case of a single binary treatment, the standard selection equation\nexhibits monotonicity that facilitates various identification strategies\n(e.g., \\citet{imbens1994identification}, \\citet{heckman2005structural},\n\\citet{VY07} to name a few). Relatedly, \\citet{vytlacil2002independence}\nshows the equivalence between imposing the selection equation with\nthreshold-crossing structure and assuming the local ATE (LATE) monotonicity.\nThis equivalence (and thus, previous identification strategies) is\ninapplicable to our setting due to the simultaneity in the first stage\n\\eqref{eq:main_model2}. To formally state this, let $\\boldsymbol{D}(\\boldsymbol{z})$\nbe a potential treatment vector, had $\\boldsymbol{Z}=\\boldsymbol{z}$\nbeen realized. When cost $\\boldsymbol{Z}=(Z_{1},Z_{2})$ increases\nfrom $\\boldsymbol{z}$ to $\\boldsymbol{z}'$, it may be that some\nmarkets witness Delta entering and United going out of business (i.e.,\n$\\boldsymbol{D}(\\boldsymbol{z})=(0,1)$ and $\\boldsymbol{D}(\\boldsymbol{z}')=(1,0)$),\nwhile other markets witness the opposite (i.e., $\\boldsymbol{D}(\\boldsymbol{z})=(1,0)$\nand $\\boldsymbol{D}(\\boldsymbol{z}')=(0,1)$). The direction of monotonicity\nis reversed in the two groups of markets, and thus, $\\Pr[\\boldsymbol{D}(\\boldsymbol{z})\\ge\\boldsymbol{D}(\\boldsymbol{z}')]\\neq1$\nand $\\Pr[\\boldsymbol{D}(\\boldsymbol{z})\\le\\boldsymbol{D}(\\boldsymbol{z}')]\\neq1$\nwhere the inequality for vectors is pair-wise inequalities, which\nviolates the LATE monotonicity.\\footnote{The same argument applies with a scalar multi-valued treatment $\\tilde{D}\\in\\{1,2,3,4\\}$,\nwhich has a one-to-one map with $\\boldsymbol{D}\\in\\{(0,0),(0,1),(1,0),(1,1)\\}$.\nThen, some markets can experience $\\tilde{D}(\\boldsymbol{z})=2$ and\n$\\tilde{D}(\\boldsymbol{z}')=3$ while others experience $\\tilde{D}(\\boldsymbol{z})=3$\nand $\\tilde{D}(\\boldsymbol{z}')=2$, and thus, it is possible to have\n$\\Pr[\\tilde{D}(\\boldsymbol{z})\\ge\\tilde{D}(\\boldsymbol{z}')]\\neq1$\nand $\\Pr[\\tilde{D}(\\boldsymbol{z})\\le\\tilde{D}(\\boldsymbol{z}')]\\neq1$.} Despite this non-monotonic pattern, Theorem \\ref{thm:mono_pattern}\nbelow restores generalized monotonicity, i.e., monotonicity in terms of\nthe algebra of sets. This generalized monotonicity, combined with\nthe compensating strategic substitutability \\eqref{eq:EQ_S2}, allows\nus to use a strategy analogous to the single-treatment case for our\nbound analysis. This also suggests that we can introduce a generalized\nversion of the LATE parameter in the current framework, although we\ndo not pursue it in this study.\n\nRelated to our study, \\citet{lee2016identifying} introduce a framework\nfor treatment effects with general non-monotonicity of selection,\nand consider the simultaneous treatment selection as one of the examples.\nAlthough they engage in a similar discussion on non-monotonicity,\ntheir approach to gain tractability for identification is different\nfrom ours. When they allow the identity of players being observed\nas in our setting, they show that their treatment measurability condition\n(Assumption 2.1) introduced to restore monotonicity is satisfied,\nprovided they assume a threshold-crossing equilibrium selection mechanism.\nIn contrast, we avoid making assumptions on equilibrium selection,\nbut require compensating variation of instruments. In addition, for\nthis particular example, they assume the first-stage is known (i.e.,\npayoff functions are known), and focus on point identification of\nthe MTE with continuous instruments.\\end{remark}\n\n\\section{General Theory\\label{sec:General-Theory}}\n\n\\subsection{Setup\\label{subsec:Model}}\n\nLet $\\boldsymbol{D}\\equiv(D_{1},...,D_{S})\\in\\mathcal{D}\\subseteq\\{0,1\\}^{S}$\nbe an $S$-vector of observed binary treatments and $\\boldsymbol{d}\\equiv(d_{1},...,d_{S})$\nbe its realization, where $S$ is fixed. We assume that $\\boldsymbol{D}$\nis predicted as a pure strategy Nash equilibrium of a complete information\ngame with $S$ players who make entry decisions or individuals who\nchoose to receive treatments.\\footnote{While this study does not consider mixed strategy equilibria, it may\nbe possible to extend the setup to incorporate mixed strategies, following\nthe argument in \\citet{CT09}.} Let $Y$ be an observed post-game outcome that results from profile\n$\\boldsymbol{D}$ of endogenous treatments. It can be an outcome common\nto all players or an outcome specific to each player. Let $(X,Z_{1},...,Z_{S})$\nbe observed exogenous covariates. We consider a model of a semi-triangular\nsystem: \n\\begin{align}\nY & =\\theta(\\boldsymbol{D},X,\\epsilon_{\\boldsymbol{D}}),\\label{eq:main_model1}\\\\\nD_{s} & =1\\left[\\nu^{s}(\\boldsymbol{D}_{-s},Z_{s})\\geq U_{s}\\right],\\mbox{\\qquad}s\\in\\{1,...,S\\},\\label{eq:main_model2}\n\\end{align}\nwhere $s$ is indices for players or interchangeably for treatments,\nand $\\boldsymbol{D}_{-s}\\equiv(D_{1},...,D_{s-1},D_{s+1},...,D_{S})$.\nWithout loss of generality, we normalize the scalar $U_{s}$ to be\ndistributed as $Unif(0,1)$, and $\\nu^{s}:\\mathbb{R}^{S-1+d_{z_{s}}}\\rightarrow(0,1]$\nand $\\theta:\\mathbb{R}^{S+d_{x}+d_{\\epsilon}}\\rightarrow\\mathbb{R}$\nare unknown functions that are nonseparable in their arguments. We\nallow the unobservables $(\\epsilon_{\\boldsymbol{D}},U_{1},...,U_{S})$\nto be arbitrarily dependent on one another. Although the notation\nsuggests that the instruments $Z_{s}$'s are player\/treatment-specific,\nthey are not necessarily required to be so for the analyses in this\nstudy; see Appendix \\ref{subsec:Common_Z} for a discussion. The exogenous\nvariables $X$ are variables excluded from all the equations for $D_{s}$.\nThe existence of $X$ is not necessary but useful for the bound analysis\nof the ATE. There may be covariates $W$ common to all the equations\nfor $Y$ and $D_{s}$, which is suppressed for succinctness. Implied\nfrom the complete information game, player $s$'s decision $D_{s}$\ndepends on the decisions of all others $\\boldsymbol{D}_{-s}$ in $\\mathcal{D}_{-s}$,\nand thus, $\\boldsymbol{D}$ is determined by a simultaneous system.\nAs before, the model \\eqref{eq:main_model1}--\\eqref{eq:main_model2}\nis incomplete because of the simultaneity in the first stage, and\nthe conventional monotonicity in the sense of \\citet{imbens1994identification}\nis not exhibited in the selection process because of simultaneity.\nThe unit of observation, a market or geographical region, is indexed\nby $i$ and is suppressed in all the expressions.\n\nThe potential outcome of receiving treatments $\\boldsymbol{D}=\\boldsymbol{d}$\ncan be written as \n\\begin{align*}\nY(\\boldsymbol{d}) & =\\theta(\\boldsymbol{d},X,\\epsilon_{\\boldsymbol{d}}),\\mbox{\\qquad}\\boldsymbol{d}\\in\\mathcal{D},\n\\end{align*}\nand $\\epsilon_{\\boldsymbol{D}}=\\sum_{\\boldsymbol{d}\\in\\mathcal{D}}1[\\boldsymbol{D}=\\boldsymbol{d}]\\epsilon_{\\boldsymbol{d}}$.\nWe are interested in the ATE and related parameters. Using the average\nstructural function (ASF) $E[Y(\\boldsymbol{d})|x]$, the ATE can be\nwritten as \n\\begin{align}\nE[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}^{\\prime})|x] & =E[\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})-\\theta(\\boldsymbol{d}^{\\prime},x,\\epsilon_{\\boldsymbol{d}^{\\prime}})],\\label{eq:ATE}\n\\end{align}\nfor $\\boldsymbol{d},\\boldsymbol{d}'\\in\\mathcal{D}$. Another parameter\nof interest is the average treatment effects on the treated or the untreated:\n$E[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}^{\\prime})|D=\\boldsymbol{d}'',z,x]$\nfor $\\boldsymbol{d}''\\in\\{ \\boldsymbol{d},\\boldsymbol{d}' \\}$.\\footnote{Technically, $\\boldsymbol{d}''$ does not necessarily have to be equal\nto $\\boldsymbol{d}$ or $\\boldsymbol{d}'$, but can take another value.} One might also be interested\nin the sign of the ATE, which in this multi-treatment case is essentially\nestablishing an ordering among the ASF's.\n\nAs an example of the ATE, we may choose $\\boldsymbol{d}=(1,...,1)$\nand $\\boldsymbol{d}'=(0,...,0)$ to measure the cancelling-out effect\nor more general nonlinear effects. Another example would be choosing\n$\\boldsymbol{d}=(1,\\boldsymbol{d}_{-s})$ and $\\boldsymbol{d}'=(0,\\boldsymbol{d}_{-s})$\nfor given $\\boldsymbol{d}_{-s}$, where we use the notation $\\boldsymbol{d}=(d_{s},\\boldsymbol{d}_{-s})$\nby switching the order of the elements for convenience. Sometimes,\nwe instead want to focus on learning about complementarity between\ntwo treatments, while averaging over the remaining $S-2$ treatments.\nThis can be examined in a more general framework of defining the ASF\nand ATE by introducing a partial potential outcome; this is discussed\nin Appendix \\ref{subsec:Partial-ATE}.\n\nIn identifying these treatment parameters, suppose we attempt to recover\nthe effect of a single treatment $D_{s}$ in model \\eqref{eq:main_model1}--\\eqref{eq:main_model2}\n\\textit{conditional on} $\\boldsymbol{D}_{-s}=\\boldsymbol{d}_{-s}$,\nand then recover the effects of multiple treatments by transitively\nusing these effects of single treatments. This strategy is not valid\nsince $\\boldsymbol{D}_{-s}$ is a function of $D_{s}$ and also because\nof multiplicity. Therefore, the approaches in the literature with\nsingle-treatment, single-agent triangular models are not directly\napplicable and a new theory is necessary in this more general setting.\n\n\\subsection{Monotonicity in Equilibria\\label{subsec:Geometry}}\n\nAs an important step in the analyses in this study, we establish that\nthe equilibria of the treatment selection process in the first-stage\ngame present a monotonic pattern when the instruments move. Specifically,\nwe consider the regions in the space of the unobservables that predict\nequilibria and establish a monotonic pattern of these regions in terms\nof instruments. The analytical characterization of the equilibrium\nregions when there are more than two players ($S>2$) can generally\nbe complicated (\\citet[p. 1800]{CT09}); however, under a mild uniformity\nassumption (Assumption M1), our result is obtained under strategic\nsubstitutability. Let $\\mathcal{Z}_{s}$ be the support of $Z_{s}$.\nWe make the following assumptions on the first-stage nonparametric\npayoff function for each $s\\in\\{1,...,S\\}$.\n\n\\begin{asSS}For every $z_{s}\\in\\mbox{\\ensuremath{\\mathcal{Z}}}_{s}$,\n$\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})$ is strictly decreasing in each\nelement of $\\boldsymbol{d}_{-s}$.\\end{asSS}\n\n\\begin{asM1}For any given $z_{s},z_{s}'\\in\\mathcal{Z}_{s}$, either\n$\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})\\geq\\nu^{s}(\\boldsymbol{d}_{-s},z_{s}')$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$, or $\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})\\leq\\nu^{s}(\\boldsymbol{d}_{-s},z_{s}')$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$.\\end{asM1}\n\nAssumption SS asserts that the agents' treatment decisions are produced\nin a game with strategic substitutability. The strictness of the monotonicity\nis not important for our purpose but convenient in making statements\nabout the equilibrium regions. In the language of \\citet{CT09}, we\nallow for heterogeneity in the \\textit{fixed competitive effects}\n(i.e., how each of other entrants affects one's payoff), as well as\nheterogeneity in how each player is affected by other entrants, which\nis ensured by the nonseparability between $\\boldsymbol{d}_{-s}$ and\n$z_{s}$ in $\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})$; this heterogeneity\nis related to the \\textit{variable competitive effects}. Assumption\nM1 is required in this multi-agent setting, and the uniformity is\nacross $\\boldsymbol{d}_{-s}$. Note that this assumption is weaker\nthan a conventional monotonicity assumption that $\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})$\nis either non-decreasing or non-increasing in $z_{s}$ for all $\\boldsymbol{d}_{-s}$.\nAssumption M1 is justifiable, especially when $z_{s}$ is chosen to\nbe of the same kind for all players. For example, in an entry game,\nif $z_{s}$ is chosen to be each player's cost shifters, the payoffs\nwould decrease in their costs for any given opponents.\n\nAs the first main result of this study, we establish the geometric\nproperty of the equilibrium regions. For $j=0,...,S$, let $\\boldsymbol{R}_{j}(\\boldsymbol{z})\\subset\\mathcal{U}\\equiv(0,1]^{S}$\ndenote the region that predicts all equilibria with $j$ treatments\nselected or $j$ entrants, defined as a subset of the space of the\nentry unobservables $\\boldsymbol{U}\\equiv(U_{1},...,U_{S})$; see\nSection \\ref{subsec:notation} in the Appendix for a formal definition.\nThen, define the region of all equilibria with \\textit{at most} $j$\nentrants as \n\\begin{align*}\n\\boldsymbol{R}^{\\le j}(\\boldsymbol{z}) & \\equiv\\bigcup_{k=0}^{j}\\boldsymbol{R}_{k}(\\boldsymbol{z}).\n\\end{align*}\nAlthough this region is hard to express explicitly in general, it\nhas a simple feature that serves our purpose. For given $j$, choose\n$z_{s},z_{s}'\\in\\mathcal{Z}_{s}$ such that \n\\begin{align}\n\\Pr[\\boldsymbol{D}=(1,...,1)|\\boldsymbol{Z}=(z_{s},\\boldsymbol{z}_{-s})] & >\\Pr[\\boldsymbol{D}=(1,...,1)|\\boldsymbol{Z}=(z_{s}',\\boldsymbol{z}_{-s})]\\label{eq:ps_condi}\n\\end{align}\nfor all $s$. This condition is to merely fix $\\boldsymbol{z},\\boldsymbol{z}'$\nthat change the \\textit{joint propensity score}, and the direction\nof change is without loss of generality. Such $\\boldsymbol{z},\\boldsymbol{z}'$\nexist by the relevance of the instruments, which is assumed below.\nLet $\\mathcal{Z}$ be the support of $\\boldsymbol{Z}\\equiv(Z_{1},...,Z_{S})$.\n\n\\begin{theorem}\\label{thm:mono_pattern}Under Assumptions SS and\nM1 and for $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$ that satisfy\n\\eqref{eq:ps_condi}, we have \n\\begin{equation}\n\\boldsymbol{R}^{\\le j}(\\boldsymbol{z})\\subseteq\\boldsymbol{R}^{\\le j}(\\boldsymbol{z}')\\text{ }\\forall j.\\label{eq:R^j(z)_vs_R^j(z')}\n\\end{equation}\n\n\\end{theorem}\n\nTheorem \\ref{thm:mono_pattern} establishes a generalized version\nof monotonicity in the treatment selection process. This theorem plays\na crucial role in calculating the bounds on the treatment parameters\nand in showing the sharpness of the bounds. Relatedly, \\citet{berry1992estimation}\nderives the probability of the event that the number of entrants is\nless than a certain value, which can be written as $\\Pr[\\boldsymbol{U}\\in\\boldsymbol{R}^{\\le j}(\\boldsymbol{z})]$\nusing our notation. However, his result is not sufficient for our\nstudy and relies on stronger assumptions, such as restricting the\npayoff functions to only depend on the number of opponents.\n\n\\subsection{Main Assumptions\\label{subsec:Assumptions}}\n\nTo characterize the bounds on the treatment parameters, we make the\nfollowing assumptions. Unless otherwise noted, the assumptions hold\nfor each $s\\in\\{1,...,S\\}$.\n\n\\begin{asIN}\\label{as:IN}$(X,\\boldsymbol{Z})\\perp(\\epsilon_{\\boldsymbol{d}},\\boldsymbol{U})$\n$\\forall\\boldsymbol{d}\\in\\mathcal{D}$.\\end{asIN}\n\n\\begin{asE} The distribution of $(\\epsilon_{\\boldsymbol{d}},\\boldsymbol{U})$\nhas strictly positive density with respect to Lebesgue measure on\n$\\mathbb{R}^{S+1}$ $\\forall\\boldsymbol{d}\\in\\mathcal{D}$. \\end{asE}\n\n\\begin{asEX}For each $\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$, $\\nu^{s}(\\boldsymbol{d}_{-s},Z_{s})|X$\nis nondegenerate.\\end{asEX}\n\nAssumptions IN, EX and all the following analyses can be understood\nas conditional on $W$, the common covariates in $X$ and $\\boldsymbol{Z}=(Z_{1},...,Z_{S})$.\nAssumption EX is related to the exclusion restriction and the relevance\ncondition of the instruments $Z_{s}$.\n\nWe now impose a shape restriction on the outcome function $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})$\nvia restrictions on \n\\[\n\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})\\equiv E[\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})|\\boldsymbol{U}=\\boldsymbol{u}]\n\\]\na.e. $\\boldsymbol{u}$. This restriction on the conditional mean is\nweaker than the one directly imposed on $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})$.\nLet $\\mathcal{X}$ be the support of $X$. Recall that we use the\nnotation $\\boldsymbol{d}=(d_{s},\\boldsymbol{d}_{-s})$ by switching\nthe order of the elements for convenience.\n\n\\begin{asM}For every $x\\in\\mathcal{X}$, either $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\geq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$\n$\\forall s$ or $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\leq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$\n$\\forall s$. Also, $Y\\in[\\underline{Y},\\overline{Y}]$.\\end{asM}\n\nAssumption M holds in, but is not restricted to, the leading case\nof binary $Y$ with a threshold crossing model that satisfies uniformity.\n\n\\begin{asM2}(i) $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})=1[\\mu(\\boldsymbol{d},x)\\geq\\epsilon_{\\boldsymbol{d}}]$\nwhere $\\epsilon_{\\boldsymbol{d}}$ is scalar and $F_{\\epsilon_{\\boldsymbol{d}}|\\boldsymbol{U}}=F_{\\epsilon_{\\boldsymbol{d}'}|\\boldsymbol{U}}$\nfor any $\\boldsymbol{d},\\boldsymbol{d}'\\in\\mathcal{D}$; (ii) for\nevery $x\\in\\mathcal{X}$, either $\\mu(1,\\boldsymbol{d}_{-s},x)\\geq\\mu(0,\\boldsymbol{d}_{-s},x)$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$ $\\forall s$ or $\\mu(1,\\boldsymbol{d}_{-s},x)\\le\\mu(0,\\boldsymbol{d}_{-s},x)$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$ $\\forall s$.\\end{asM2}\n\nAssumption M$^{*}$ implies Assumption M. The second statement in\nAssumption M is satisfied with binary $Y$.\\footnote{Another example would be when $Y\\in[0,1]$, as in Example \\ref{example3}.}\nThe first statement in Assumption M can be stated in two parts, corresponding\nto (i) and (ii) of Assumption M$^{*}$: (a) for every $x$ and $\\boldsymbol{d}_{-s}$,\neither $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\geq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$, or $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\leq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$; (b) for every $x$, each inequality statement\nin (a) holds for all $\\boldsymbol{d}_{-s}$. For an outcome function\nwith a scalar index, $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})=\\tilde{\\theta}(\\mu(\\boldsymbol{d},x),\\epsilon_{\\boldsymbol{d}})$,\npart (a) is implied by $\\epsilon_{\\boldsymbol{d}}=\\epsilon_{\\boldsymbol{d}'}=\\epsilon$\n(or more generally, $F_{\\epsilon_{\\boldsymbol{d}}|\\boldsymbol{U}}=F_{\\epsilon_{\\boldsymbol{d}'}|\\boldsymbol{U}}$)\nfor any $\\boldsymbol{d},\\boldsymbol{d}'\\in\\mathcal{D}$ and $E[\\tilde{\\theta}(t,\\epsilon_{\\boldsymbol{d}})|\\boldsymbol{U}=\\boldsymbol{u}]$\nbeing strictly increasing (decreasing) in $t$ a.e. $\\boldsymbol{u}$.\\footnote{A single-treatment version of the latter assumption appears in \\citet{VY07}\n(Assumption A-4), which is weaker than assuming $\\tilde{\\theta}(t,\\epsilon)$\nis strictly increasing (decreasing) a.e. $\\epsilon$; see \\citet{VY07}\nfor related discussions.} Functions that satisfy the latter assumption include strictly monotonic\nfunctions, such as transformation models $\\tilde{\\theta}(t,\\epsilon)=r(t+\\epsilon)$\nwith $r(\\cdot)$ being possibly unknown strictly increasing, or their\nspecial case $\\tilde{\\theta}(t,\\epsilon)=t+\\epsilon$, allowing continuous\ndependent variables; and functions that are not strictly monotonic,\nsuch as models for limited dependent variables, $\\tilde{\\theta}(t,\\epsilon)=1[t\\ge\\epsilon]$\nor $\\tilde{\\theta}(t,\\epsilon)=1[t\\ge\\epsilon](t-\\epsilon)$. However,\nthere can be functions that violate the latter assumption but satisfy\npart (a). For example, consider a threshold crossing model with a\nrandom coefficient: $\\theta(\\boldsymbol{d},x,\\epsilon)=1[\\phi(\\epsilon)\\boldsymbol{d}\\beta^{\\top}\\geq x\\gamma^{\\top}]$,\nwhere $\\phi(\\epsilon)$ is nondegenerate. When $\\beta_{s}\\geq0$,\nthen $E[\\theta(1,\\boldsymbol{d}_{-s},x,\\epsilon)-\\theta(0,\\boldsymbol{d}_{-s},x,\\epsilon)|\\boldsymbol{U}=\\boldsymbol{u}]=\\Pr\\left[\\frac{x\\gamma^{\\top}}{\\beta_{s}+\\boldsymbol{d}_{-s}\\beta_{-s}^{\\top}}\\leq\\phi(\\epsilon)\\leq\\frac{x\\gamma^{\\top}}{\\boldsymbol{d}_{-s}\\beta_{-s}^{\\top}}|\\boldsymbol{U}=\\boldsymbol{u}\\right]$,\nand thus, nonnegative a.e. $\\boldsymbol{u}$, and vice versa. Part\n(a) also does not impose any monotonicity of $\\theta$ in $\\epsilon_{\\boldsymbol{d}}$\n(e.g., $\\epsilon_{\\boldsymbol{d}}$ can be a vector).\n\nPart (b) of Assumption M imposes uniformity, as we deal with more\nthan one treatment. Uniformity is required across different values\nof $\\boldsymbol{d}_{-s}$ and $s$. For instance, in the empirical\napplication of this study, this assumption seems reasonable, since\nan airline's entry is likely to increase the expected pollution regardless\nof the identity or the number of existing airlines. On the other hand,\nin Example \\ref{example3} in the Appendix regarding media and political\nbehavior, this assumption may rule out the ``over-exposure'' effect\n(i.e., too much media exposure diminishes the incumbent's chance\nof being re-elected). In any case, knowledge on the direction of the\nmonotonicity is not necessary in this assumption, unlike \\citet{manski1997monotone}\nor \\citet{Man13}, where the semi-monotone treatment response is assumed\nfor possible multiple treatments.\n\nLastly, we require that there exists variation in $\\boldsymbol{Z}$\nthat offsets the effect of strategic substitutability. Similar as\nbefore, using the notation $\\boldsymbol{d}_{-s}=(d_{s'},\\boldsymbol{d}_{-(s,s')})$\nwhere $\\boldsymbol{d}_{-(s,s')}$ is $\\boldsymbol{d}$ without $s$-th\nand $s'$-th elements, note that Assumption SS can be restated as\n$\\nu^{s}(0,\\boldsymbol{d}_{-(s,s')},z_{s})>\\nu^{s}(1,\\boldsymbol{d}_{-(s,s')},z_{s})$\nfor every $z_{s}$. Given this, we assume the following.\n\n\\begin{asEQ}There exist $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$,\nsuch that $\\nu^{s}(0,\\boldsymbol{d}_{-(s,s')},z_{s}')\\le\\nu^{s}(1,\\boldsymbol{d}_{-(s,s')},z_{s})$\n$\\forall\\boldsymbol{d}_{-(s,s')}$ $\\forall s,s'$.\\end{asEQ}\n\nFor example, in an entry game with $Z_{s}$ being cost shifters, Assumption\nEQ may hold with $z_{s}'>z_{s}$ $\\forall s$. In this example, players\nmay become less profitable with an increase in cost from government\nregulation. In particular, players' decreased profits cannot be overturned\nby the market being less competitive, as one player is absent due\nto unprofitability. Recall that Assumption EQ is illustrated in Figure\n\\ref{fig:As_EQ-1} with $\\nu^{s}(0,z_{s}')=\\gamma_{s}z_{s}'<\\nu^{s}(1,z_{s})=\\delta_{-s}+\\gamma_{s}z_{s}$\nfor $s=1,2$. Assumption EQ is key for our analysis. To see this,\nlet $R_{j}^{M}(\\cdot)$ denote the region that predicts multiple equilibria\nwith $j$ treatments selected or $j$ entrants. In the proof of a\nlemma that follows, we show that Assumption EQ holds if and only if\n$R_{j}^{M}(\\boldsymbol{z})\\cap R_{j}^{M}(\\boldsymbol{z}')=\\emptyset$.\nThat is, we can at least ensure that there is no market where firms'\ndecisions change from one realization of multiple equilibria to another\nrealization of multiple equilibria with the same number of entrants.\nTo the extent of our analysis, this liberates us from concerns about\nthe regions of multiple equilibria and about a possible change in\nequilibrium selection when changing $\\boldsymbol{Z}$.\\footnote{In Section \\ref{subsec:Group}, we discuss an assumption, partial\nconditional symmetry, which can be imposed alternative to Assumption\nEQ.} Assumption EQ has a simple testable sufficient condition, provided\nthat the unobservables in the payoffs are mutually independent.\n\n\\begin{asEQ2}There exist $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$,\nsuch that \n\\begin{align}\n\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j-2}|\\boldsymbol{z}'] & >2-\\sqrt{2}.\\label{eq:EQ_suff}\n\\end{align}\nfor all $\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}$, $\\boldsymbol{d}^{j-2}\\in\\mathcal{D}^{j-2}$\nand $2\\le j\\le S$.\n\n\\end{asEQ2}\n\nWhen $S=2$, the condition is stated as $\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}=(0,0)|\\boldsymbol{z}']>2-\\sqrt{2}$.\nAs is detailed in the proof, this essentially restricts the sum of\nradii of two circular isoquant curves to be less than the length of\nthe diagonal of $\\mathcal{U}$: $(1-\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z}])+(1-\\Pr[\\boldsymbol{D}=(0,0)|\\boldsymbol{z}'])<\\sqrt{2}$.\nThis ensures the required variation in Assumption EQ.\n\n\\begin{lemma}\\label{lem:asy_to_asy_star}Under Assumptions SS, M1,\nand $U_{s}\\perp U_{t}$ for all $s\\neq t$, Assumption EQ$^{*}$ implies\nAssumption EQ.\n\n\\end{lemma}\n\nThe mutual independence of $U_{s}$'s (conditional on $W$) is useful\nin inferring the relationship between players' interaction and instruments\nfrom the observed choices of players. The intuition for the sufficiency\nof Assumption EQ$^{*}$ is as follows. As long as there is no dependence\nin unobserved types, \\eqref{eq:EQ_suff} dictates that the variation\nof $\\boldsymbol{Z}$ is large enough to offset strategic substitutability,\nbecause otherwise, the payoffs of players cannot move in the same\ndirection, and thus, will not result in the same decisions. The requirement\nof $\\boldsymbol{Z}$ variation in \\eqref{eq:EQ_suff} is significantly\nweaker than the large support assumption invoked for an identification\nat infinity argument to overcome the problem of multiple equilibria.\n\n\\subsection{Partial Identification of the ATE\\label{subsec:Partial-Identification}}\n\nUnder the above assumptions, we now present a generalized version\nof the sign matching results \\eqref{eq:sign_match} and \\eqref{eq:sign_match_x}\nin Section \\ref{sec:stylized_ex}. We need to introduce additional\nnotation. Let $\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}$ denote an equilibrium\nprofile with $j$ treatments selected or $j$ entrants, i.e., a vector\nof $j$ ones and $S-j$ zeros, where $\\mathcal{D}^{j}$ is a set of\nall equilibrium profiles with $j$ treatments selected. For realizations\n$x$ of $X$ and $\\boldsymbol{z},\\boldsymbol{z}'$ of $\\boldsymbol{Z}$,\ndefine \n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv E[Y|\\boldsymbol{z},x]-E[Y|\\boldsymbol{z}',x],\\label{eq:h(zzx)}\\\\\nh_{\\boldsymbol{d}^{j}}(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]\\nonumber \\\\\n & -E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z}',x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}'].\\label{eq:hj}\n\\end{align}\nSince $\\sum_{j=0}^{S}\\sum_{\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\cdot]=1$,\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)=\\sum_{j=0}^{S}\\sum_{\\boldsymbol{d}^{j}}h_{\\boldsymbol{d}^{j}}(\\boldsymbol{z},\\boldsymbol{z}',x)$.\nLet $\\tilde{\\boldsymbol{x}}=(x_{0},...,x_{S})$ be an $(S+1)$-dimensional\narray of (possibly different) realizations of $X$, i.e., each $x_{j}$\nfor $j=0,...,S$ is a realization of $X$, and define \n\\[\n\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})\\equiv\\sum_{j=0}^{S}\\sum_{\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}}h_{\\boldsymbol{d}^{j}}(\\boldsymbol{z},\\boldsymbol{z}',x_{j}).\n\\]\nFor $1\\le k\\le j$, define a \\textit{reduction} of $\\boldsymbol{d}^{j}=(d_{1}^{j},...,d_{S}^{j})$\nas $\\boldsymbol{d}^{j-k}=(d_{1}^{j-k},...,d_{S}^{j-k})$, such that\n$d_{s}^{j-k}\\le d_{s}^{j}$ $\\forall s$. Symmetrically, for $1\\le k\\le S-j$,\ndefine an \\textit{extension} of $\\boldsymbol{d}^{j}$ as $\\boldsymbol{d}^{j+k}=(d_{1}^{j+k},...,d_{S}^{j+k})$,\nsuch that $d_{s}^{j+k}\\ge d_{s}^{j}$ $\\forall s$. For example, given\n$\\boldsymbol{d}^{2}=(1,1,0)$, a reduction $\\boldsymbol{d}^{1}$ is\neither $(1,0,0)$ or $(0,1,0)$ but not $(0,0,1)$, a reduction $\\boldsymbol{d}^{0}$\nis $(0,0,0)$, and an extension $\\boldsymbol{d}^{3}$ is $(1,1,1)$.\nLet $\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$ and $\\mathcal{D}^{>}(\\boldsymbol{d}^{j})$\nbe the set of all reductions and extensions of $\\boldsymbol{d}^{j}$,\nrespectively, and let $\\mathcal{D}^{\\le}(\\boldsymbol{d}^{j})\\equiv\\mathcal{D}^{<}(\\boldsymbol{d}^{j})\\cup\\{\\boldsymbol{d}^{j}\\}$\nand $\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})\\equiv\\mathcal{D}^{>}(\\boldsymbol{d}^{j})\\cup\\{\\boldsymbol{d}^{j}\\}$.\nRecall $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})\\equiv E[\\theta(\\boldsymbol{d},x,\\epsilon)|\\boldsymbol{U}=\\boldsymbol{u}]$.\nNow, we state the main lemma of this section.\n\n\\begin{lemma}\\label{lem:sign_match_gen}In model \\eqref{eq:main_model1}--\\eqref{eq:main_model2},\nsuppose Assumptions SS, M1, IN, E, EX, and M hold, and $h(\\boldsymbol{z},\\boldsymbol{z}',x)$\nand $h(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})$ are\nwell-defined. For $\\boldsymbol{z},\\boldsymbol{z}'$ such that \\eqref{eq:ps_condi}\nand Assumption EQ hold, and for $j=1,...,S$, it satisfies that\\\\\n (i) $sgn\\{h(\\boldsymbol{z},\\boldsymbol{z}',x)\\}=sgn\\left\\{ \\vartheta(\\boldsymbol{d}^{j},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}^{j-1},x;\\boldsymbol{u})\\right\\} $\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}^{j-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$;\\\\\n (ii) for $\\iota\\in\\{-1,0,1\\}$, if $sgn\\{\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})\\}=sgn\\{-\\vartheta(\\boldsymbol{d}^{k},x_{k};\\boldsymbol{u})+\\vartheta(\\boldsymbol{d}^{k-1},x_{k-1};\\boldsymbol{u})\\}=\\iota$\n$\\forall\\boldsymbol{d}^{k-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{k})$\n$\\forall k\\neq j$ ($k\\ge1$), then $sgn\\{\\vartheta(\\boldsymbol{d}^{j},x_{j};\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}^{j-1},x_{j-1};\\boldsymbol{u})\\}=\\iota$\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}^{j-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$.\\end{lemma}\n\nParts (i) and (ii) parallel \\eqref{eq:sign_match} and \\eqref{eq:sign_match_x},\nrespectively. Using Lemma \\ref{lem:sign_match_gen}, we can learn\nabout the ATE. First, note that the sign of the ATE is identified\nby Lemma \\ref{lem:sign_match_gen}(i), since $E[Y(\\boldsymbol{d})|x]=E[\\vartheta(\\boldsymbol{d},x;\\boldsymbol{U})]$.\nNext, we establish the bounds on $E[Y(\\boldsymbol{d}^{j})|x]$ for\ngiven $\\boldsymbol{d}^{j}$ for some $j=0,...,S$.\n\nWe first present the bounds using the variation in $\\boldsymbol{Z}$\nonly, i.e., by using Lemma \\ref{lem:sign_match_gen}(i). To this end,\nwe fix $X=x$ and suppress it in all relevant expressions. To gain\nefficiency we define the integrated version of $h$ as \n\\begin{align}\nH(x) & \\equiv E\\left[h(\\boldsymbol{Z},\\boldsymbol{Z}',x)\\left|(\\boldsymbol{Z},\\boldsymbol{Z}')\\in\\mathcal{Z}_{EQ,j}\\text{ }\\forall j=0,...,S-1\\right.\\right],\\label{eq:H}\n\\end{align}\nwhere $\\mathcal{Z}_{EQ,j}$ is the set of $(\\boldsymbol{z},\\boldsymbol{z}')$\nthat satisfy \\eqref{eq:ps_condi} and Assumption EQ given $j$, and\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)=0$ whenever it is not well-defined.\nWe focus on the case $H(x)>0$; $H(x)<0$ is symmetric and $H(x)=0$\nis straightforward. Using Lemma \\ref{lem:sign_match_gen}(i), one\ncan readily show that $L_{\\boldsymbol{d}^{j}}(x)\\le E[Y(\\boldsymbol{d}^{j})|x]\\le U_{\\boldsymbol{d}^{j}}(x)$\nwith \n\\begin{align}\nU_{\\boldsymbol{d}^{j}}(x) & \\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\boldsymbol{z},x]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}\\backslash\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\boldsymbol{z},x]\\Biggr\\},\\label{eq:U_dj}\\\\\nL_{\\boldsymbol{d}^{j}}(x) & \\equiv\\sup_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\le}(\\boldsymbol{d}^{j})|\\boldsymbol{z},x]\\Biggr\\}.\\label{eq:L_dj}\n\\end{align}\nWe can simplify these bounds and show that they are sharp under the\nfollowing assumption.\n\n\\begin{asC}(i) $\\mu_{\\boldsymbol{d}}(\\cdot)$ and $\\nu_{\\boldsymbol{d}_{-s}}(\\cdot)$\nare continuous; (ii) $\\mathcal{Z}$ is compact.\\end{asC}\n\nUnder Assumption C, for given $\\boldsymbol{d}^{j}$, there exist vectors\n$\\bar{\\boldsymbol{z}}\\equiv(\\bar{z}_{1},...,\\bar{z}_{S})$ and $\\underline{\\boldsymbol{z}}\\equiv(\\underline{z}_{1},...,\\underline{z}_{S})$\nthat satisfy \n\\begin{equation}\n\\begin{array}{c}\n\\bar{\\boldsymbol{z}}=\\arg\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\max_{\\boldsymbol{d}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}],\\\\\n\\underline{\\boldsymbol{z}}=\\arg\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\min_{\\boldsymbol{d}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}].\n\\end{array}\\label{eq:max_min_pM}\n\\end{equation}\nThe following is the first main result of this study, which establishes\nthe sharp bounds on $E[Y(\\boldsymbol{d}^{j})|x]$, where $X=x$ is\nfixed in the model.\n\n\\begin{theorem}\\label{thm:sharp}Given model \\eqref{eq:main_model1}--\\eqref{eq:main_model2}\nwith fixed $X=x$, suppose Assumptions SS, M1, IN, E, EX, M$^{*}$,\nEQ and C hold. In addition, suppose $H(x)$ is well-defined and $H(x)\\ge0$.\nThen, the bounds $U_{\\boldsymbol{d}^{j}}$ and $L_{\\boldsymbol{d}^{j}}$\nin \\eqref{eq:U_dj} and \\eqref{eq:L_dj} simplify to \n\\begin{align*}\nU_{\\boldsymbol{d}^{j}}(x) & =\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\bar{\\boldsymbol{z}},x]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}\\backslash\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\bar{\\boldsymbol{z}},x],\\\\\nL_{\\boldsymbol{d}^{j}}(x) & =\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\le}(\\boldsymbol{d}^{j})|\\underline{\\boldsymbol{z}},x],\n\\end{align*}\nand these bounds are sharp.\n\n\\end{theorem}\n\nWith binary $Y$ (Assumption M$^{*}$), sharp bounds on the mean treatment\nparameters can be obtained, which is reminiscent of the findings of\nstudies that consider single-treatment models. However, our analysis\nis substantially different from earlier studies. In a single treatment\nmodel, \\citet{SV11} use the propensity score as a scalar conditioning\nvariable, which summarizes all the exogenous variation in the selection\nprocess and is convenient for simplifying the bounds and proving sharpness.\nHowever, in the context of our paper this approach is invalid, since\n$\\Pr[D_{s}=1|Z_{s}=z_{s},\\boldsymbol{D}_{-s}=\\boldsymbol{d}_{-s}]$\ncannot be written in terms of a propensity score of player $s$ as\n$\\boldsymbol{D}_{-s}$ is endogenous. We instead use the vector $\\boldsymbol{Z}$\nas conditioning variables and establish partial ordering for the relevant\nconditional probabilities (that define the lower and upper bounds)\nwith respect to the joint propensity score $\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{Z}=\\boldsymbol{z}]$\n$\\forall\\boldsymbol{d}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})$.\n\nWhen the variation of $X$ is additionally exploited in the model,\nthe bounds will be narrower than the bounds in Theorem \\ref{thm:sharp}.\nWe now proceed with this case, utilizing Lemma \\ref{lem:sign_match_gen}\n(i) and (ii). First, analogous to \\eqref{eq:H}, we define the integrated\nversion of $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})$\nas \n\\begin{align*}\n\\tilde{H}(\\tilde{\\boldsymbol{x}}) & \\equiv E\\left[\\tilde{h}(\\boldsymbol{Z},\\boldsymbol{Z}',\\tilde{\\boldsymbol{x}})\\left|(\\boldsymbol{Z},\\boldsymbol{Z}')\\in\\mathcal{Z}_{EQ,j}\\text{ }\\forall j=0,...,S-1\\right.\\right],\n\\end{align*}\nwhere $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})=0$\nwhenever it is not well-defined. Then, we define the following sets\nof two consecutive elements $(x_{j},x_{j-1})$ of $\\boldsymbol{x}$\nthat satisfy the conditions in Lemma \\ref{lem:sign_match_gen}: for\n$j=1,...,S$, define $\\mathcal{X}_{j,j-1}^{0}(\\iota)\\equiv\\{(x_{j},x_{j-1}):sgn\\{\\tilde{H}(\\tilde{\\boldsymbol{x}})\\}=\\iota,x_{0}=\\cdots=x_{S}\\}$\nand for $t\\ge1$, \n\\begin{align*}\n\\mathcal{X}_{j,j-1}^{t}(\\iota) & \\equiv\\{(x_{j},x_{j-1}):sgn\\{\\tilde{H}(\\tilde{\\boldsymbol{x}})\\}=\\iota,(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}^{t-1}(-\\iota)\\mbox{ \\ensuremath{\\forall}}k\\neq j\\}\\cup\\mathcal{X}_{j,j-1}^{t-1}(\\iota),\n\\end{align*}\nwhere the sets are understood to be empty whenever $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})$\nis not well-defined for any $p_{M^{\\le j}}(\\boldsymbol{z})}(\\boldsymbol{d}^{j})$,\n\\begin{align}\n\\mathcal{X}_{\\boldsymbol{d}^{j}}^{L}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(-1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j'+1\\leq k\\leq j,x_{j}=x\\right\\} \\nonumber \\\\\n & \\cup\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j+1\\leq k\\leq j',x_{j}=x\\right\\} ,\\label{eq:script_X_L}\\\\\n\\mathcal{X}_{\\boldsymbol{d}^{j}}^{U}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j'+1\\leq k\\leq j,x_{j}=x\\right\\} \\nonumber \\\\\n & \\cup\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(-1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j+1\\leq k\\leq j',x_{j}=x\\right\\} .\\label{eq:script_X_U}\n\\end{align}\nThe following theorem summarizes our results:\n\n\\begin{theorem}\\label{thm:main} In model \\eqref{eq:main_model1}--\\eqref{eq:main_model2},\nsuppose the assumptions of Lemma \\ref{lem:sign_match_gen} hold. Then\nthe sign of the ATE is identified, and the upper and lower bounds\non the ASF and ATE with $\\boldsymbol{d},\\tilde{\\boldsymbol{d}}\\in\\mathcal{D}$\nare \n\\begin{align*}\nL_{\\boldsymbol{d}}(x) & \\leq E[Y(\\boldsymbol{d})|x]\\leq U_{\\boldsymbol{d}}(x)\n\\end{align*}\nand $L_{\\boldsymbol{d}}(x)-U_{\\tilde{\\boldsymbol{d}}}(x)\\leq E[Y(\\boldsymbol{d})-Y(\\tilde{\\boldsymbol{d}})|x]\\leq U_{\\boldsymbol{d}}(x)-L_{\\tilde{\\boldsymbol{d}}}(x)$,\nwhere for any given $\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}\\subset\\mathcal{D}$\nfor some $j$, \n\\begin{align*}\nU_{\\boldsymbol{d}^{j}}(x) & \\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{ E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}^{j}\\backslash\\{\\boldsymbol{d}^{j}\\}|\\boldsymbol{z}]\\overline{Y}\\\\\n & +\\sum_{\\boldsymbol{d}^{\\prime}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})\\cup\\mathcal{D}^{>}(\\boldsymbol{d}^{j})}\\inf_{x'\\in\\mathcal{X}_{\\boldsymbol{d}^{j}}^{U}(x;\\boldsymbol{d}')}E[Y|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x']\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}]\\Biggr\\},\\\\\nL_{\\boldsymbol{d}^{j}}(x) & \\equiv\\sup_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{ E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}^{j}\\backslash\\{\\boldsymbol{d}^{j}\\}|\\boldsymbol{z}]\\underline{Y}\\\\\n & +\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})\\cup\\mathcal{D}^{>}(\\boldsymbol{d}^{j})}\\sup_{x'\\in\\mathcal{X}_{\\boldsymbol{d}^{j}}^{L}(x;\\boldsymbol{d}')}E[Y|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x']\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}]\\Biggr\\}.\n\\end{align*}\n\\end{theorem}\n\nSee Sections \\ref{sec:Monte-Carlo-Studies} and \\ref{sec:Empirical-Application}\nfor concrete examples of the expression of $U_{\\boldsymbol{d}^{j}}(x)$\nand $L_{\\boldsymbol{d}^{j}}(x)$. The terms $\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{'}|\\boldsymbol{z}]\\overline{Y}$\nand $\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{'}|\\boldsymbol{z}]\\underline{Y}$\nappear in the expression of the bounds because Lemma \\ref{lem:sign_match_gen}\ncannot establish an order between $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})$'s\nfor $\\boldsymbol{d}\\in\\mathcal{D}^{j}$, which is related to the complication\ndue to multiple equilibria, which occurs for $\\boldsymbol{d}\\in\\mathcal{D}^{j}$.\nWhen the variation in $\\boldsymbol{Z}$ is only used in deriving the\nbounds, $\\mathcal{X}_{k,k-1}(\\iota)$ should simply reduce to $\\mathcal{X}_{k,k-1}^{0}(\\iota)$\nin the definition of $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{L}(x;\\boldsymbol{d}')$\nand $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{U}(x;\\boldsymbol{d}')$. When\n$Y$ is binary with no $X$, such bounds are equivalent to \\eqref{eq:U_dj}\nand \\eqref{eq:L_dj}. The variation in $X$ given $\\boldsymbol{Z}$\nyields substantially narrower bounds than the sharp bounds established\nin Theorem \\ref{thm:sharp} under Assumption C. However, the resulting\nbounds are not automatically implied to be sharp from Theorem \\ref{thm:sharp},\nsince they are based on a different DGP and the additional exclusion\nrestriction.\n\n\\begin{remark}\\label{rem:mourifie}Maintaining that $Y$ is binary,\nsharp bounds on the ATE with variation in $X$ can be derived assuming\nthat the signs of $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}',x';\\boldsymbol{u})$\nare identified for $\\boldsymbol{d}\\in\\mathcal{D}$ and $\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d})$\nand $x,x'\\in\\mathcal{X}$ via Lemma \\ref{lem:sign_match_gen}. To\nsee this, define \n\\begin{align*}\n\\mathcal{\\tilde{X}}_{\\boldsymbol{d}}^{U}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x':\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}',x';\\boldsymbol{u})\\leq0\\text{ a.e. }\\boldsymbol{u}\\right\\} ,\\\\\n\\mathcal{\\tilde{X}}_{\\boldsymbol{d}}^{L}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x':\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}',x';\\boldsymbol{u})\\geq0\\text{ a.e. }\\boldsymbol{u}\\right\\} ,\n\\end{align*}\nwhich are identified by assumption. Then, by replacing $\\mathcal{X}_{\\boldsymbol{d}}^{i}(x;\\boldsymbol{d}')$\nwith $\\mathcal{\\tilde{X}}_{\\boldsymbol{d}}^{i}(x;\\boldsymbol{d}')$\n(for $i\\in\\{U,L\\}$) in Theorem \\ref{thm:main}, we may be able to\nshow that the resulting bounds are sharp. Since Lemma \\ref{lem:sign_match_gen}\nimplies that $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')\\subset\\mathcal{\\tilde{X}}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')$\nbut not necessarily $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')\\supset\\mathcal{\\tilde{X}}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')$,\nthese modified bounds and the original bounds in Theorem \\ref{thm:main}\ndo not coincide. This contrasts with the result of \\citet{SV11} for\na single-treatment model, and the complication lies in the fact that\nwe deal with an incomplete model with a vector treatment. When there\nis no $X$, Lemma \\ref{lem:sign_match_gen}(i) establishes equivalence\nbetween the two signs, and thus, $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')=\\mathcal{\\tilde{X}}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')$\nfor $i\\in\\{U,L\\}$, which results in Theorem \\ref{thm:sharp}. Relatedly,\nwe can also exploit variation in $W$, namely, variables that are\ncommon to both $X$ and $\\boldsymbol{Z}$ (with or without exploiting\nexcluded variation of $X$). This is related to the analysis of \\citet{Chi10}\nand \\citet{mourifie2015sharp} in a single-treatment setting. One\ncaveat of this approach is that, similar to these papers, we need\nto additionally assume that $W\\perp(\\epsilon,\\boldsymbol{U})$.\\end{remark}\n\n\\begin{remark}When $X$ does not have enough variation, we can calculate\nthe bounds on the ATE. To see this, suppose we do not use the variation\nin $X$ and suppose $H(x)\\ge0$. Then $\\vartheta(\\boldsymbol{d}^{j},x;\\boldsymbol{u})\\ge\\vartheta(\\boldsymbol{d}^{j-1},x;\\boldsymbol{u})$\n$\\forall\\boldsymbol{d}^{j-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$\n$\\forall j=1,...,S$ by Lemma \\ref{lem:sign_match_gen}(i) and by\ntransitivity, $\\vartheta(\\boldsymbol{d}^{'},x;\\boldsymbol{u})\\ge\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})$\nwith $\\boldsymbol{d}'$ being an extension of $\\boldsymbol{d}$. Therefore,\nwe have \n\\begin{align}\nE[Y(\\boldsymbol{d})|x] & \\le E[Y|\\boldsymbol{D}=\\boldsymbol{d},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}]+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{>}(\\boldsymbol{d})}E[Y|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}]\\nonumber \\\\\n & +\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}\\backslash\\mathcal{D}^{\\ge}(\\boldsymbol{d})}E[Y(\\boldsymbol{d}^{j})|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}].\\label{eq:interim_bd}\n\\end{align}\nWithout using variation in $X$, we can bound the last term in \\eqref{eq:interim_bd}\nby $Y\\in[\\underline{Y},\\overline{Y}]$. This is done above with $\\theta(\\boldsymbol{d},x,\\epsilon)=1[\\mu(\\boldsymbol{d},x)\\geq\\epsilon_{\\boldsymbol{d}}]$\nand $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})=F_{\\epsilon|\\boldsymbol{U}}(\\mu(\\boldsymbol{d},x)|\\boldsymbol{u})$.\n\n\\end{remark}\n\n\\section{Numerical Study\\label{sec:Monte-Carlo-Studies}}\n\nTo illustrate the main results of this study in a simulation exercise,\nwe calculate the bounds on the ATE using the following data generating\nprocess: \n\\begin{align*}\nY_{\\boldsymbol{d}} & =1[\\tilde{\\mu}_{\\boldsymbol{d}}+\\beta X\\ge\\epsilon],\\\\\nD_{1} & =1[\\delta_{2}D_{2}+\\gamma_{1}Z_{1}\\ge V_{1}],\\\\\nD_{2} & =1[\\delta_{1}D_{1}+\\gamma_{2}Z_{2}\\ge V_{2}],\n\\end{align*}\nwhere $(\\epsilon,V_{1},V_{2})$ are drawn, independent of $(X,\\boldsymbol{Z})$,\nfrom a joint normal distribution with zero means and each correlation\ncoefficient being $0.5$. We draw $Z_{s}$ ($s=1,2$) and $X$ from\na multinomial distribution, allowing $Z_{s}$ to take two values,\n$\\mathcal{Z}_{s}=\\{-1,1\\}$, and $X$ to take either three values,\n$\\mathcal{X}=\\{-1,0,1\\}$, or fifteen values, $\\mathcal{X}=\\{-1,-\\frac{6}{7},-\\frac{5}{7},...,\\frac{5}{7},\\frac{6}{7},1\\}$.\nBeing consistent with Assumption M, we choose $\\tilde{\\mu}_{11}>\\tilde{\\mu}_{10}$\nand $\\tilde{\\mu}_{01}>\\tilde{\\mu}_{00}$. Let $\\tilde{\\mu}_{10}=\\tilde{\\mu}_{01}$.\nWith Assumption SS, we choose $\\delta_{1}<0$ and $\\delta_{2}<0$.\nWithout loss of generality, we choose positive values for $\\gamma_{1}$,\n$\\gamma_{2}$, and $\\beta$. Specifically, $\\tilde{\\mu}_{11}=0.25$,\n$\\tilde{\\mu}_{10}=\\tilde{\\mu}_{01}=0$ and $\\tilde{\\mu}_{00}=-0.25$.\nFor default values, $\\delta_{1}=\\delta_{2}\\equiv\\delta=-0.1$, $\\gamma_{1}=\\gamma_{2}\\equiv\\gamma=1$\nand $\\beta=0.5$.\n\nIn this exercise, we focus on the ATE $E[Y(1,1)-Y(0,0)\\vert X=0]$,\nwhose true value is $0.2$ given our choice of parameter values. For\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)$, we consider $\\boldsymbol{z}=(1,1)$\nand $\\boldsymbol{z}'=(-1,-1)$. Note that $H(x)=h(\\boldsymbol{z},\\boldsymbol{z}',x)$\nand $\\tilde{H}(x,x',x'')=\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x,x',x'')$,\nsince $Z_{s}$ is binary. Then, we can derive the sets $\\mathcal{X}_{\\boldsymbol{d}}^{U}(0;\\boldsymbol{d}')$\nand $\\mathcal{X}_{\\boldsymbol{d}}^{L}(0;\\boldsymbol{d}')$ for each\n$\\boldsymbol{d}\\in\\{(1,1),(0,0)\\}$ and $\\boldsymbol{d}'\\neq\\boldsymbol{d}$\nin Theorem \\ref{thm:main}.\n\nBased on our design, $H(0)>0$, and thus, the bounds when we use $Z$\nonly are, with $x=0$, \n\\[\n\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]\\le\\Pr[Y(0,0)=1\\vert x]\\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1\\vert\\boldsymbol{z},x],\n\\]\nand \n\\[\n\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1\\vert\\boldsymbol{z},x]\\le\\Pr[Y(1,1)=1\\vert x]\\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]\\right\\} .\n\\]\nUsing both $\\boldsymbol{Z}$ and $X$, we obtain narrower bounds.\nFor example, when $\\left|\\mathcal{X}\\right|=3$, with $\\tilde{H}(0,-1,-1)<0$,\nthe lower bound on $\\Pr[Y(0,0)=1\\vert X=0]$ becomes \n\\[\n\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},0]+\\Pr[Y=1,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},-1]\\right\\} .\n\\]\nWith $\\tilde{H}(1,1,0)<0$, the upper bound on $\\Pr[Y(1,1)=1\\vert X=0]$\nbecomes \n\\[\n\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},0]+\\Pr[Y=1,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},1]+\\Pr[\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},0]\\right\\} .\n\\]\nFor comparison, we calculate the bounds in \\citet{manski1990nonparametric}\nusing $\\boldsymbol{Z}$. These bounds are given by \n\\begin{align*}\n & \\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]\\le\\Pr[Y(0,0)=1\\vert x]\\\\\n & \\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z}]\\right\\} ,\n\\end{align*}\nand \n\\begin{align*}\n & \\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]\\le\\Pr[Y(1,1)=1\\vert x]\\\\\n & \\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z}]\\right\\} .\n\\end{align*}\nWe also compare the estimated ATE using a standard linear IV model\nin which the nonlinearity of the true DGP is ignored:\n\\begin{align*}\nY & =\\pi_{0}+\\pi_{1}D_{1}+\\pi_{2}D_{2}+\\beta X+\\epsilon,\\\\\n\\left(\\begin{array}{c}\nD_{1}\\\\\nD_{2}\n\\end{array}\\right) & =\\left(\\begin{array}{c}\n\\gamma_{10}\\\\\n\\gamma_{20}\n\\end{array}\\right)+\\left(\\begin{array}{cc}\n\\gamma_{11} & \\gamma_{12}\\\\\n\\gamma_{21} & \\gamma_{22}\n\\end{array}\\right)\\left(\\begin{array}{c}\nZ_{1}\\\\\nZ_{2}\n\\end{array}\\right)+\\left(\\begin{array}{c}\nV_{1}\\\\\nV_{2}\n\\end{array}\\right).\n\\end{align*}\nHere, the first stage is the reduced-form representation of the linear\nsimultaneous equations model for strategic interaction. Under this\nspecification, the ATE becomes $E[Y(1,1)-Y(0,0)\\vert X=0]=\\pi_{1}+\\pi_{2}$,\nwhich is estimated via two-stage least squares (TSLS).\n\nThe bounds calculated for the ATE are shown in Figures \\ref{fig:sim1}--\\ref{fig:sim4}.\nFigure \\ref{fig:sim1} shows how the bounds on the ATE change, as\nthe value of $\\gamma$ changes from $0$ to $2.5$. The larger $\\gamma$\nis, the stronger the instrument $\\boldsymbol{Z}$ is. The first conspicuous\nresult is that the TSLS estimate of the ATE is biased because of the\nproblem of misspecification. Next, as expected, Manski's bounds and\nour proposed bounds converge to the true value of the ATE as the instrument\nbecomes stronger. Overall, our bounds, with or without exploiting\nthe variation of $X$, are much narrower than Manski's bounds.\\footnote{Although we do not make a rigorous comparison of the assumptions here,\nnote that the bounds by \\citet{MP00} under the semi-MTR is expected\nto be similar to ours. However, their bounds need to assume the direction\nof the monotonicity.} Notice that the sign of the ATE is identified in the whole range\nof $\\gamma$, as predicted by the first part of Theorem \\ref{thm:main},\nin contrast to Manski's bounds. Using the additional variation in\n$X$ with $\\left|\\mathcal{X}\\right|=3$ decreases the width of the\nbounds, particularly with the smaller upper bounds on the ATE in this\nsimulation design. Figure \\ref{fig:sim2} depicts the bounds using\n$X$ with $\\left|\\mathcal{X}\\right|=15$, which yields narrower bounds\nthan when $\\left|\\mathcal{X}\\right|=3$, and substantially narrower\nthan those only using $\\boldsymbol{Z}$.\n\nFigure \\ref{fig:sim3} shows how the bounds change as the value of\n$\\beta$ changes from $0$ to $1.5$, where a larger $\\beta$ corresponds\nto a stronger exogenous variable $X$. The jumps in the upper bound\nare associated with the sudden changes in the signs of $\\tilde{H}(-1,0,-1)$\nand $\\tilde{H}(0,1,1)$. At least in this simulation design, the strength\nof $X$ is not a crucial factor for obtaining narrower bounds. In\nfact, based on other simulation results (omitted in the paper), we\nconclude that the number of values $X$ can take matters more than\nthe dispersion of $X$ (unless we pursue point identification of the\nATE).\n\nFinally, Figure \\ref{fig:sim4} shows how the width of the bounds\nis related to the extent to which the opponents' actions $D_{-s}$\naffect one's payoff, captured by $\\delta$. We vary the value of $\\delta$\nfrom $-2$ to $0$, and when $\\delta=0$, the players solve a single-agent\noptimization problem. Thus, heuristically, the bound at this point\nwould be similar to the ones that can be obtained when \\citet{SV11}\nis extended to a multiple-treatment setting with no simultaneity.\nIn the figure, as the value of $\\delta$ becomes smaller, the bounds\nget narrower.\n\n\\section{Empirical Application: Airline Markets and Pollution\\label{sec:Empirical-Application}}\n\nAircrafts are a major source of emissions, and thus, quantifying the\ncausal effect of air transport on pollution is of importance to policy\nmakers. Therefore, in this section, we take the bounds proposed in\nSection \\ref{subsec:Partial-Identification} to data on airline market\nstructure and air pollution in cities in the U.S.\n\nIn 2013, aircrafts were responsible for about 3 percent of total U.S.\ncarbon dioxide emissions and nearly 9 percent of carbon dioxide emissions\nfrom the U.S. transportation sector, and it is one of the fastest\ngrowing sources.\\footnote{See \\texttt{https:\/\/www.c2es.org\/content\/reducing-carbon-dioxide-emissions-from-aircraft\/7\/}}\nAirplanes remain the single largest source of carbon dioxide emissions\nwithin the U.S. transportation sector, which is not yet subject to\ngreenhouse gas regulations. In addition to aircrafts, airport land\noperations are also a big source of pollution, making airports one\nof the major sources of air pollution in the U.S. For example, 43\nof the 50 largest airports are in ozone non-attainment areas and 12\nare in particulate matter non-attainment areas.\\footnote{Ozone is not emitted directly but is formed when nitrogen oxides and\nhydrocarbons react in the atmosphere in the presence of sunlight.\nIn United States environmental law, a non-attainment area is an area\nconsidered to have air quality worse than the National Ambient Air\nQuality Standards as defined in the Clean Air Act.}\n\nThere is growing literature showing the effects of air pollution on\nvarious health outcomes (see, \\citet{schlenker2015airports}, \\citet{cg2003},\n\\citet{kms2011}). In particular, \\citet{schlenker2015airports} show\nthat the causal effect of airport pollution on the health of local\nresidents---using a clever instrumental variable approach---is sizable.\nTheir study focuses on the 12 major airports in California and implicitly\nassume that the level of competition (or market structure) is fixed.\nUsing high-frequency data, they exploit weather shocks in the East\ncoast---that might affect airport activity in California through\nnetwork effects---to quantify the effect of airport pollution on\nrespiratory and cardiovascular health complications. In contrast,\nwe take the link between airport pollution and health outcomes as\ngiven and are interested in quantifying the effects of different market\nstructures of the airline industry on air pollution.\\footnote{In this section, we refer to market structure as the particular configuration\nof airlines present in the market. In other words, market structure\nnot only refers to the number of firms competing in a given market\nbut to the actual identities of the firms. Thus, we will regard a\nmarket in which, say, United and American operate as different from\na market in which Southwest and Delta operate, despite both markets\nhaving two carriers.} We explicitly allow market structure to be determined endogenously,\nas the outcome of an entry game in which airlines behave strategically\nto maximize their profits and the resulting pollution in this market\nis not internalized by the firms. Understanding these effects can\nthen help inform the policy discussion on pollution regulation. Given\nthat we treat market structure as endogenous, one cannot simply run\na regression of a measure of pollution on market structure (or the\nnumber of airlines present in a market) to obtain the causal effect,\nif there are unobserved market characteristics that affect both firm\ncompetition and pollution outcomes. For example, if at both ends of\na city-pair there are firms from a high-polluting industry which engage\nin a lot of business travel, it would drive both pollution and the\nentry of airlines in the market. Therefore, we use the method presented\nin Section \\ref{subsec:Partial-Identification}.\n\nIn each market, we assume that a set of airlines chooses to be in\nor out as part of a simultaneous entry game of perfect information,\nas introduced in Section \\ref{subsec:Model}. Therefore, we treat\nmarket structure as our endogenous treatment. We then model air pollution\nas a function of the airline market structure as in equation (\\ref{eq:main_model1}),\nwhere $Y$ is a measure of air pollution at the airport level (including\nboth aircraft and land operation pollution), the vector $\\boldsymbol{D}$\nrepresents the market structure, and $X$ includes market specific\ncovariates that affect pollution directly (i.e., not through airline\nactivity), such as weather shocks or the share of pollution-related\nactivity in the local economy.\\footnote{Note that our definition of market is a city-pair; hence, all of our\nvariables are, in fact, weighted averages over the two cities.} Additionally, we allow for market-level covariates, $\\boldsymbol{W}$,\nwhich affect both the participation decisions and pollution (e.g.,\nthe size of the market as measured by population or the level of economic\nactivity). As instruments, $Z_{s}$, we consider a firm-market proxy\nfor cost introduced in \\citet{CT09}. We discuss the definition and\nconstruction of the variables in detail below.\n\nOur objective is to estimate the effect of a change in market structure\non air pollution, $E[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}')]$. For\nexample, we might be interested in the average effect on pollution\nof moving from two airlines operating in the market to three, or how\nthe pollution level changes on average when Delta is a monopolist\nversus a situation in which Delta faces competition from American.\nFollowing entry, firms compete by choosing their pricing, frequency,\nand which airplanes to operate. Different market structures will have\ndifferent impacts on these variables, which in turn, affect the level\nof pollution. Note that the effects might be asymmetric. That is,\nfor a given number of entrants, their identities are important to\ndetermine the pollution level. For example, when comparing the effect\nof a monopoly on pollution, we find that the airline operating plays\na role.\n\nTo illustrate our estimation procedure, we consider three types of\nATE exercises. The first examines the effects on pollution from a\nmonopolist airline vis-a-vis a market that is not served by any airline.\nThe second set of exercises examine the total effect of the industry\non pollution under all possible market configurations. Finally, the\nthird type of exercises examine how the (marginal) effect of a given\nairline changes when the firm faces different levels of competition.\nNotice that regardless of the exercise we run, we quantify ``reduced-form''\neffects, in that they summarize structural effects resulting from\na given market structure. The idea is that given the market structure,\nprices are determined, and given demand, ultimately the frequency\nof flights in the market is determined, which in fact, causes pollution.\n\nIn the rest of this section, we first describe our data sources, then\nshow results for three different ATE exercises, and conclude with\na brief discussion relating our results to potential policy recommendations.\n\n\\subsection{Data}\n\nFor our analysis, we combine data spanning the period 2000--2015\nfrom two sources: airline data from the U.S. Department of Transportation\nand pollution data from the Environmental Protection Agency (EPA).\n\n\\textbf{Airline Data.} Our first data source contains airline information\nand combines publicly available data from the Department of Transportation's\nOrigin and Destination Survey (DB1B) and Domestic Segment (T-100)\ndatabase. These datasets have been used extensively in the literature\nto analyze the airline industry (see, e.g., \\citet{borenstein89},\n\\citet{berry1992estimation}, \\citet{CT09}, and more recently, \\citet{robertssweeting2013}\nand \\citet{ciliberto2015market}). The DB1B database is a quarterly\nsample of all passenger domestic itineraries. The dataset contains\ncoupon-specific information, including origin and destination airports,\nnumber of coupons, the corresponding operating carriers, number of\npassengers, prorated market fare, market miles flown, and distance.\nThe T-100 dataset is a monthly census of all domestic flights broken\ndown by airline, and origin and destination airports.\n\nOur time-unit of analysis is a quarter and we define a market as the\nmarket for air connection between a pair of airports (regardless of\nintermediate stops) in a given quarter.\\footnote{In cities that operate more than one airport, we assume that flights\nto different airports in the same metropolitan area are in separate\nmarkets.} We restrict the sample to include the top 100 metropolitan statistical\nareas (MSA's), ranked by population at the beginning of our sample\nperiod. We follow \\citet{berry1992estimation} and \\citet{CT09} and\ndefine an airline as actively serving a market in a given quarter,\nif we observe at least 90 passengers in the DB1B survey flying with\nthe airline in the corresponding quarter.\\footnote{This corresponds to approximately the number of passengers that would\nbe carried on a medium-size jet operating once a week.} We exclude from our sample city pairs in which no airline operates\nin the whole sample period. Notice that we do include markets that\nare temporarily not served by any airline. This leaves us with 181,095\nmarket-quarter observations.\n\nIn our analysis, we allow for airlines to have a heterogeneous effect\non pollution, and to simplify computation, in each market we allow\nfor six potential participants: American (AA), Delta (DL), United\n(UA), Southwest (WN), a medium-size airline, and a low-cost carrier.\\footnote{That is, to limit the number of potential market structures, we lump\ntogether all the low cost carriers into one category, and Northwest,\nContinental, America West, and USAir under the medium airline type.} The latter is not a bad approximation to the data in that we rarely\nobserve more than one medium-size or low-cost in a market but it assumes\nthat all low-cost airlines have the same strategic behavior, and so\ndo the medium airlines. Table \\ref{tab:marketstructure} shows the\nnumber of firms in each market broken down by size as measured by\npopulation. As the table shows, market size alone does not explain\nmarket structure, a point first made by \\citet{CT09}.\n\n\\begin{table}[t!]\n\\caption{Distribution of the Number of Carriers by Market Size}\n\\label{tab:marketstructure} \\centering{}%\n\\begin{tabular}{lrrrr}\n\\hline \n & \\multicolumn{3}{c}{\\rule{0ex}{2.5ex}Market size} & \\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}\\# firms & Large & Medium & Small & Total\\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}0 & 7.96 & 8.20 & 8.62 & 8.18\\tabularnewline\n1 & 41.18 & 22.53 & 20.58 & 30.30\\tabularnewline\n2 & 28.14 & 23.41 & 21.25 & 25.04\\tabularnewline\n3 & 12.65 & 20.00 & 16.67 & 16.05\\tabularnewline\n4 & 7.65 & 14.72 & 15.17 & 11.51\\tabularnewline\n5 & 1.98 & 9.90 & 16.48 & 7.80\\tabularnewline\n6+ & 0.52 & 1.23 & 2.21 & 1.12\\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}\\# markets & 79,326 & 64,191 & 37,578 & 181,095\\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\nIn our application, we consider two instruments for the entry decisions.\nThe first is the \\emph{airport presence} of an airline proposed by\n\\citet{berry1992estimation}. For a given airline, this variable is\nconstructed as the number of markets it serves out of an airport as\na fraction of the total number of markets served by all airlines out\nof the airport. A hub-and-spoke network allows firms to exploit demand-side\nand cost-side economies, which should affect the firm's profitability.\nWhile \\citet{berry1992estimation} assumes that an airline's airport\npresence only affects its own profits (and hence, is excluded from\nrivals' profits), \\citet{CT09} argue that this may not be the case\nin practice, since airport presence might be a measure of product\ndifferentiation, rendering it likely to enter the profit function\nof all firms through demand. While an instrument that enters all of\nthe profit functions is fine in our context (see Appendix \\ref{subsec:Common_Z}),\nwe also consider the instrument proposed by \\citet{CT09}, which captures\nshocks to the fixed cost of providing a service in a market. This\nvariable, which they call \\emph{cost}, is constructed as the percentage\nof the nonstop distance that the airline must travel in excess of\nthe nonstop distance, if the airline uses a connecting instead of\na nonstop flight.\\footnote{Mechanically, the variable is constructed as the difference between\nthe sum of the distances of a market's endpoints and the closest hub\nof an airline, and the nonstop distance between the endpoints, divided\nby the nonstop distance.} Arguably, this variable only affects its own profits and is excluded\nfrom rivals' profits.\n\n\\begin{table}[t!]\n\\caption{Airline Summary Statistics}\n\\label{tab:airlinessumstat} \\centering{}%\n\\begin{tabular}{llccccccc}\n\\hline \n\\rule{0ex}{2.5ex} & & American & Delta & United & Southwest & medium & low-cost & \\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}Market presence (0\/1) & mean & 0.44 & 0.57 & 0.28 & 0.25 & 0.56 & 0.17 & \\tabularnewline\n & sd & 0.51 & 0.51 & 0.46 & 0.44 & 0.51 & 0.38 & \\tabularnewline\nAirport presence (\\%) & mean & 0.43 & 0.56 & 0.27 & 0.25 & 0.39 & 0.10 & \\tabularnewline\n & sd & 0.17 & 0.18 & 0.16 & 0.18 & 0.14 & 0.08 & \\tabularnewline\nCost (\\%) & mean & 0.71 & 0.41 & 0.76 & 0.29 & 0.22 & 0.04 & \\tabularnewline\n & sd & 1.56 & 1.28 & 1.43 & 0.83 & 0.60 & 0.17 & \\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:airlinessumstat} presents the summary statistics of\nthe airline related variables. Of the leading airlines, we see that\nAmerican and Delta are present in about half of the markets, while\nUnited and Southwest are only present in about a quarter of the markets.\nAmerican and Delta tend to dominate the airports in which they operate\nmore than United and Southwest. From the cost variable, we see that\nboth American and United tend to operate a hub-and-spoke network,\nwhile Southwest (and to a lesser extent Delta) operates most markets\nnonstop.\n\n\\textbf{Pollution Data.} The second component of our dataset is the\nair pollution data. The EPA compiles a database of outdoor concentrations\nof pollutants measured at more than 4,000 monitoring stations throughout\nthe U.S., owned and operated mainly by state environmental agencies.\nEach monitoring station is geocoded, and hence, we are able to merge\nthese data with the airline dataset by matching all the monitoring\nstations that are located within a 10km radius of each airport in\nour first dataset.\n\nThe principal emissions of aircraft include the greenhouse gases carbon\ndioxide ($\\text{CO}_{2}$) and water vapor ($\\text{H}_{2}\\text{O}$),\nwhich have a direct impact on climate change. Aircraft jet engines\nalso produce nitric oxide ($\\text{NO}$) and nitrogen dioxide ($\\text{NO}_{2}$)\n(which together are termed nitrogen oxides ($\\text{NO}_{\\text{x}}$)),\ncarbon monoxide (CO), oxides of sulphur ($\\text{SO}_{\\text{x}}$),\nunburned or partially combusted hydrocarbons (also known as volatile\norganic compounds or VOC's), particulates, and other trace compounds\n(see, \\citet{FAA2015}). In addition, ozone ($\\text{O}_{3}$) is formed\nby the reaction of VOC's and $\\text{NO}_{\\text{x}}$ in the presence\nof heat and sunlight. The set of pollutants other than $\\text{CO}_{2}$\nare more pernicious in that they can harm human health directly and\ncan result in respiratory, cardiovascular, and neurological conditions.\nResearch to date indicates that fine particulate matter (PM) is responsible\nfor the majority of the health risks from aviation emissions, although\nozone has a substantial health impact too.\\footnote{See \\citet{FAA2015}.}\nTherefore, as our measure of pollution, we will consider both.\n\nOur measure of ozone is a quarterly mean of daily maximum levels in\nparts per million. In terms of PM, as a general rule, the smaller\nthe particle the further it travels in the atmosphere, the longer\nit remains suspended in the atmosphere, and the more risk it poses\nto human health. PM that measure less than 2.5 micrometer can be readily\ninhaled, and thus, potentially pose increased health risks. The variable\nPM2.5 is a quarterly average of daily averages and is measured in\nmicrograms\/cubic meter. For each airport in our sample, we take an\naverage (weighted by distance to the airport) of the data from all\nair monitoring stations within a 10km radius. The top panel of Table\n\\ref{tab:marketsumstats} shows the summary statistics of the pollution\nmeasures.\n\n\\begin{table}[t!]\n\\caption{Market-level Summary Statistics}\n\\label{tab:marketsumstats} \\centering{}%\n\\begin{tabular}{lcc}\n\\hline \n\\rule{0ex}{2.5ex} & Mean & Std. Dev.\\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}Pollution & & \\tabularnewline\n\\hspace{2ex}Ozone ($\\text{O}_{3}$) & .0477 & .0056\\tabularnewline\n\\hspace{2ex}Particulate matter (PM2.5) & 8.3881 & 2.5287 \\tabularnewline\n\\rule{0ex}{2.5ex}Other controls & & \\tabularnewline\n\\hspace{2ex}Market size (pop.) & 2307187.8 & 1925533.4\\tabularnewline\n\\hspace{2ex}Income (per capita) & 34281.6 & 4185.5\\tabularnewline\n\\rule{0ex}{2.5ex}\\# of markets & 181,095 & \\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\n\\textbf{Other Market-Level Controls.} We also include in our analysis\nmarket-level covariates that may affect both market structure and\npollution levels. In particular, we construct a measure of market\nsize by computing the (geometric) mean of the MSA populations at the\nmarket endpoints and a measure of economic activity by computing the\naverage per capita income at the market endpoints, using data from\nthe Regional Economic Accounts of the Bureau of Economic Analysis.\n\nFinally, as we mentioned in Section \\ref{subsec:Partial-Identification},\nhaving access to data on a variable that affects pollution but is\nexcluded from the airline participation decisions can greatly help\nin calculating the bounds of the ATE. Therefore, we construct a variable\nthat measures the economic activity of pollution related industries\n(manufacturing, construction, and transportation other than air transportation)\nin a given market (MSA) as a fraction of total economic activity in\nthat market, again, using data from the Regional Economic Accounts\nof the Bureau of Economic Analysis.\nOur implicit assumption is as follows. The size of the market, among\nother things, determines whether a firm might enter it, but not the\ntype of economic activity in the cities.\nThe idea is that conditional on the market GDP, a market with a higher\nshare of polluting industries will have a higher level of pollution\nbut this share would not affect the airline market structure.\n\n\\subsection{Estimation and Results}\n\n\nTo simplify the estimation, we discretize all continuous variables\ninto binary variables (taking a value of 0 (1) if the corresponding\ncontinuous variable is below (above) its median). Using the notation\nfrom Section \\ref{subsec:Model}, let the elements of the treatment\nvector $\\boldsymbol{d}=(d_{\\text{DL}},d_{\\text{AA}},d_{\\text{UA}},d_{\\text{WN}},d_{\\text{med}},d_{\\text{low}})$\nbe either 0 or 1, indicating whether each firm is active in the market.\nWe compute the upper and lower bounds on the ATE using the result\nfrom Theorem \\ref{thm:main} and the fact that our $Y$ variable is\nbinary. Specifically, given two treatment vectors $\\boldsymbol{d}$\nand $\\tilde{\\boldsymbol{d}}$ we can bound the ATE \n\\begin{align*}\nL(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w) & \\leq E[Y(\\boldsymbol{d})-Y(\\tilde{\\boldsymbol{d}})|x,w]\\leq U(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w)\n\\end{align*}\nwhere the upper bound can be characterized by \n\\begin{align*}\nU(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w) & \\equiv\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z},x,w]+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{j}\\backslash\\{\\boldsymbol{d}\\}}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{Z}=\\boldsymbol{z},W=w]\\\\\n & \\quad+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d})\\cup\\mathcal{D}^{>}(\\boldsymbol{d})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{Z}=\\boldsymbol{z},X=x'(\\boldsymbol{d}'),W=w]\\\\\n & \\quad-\\text{Pr}[Y=1,\\boldsymbol{D}=\\tilde{\\boldsymbol{d}}|\\boldsymbol{Z}=\\boldsymbol{z},X=x,W=w]\\\\\n & \\quad-\\sum_{\\boldsymbol{d}''\\in\\mathcal{D}^{<}(\\tilde{\\boldsymbol{d}})\\cup\\mathcal{D}^{>}(\\tilde{\\boldsymbol{d}})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}''|\\boldsymbol{Z}=\\boldsymbol{z},X=x''(\\boldsymbol{d}''),W=w]\n\\end{align*}\nfor every $\\boldsymbol{z}$, $x'(\\boldsymbol{d}')\\in\\mathcal{X}_{\\boldsymbol{d}}^{U}(x;\\boldsymbol{d}')$\nfor $\\boldsymbol{d}'\\neq\\boldsymbol{d}$, and $x''(\\boldsymbol{d}'')\\in\\mathcal{X}_{\\tilde{\\boldsymbol{d}}}^{L}(x;\\boldsymbol{d}'')$\nfor $\\boldsymbol{d}''\\neq\\tilde{\\boldsymbol{d}}$. Similarly, the\nlower bound can be characterized by \n\\begin{align*}\nL(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w) & \\equiv\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{Z}=\\boldsymbol{z},X=x,W=w]\\\\\n & \\quad+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d})\\cup\\mathcal{D}^{>}(\\boldsymbol{d})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{Z}=\\boldsymbol{z},X=x'(\\boldsymbol{d}'),W=w]\\\\\n & \\quad-\\text{Pr}[Y=1,\\boldsymbol{D}=\\tilde{\\boldsymbol{d}}|\\boldsymbol{Z}=\\boldsymbol{z},X=x,W=w]-\\sum_{\\boldsymbol{d}''\\in\\mathcal{D}^{j}\\backslash\\{\\tilde{\\boldsymbol{d}}\\}}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}''|\\boldsymbol{Z}=\\boldsymbol{z},W=w]\\\\\n & \\quad-\\sum_{\\boldsymbol{d}''\\in\\mathcal{D}^{<}(\\tilde{\\boldsymbol{d}})\\cup\\mathcal{D}^{>}(\\tilde{\\boldsymbol{d}})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}''|\\boldsymbol{Z}=\\boldsymbol{z},X=x''(\\boldsymbol{d}''),W=w]\n\\end{align*}\nfor every $\\boldsymbol{z}$, $x'(\\boldsymbol{d}')\\in\\mathcal{X}_{\\boldsymbol{d}}^{L}(x;\\boldsymbol{d}')$\nfor $\\boldsymbol{d}'\\neq\\boldsymbol{d}$, and $x''(\\boldsymbol{d}'')\\in\\mathcal{X}_{\\tilde{\\boldsymbol{d}}}^{U}(x;\\boldsymbol{d}'')$\nfor $\\boldsymbol{d}''\\neq\\tilde{\\boldsymbol{d}}$. We estimate the\npopulation objects above using their sample counterparts. We experimented\nwith both measures of pollution discussed earlier and obtain qualitatively\nand quantitatively similar results in all cases, which is not surprising\ngiven that the two pollution measures are highly correlated. In order\nto save space, we only show results using PM2.5 as our outcome variable.\nWe also experimented with several specifications of the covariates,\n$\\boldsymbol{X}$ and $\\boldsymbol{W}$, and instruments, $\\boldsymbol{Z}$.\nIn particular, we tried different discretizations of each variable\n(including allowing for more than two points in their supports and\ndifferent cutoffs). Clearly, there is a limit to how finely we can\ncut the data even with a large sample size such as ours. The coarser\ndiscretization occurs when each covariate (and instrument) is binary\nand it seems to produce reasonable results; hence, we stick with this\ndiscretization in all of our exercises. Again, aiming at the most\nparsimonious model, and after some experimentation, we obtained reasonable\nresults when both $\\boldsymbol{X}$ and $\\boldsymbol{W}$ are scalars\n(share of pollution related industries in the market and total GDP\nin the market, respectively).\n\nWe also compute confidence sets by deriving unconditional moment inequalities\nfrom our conditional moment inequalities and implementing the Generalized\nMoment Selection test proposed by \\citet{as2010}. The confidence\nsets are obtained by inverting the test.\\footnote{For details of this procedure, see \\citet{dm2016}.}\n\n\\begin{figure*}[!t]\n\\begin{centering}\n\\includegraphics[scale=0.7\n{fig2_all_cs_monop_CROP.pdf}\\caption{Effect of a Monopolistic Market Structure}\n\\label{fig:emp_monop} \n\\par\\end{centering}\n\\medskip{}\n\n\\begin{small} This plot shows the ATEs of a change in market structure\nfrom no airline serving a market to a monopolist serving it. The solid\nblack intervals are our estimates of the identified sets and the thin\nred lines are the 95\\% confidence sets. \\end{small} \n\\end{figure*}\n\n\\textbf{Monopoly Effects.} Here we examine a very simple ATE of a\nchange in market structure from no airline serving a market to a monopolist\nserving it. Intuitively, we want to understand the change in the probability\nof being a high-pollution market when an airline starts operating\non it. Recall that we allow each firm to have different effects on\npollution; hence, we estimate the effects of each one of the six firms\nin our data becoming a monopolist. Thus, we are interested in the\nATEs of the form \n\\[\nE[Y(\\boldsymbol{d}_{\\text{monop}})-Y(\\boldsymbol{d}_{\\text{noserv}})|X,W]\n\\]\nwhere $\\boldsymbol{d}_{\\text{monop}}$ is one of the six vectors in\nwhich only one element is 1 and the rest are 0's, and $\\boldsymbol{d}_{\\text{noserv}}$\nis a vector of all 0's. The results are shown in Figure \\ref{fig:emp_monop},\nwhere the solid black intervals are our estimates of the identified\nsets and the thin red lines are the 95\\% confidence sets. We see that\nall ATEs are positive and statistically significant different from\n0, except for the medium-size carriers. While there no major differences\non the effects of the leader carriers, with the exception of Delta\nwhich seems to induce a higher increase in the probability of high\npollution, the medium and low-cost carriers induce a smaller effect.\n\n\\begin{figure*}[!t]\n\\begin{centering}\n\\includegraphics[scale=0.7\n{fig1_all_cs_num_entrants_CROP.pdf}\\caption{Total Market Structure Effect}\n\\label{fig:emp_numentrants} \n\\par\\end{centering}\n\\medskip{}\n\n\\begin{small} This plot shows the ATEs of the airline industry under\nall possible market configurations. The solid black intervals are\nour estimates of the identified sets and the thin red lines are the\n95\\% confidence sets. The bars in each cluster correspond to all possible\nmarket configurations, respectively. \\end{small} \n\\end{figure*}\n\n\\textbf{Total Market Structure Effect.} We now turn to our second\nset of exercises. Here, we quantify the effect of the airline industry\non the likelihood of a market having high levels of pollution. To\ndo so, we estimate ATEs of the form \n\\[\nE[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}_{\\text{noserv}})|X,W]\n\\]\nfor all potential market configurations $\\boldsymbol{d}$, and where,\nas before, $\\boldsymbol{d}_{\\text{noserv}}$ is a vector of all 0's.\nFigure \\ref{fig:emp_numentrants} depicts the results. The left-most\nset of intervals corresponds to the 6 different monopolistic market\nstructures, and by construction, coincide with those from Figure \\ref{fig:emp_monop}.\nThe next set corresponds to all possible duopolistic structures, which\nhas 15 possibilities, and so on. Not surprisingly, we observe that\nthe effect on the probability of being a high-pollution market is\nincreasing in the number of firms operating in the market. More interesting\nis the non-linearity of the effect: the effect increases at a decreasing\nrate. This would be consistent with a model in which firms \\emph{accommodate}\nnew entrants by decreasing their frequency, which is analogous to\nthe prediction of a Cournot competition model, as we increase the\nnumber of firms. To further investigate this point, in the next set\nof exercises, we examine the effect of one firm as we change the competition\nit faces.\n\n\\begin{figure*}[!t]\n\\begin{centering}\n\\includegraphics[scale=0.7\n{fig3_all_cs_compet_effect_DL_CROP.pdf}\\caption{Marginal Effect of Delta under Different Market Structures}\n\\label{fig:emp_competDL} \n\\par\\end{centering}\n\\medskip{}\n\n\\begin{small} This plot shows the ATEs of Delta entering the market\ngiven all possible rivals market configurations. The solid black intervals\nare our estimates of the identified sets and the thin red lines are\nthe 95\\% confidence sets. The bars in each cluster correspond to all\npossible market configurations, respectively. \\end{small} \n\\end{figure*}\n\n\\textbf{Marginal Carrier Effect.} In our last set of exercises, we\nare interested in investigating how the marginal effect (i.e., the\neffect of introducing one more firm into the market) changes under\ndifferent configurations of the market structure. Say we are interested\nin the effect of Delta entering the market on pollution, given that\nthe current market structure (excluding Delta) is $\\boldsymbol{d}_{\\text{--DL}}=(d_{\\text{AA}},d_{\\text{UA}},d_{\\text{WN}},d_{\\text{med}},d_{\\text{low}})$.\nThen, we want to estimate \n\\[\nE[Y(1,\\boldsymbol{d}_{\\text{--DL}})-Y(0,\\boldsymbol{d}_{\\text{--DL}})|X,W].\n\\]\n\nFigure \\ref{fig:emp_competDL} shows the identified sets and confidence\nsets of the marginal effect of Delta on the probability of high pollution\nunder all possible market configuration for Delta's rivals. We obtain\nqualitatively similar results when estimating the marginal effects\nof the other five carriers, and hence, we omit the graphs to save\nspace. In the Figure, the left-most exercise is the effect of Delta\nas a monopolist, and coincides, by construction, with the left-most\nexercise in Figure \\ref{fig:emp_monop}. The second exercise (from\nthe left) corresponds to the additional effect of Delta on pollution\nwhen there is already one firm operating in the market, which yields\nfive different possibilities. The next exercise shows the effect of\nDelta when there are two firms already operating in the market yielding\n10 possibilities, and so on. Again, the estimated marginal ATEs in\nall cases are positive and statistically significant. Interestingly,\nalthough we cannot entirely reject the null hypothesis that all the\neffects are the same, it seems that the marginal effect of Delta is\ndecreasing in the number of rivals it faces. Intuitively, this suggests\na situation in which Delta enters the market and operates with a frequency\nthat is decreasing with the number of rivals (again, as we would expect\nin a Cournot competition model) and is consistent with the findings\nin our previous set of exercises.\n\nThe conclusions from the \\emph{total market} and \\emph{marginal} ATEs\nare also interesting from a policy perspective. For example, a merger\nof two airlines in which duplicate routes are eliminated would imply\na decrease in total pollution in the affected markets, but by less\nthan what one would have naively anticipated from removing one airline\nwhile keeping everything else constant.\\footnote{Note that, however, to the extent that a merger alters the way the merged firm behaves post entry, the treatment effects we estimate will not be informative. In other words, our model can only speak to the effects of a merger on pollution that only affects behavior through entry.} In other words, there are\ntwo effects of removing an airline from a market. The first is a direct\naffect: pollution decreases by the amount of pollution by the carrier\nthat is no longer present in the market. However, the remaining firms\nin the market will react strategically to the new market structure.\nIn our exercises, we find that this indirect effect implies an increase\nin pollution. The overall effect is a net decrease in pollution. Moreover,\ngiven the non-linearities of the ATEs we estimate it looks like the\noverall effect, while negative, might be negligible in markets with\nfour or more competitors. While it is unclear that merger analysis, which is typically concerned\nwith price increases post-merge or cost savings of the merging firms,\nshould also consider externalities such as pollution, (social) welfare\nanalysis should. Hence, our findings may serve as guidance to policy\ndiscussion on air traffic regulation.\n\n\\medskip{}\n\n \\bibliographystyle{ecta}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}