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+{"text":"\\section{Symmetry breaking in quantum dots}\n\nTwo-dimensional (2D) quantum dots (QDs), created at semiconductor interfaces \nthrough the use of lithographic and gate-voltage techniques, with refined \ncontrol of their size, shape, and number of electrons, are often referred to as\n``artificial atoms'' \\cite{kast,taru2,asho}. These systems which, with the use \nof applied magnetic fields, are expected to have future applications as \nnanoscale logic gates and switching devices, have been in recent years the \nsubject of significant theoretical and experimental research efforts. As \nindicated above, certain analogies have been made between these man-made systems\nand their natural counterparts, suggesting that the physics of electrons in the \nformer is similar to that underlying the traditional description of natural \natoms -- pertaining particularly to electronic shells and the Aufbau principle \nin atoms (where electrons are taken to be moving in a spherically averaged \neffective central mean-field potential). \n\nThe above-mentioned analogy has been theoretically challenged recently \n\\cite{yyl1,yyl2} on the basis of calculations that showed evidence for \nformation, under favorable conditions (that are readily achieved in the \nlaboratory), of ``electron molecules,'' which are alternatively called Wigner \nmolecules (WMs) after the physicist who predicted formation of electron crystals\nin extended systems \\cite{wign}. These \nspin-and-space (sS) unrestricted Hartree-Fock (UHF) calculations (denoted in the\nfollowing as sS-UHF or simply UHF) of electrons confined in 2D QDs by a \nparabolic external potential led to the discovery of spontaneous symmetry \nbreaking in QDs, manifested in the appearance of distinct interelectronic \nspatial (crystalline) correlations (even in the absence of magnetic fields). \nSuch symmetry breaking may indeed be expected to occur based on the interplay \nbetween the interelectron repulsion, $Q$, and the zero-point kinetic energy, \n$K$. It is customary to take $Q=e^2\/\\kappa l_0$ and $K \\equiv \\hbar \\omega_0$, \nwhere $l_0=(\\hbar\/m^*\\omega_0)^{1\/2}$ is the spatial extent of an electron in the\nlowest state of the parabolic confinement; $m^*$ is the electron effective mass,\n$\\kappa$ is the dielectric constant, and $\\omega_0$ is the frequency that \ncharacterizes the parabolic (harmonic) confining potential. Thus, defining the \nWigner parameter as $R_W = Q\/K$, one may expect symmetry breaking to occur when \nthe interelectron repulsion dominates, i.e., for $R_W > 1$. Under such \ncircumstances, an appropriate solution of the Schr\\\"{o}dinger equation \nnecessitates consideration of wave functions with symmetries that are lower than\nthat of the circularly symmetric QD Hamiltonian. Such solutions may be found \nthrough the use of the sS-UHF method, where all restrictions on the symmetries \nof the wave functions are lifted. On the other hand, when $R_W < 1$, no symmetry\nbreaking is expected, and the sS-UHF solution collapses onto that obtained via \nthe restricted Hartree-Fock (RHF) method, and the aforementioned circularly \nsymmetric ``artificial-atom'' analogy maintains. From the above we note that the\nstate of the system may be controlled and varied through the choice of materials\n(i.e., $\\kappa$) and\/or the strength of the confinement ($\\omega_0$), since  \n$R_W \\propto 1\/(\\kappa \\sqrt{\\omega_0})$. \n\n\\begin{figure}[t]\n\\centering\\includegraphics[width=12cm]{tnt_2005_fig1.eps}\n\\caption{The calculated spectrum of a two-electron parabolic quantum dot, with \n$R_W=200$. The quantum numbers are $(N, M, n, m)$ with $N$ corresponding to the \nnumber of radial nodes in the center of mass (CM) wavefunction, and $M$ is the \nCM azimuthal quantum number. The integers $n$ and $m$ are the corresponding \nquantum numbers for the electrons' relative motion (RM) and the total energy is \ngiven by $E_{NM,nm} = E^{cm}_{NM} + E^{rm}(n,|m|)$. The spectrum may be \nsummarized by the ``spectral rule'' given in the figure, with $C = 0.037$, the \nphonon stretching vibration $\\hbar \\omega_s = 3.50$, and the phonon for the \nbending vibration coincides with that of the CM motion, i.e., $\\hbar \\omega_b \n= \\hbar \\omega_0 = 2$. All energies are in units of $\\hbar \\omega_0\/2$, where \n$\\omega_0$ is the parabolic confinement frequency.\n}\n\\end{figure}\n\n\\subsection{Two-electron circular dots}\n\nTo illustrate the formation of ``electron molecules,'' we show first exact \nresults obtained for a two-electron QD, through separation of the center-of-mass\nand inter-electron relative-distance degrees of freedom \\cite{yyl3}. The \nspectrum obtained for $R_W = 200$ (Fig.\\ 1), exhibits features that are \ncharacteristic of a collective rovibrational dynamics, akin to that of a natural\n``rigid'' triatomic molecule with an infinitely heavy middle particle \nrepresenting the center of mass of the dot. This spectrum transforms to that of \na ``floppy'' molecule for smaller value of $R_W$ (i.e., for stronger \nconfinements characterized by a larger value of $\\omega_0$, and\/or for weaker \ninter-electron repulsion), ultimately converging to the independent-particle \npicture associated with the \ncircular central mean-field of the QD. Further evidence for the formation of the\nelectron molecule was found through examination of the conditional probability \ndistribution (CPD); that is, the anisotropic pair correlation \n$P({\\bf r}, {\\bf r}_0)$, which expresses the probability of finding a particle \nat ${\\bf r}$ given that the ``observer'' (reference point) is riding on another \nparticle at ${\\bf r}_0$ \\cite{yyl2,yyl3},\n\\begin{equation}\nP({\\bf r},{\\bf r}_0) =\n\\langle \\Psi | \n\\sum_{i=1}^N \\sum_{j \\neq i}^N  \\delta({\\bf r}_i -{\\bf r})\n\\delta({\\bf r}_j-{\\bf r}_0) \n| \\Psi \\rangle  \/ \\langle \\Psi | \\Psi \\rangle.\n\\label{cpds}\n\\end{equation} \nHere $\\Psi ({\\bf r}_1, {\\bf r}_2, ..., {\\bf r}_N)$ \ndenotes the many-body wave function under consideration.\nIt is instructive to note here certain \nsimilarities between the formation of a ``two-electron molecule'' in man-made \nquantum dots, and the collective (rovibrational) features observed in the \nelectronic spectrum of doubly-excited helium atoms \\cite{kell1,kell2,berr}.\n\nFor confined (finite) systems with a larger number of particles, one must resort\nto approximate computational schemes. Of particular interest are methodologies \nthat permit systematic evaluation of high-accuracy solutions to these many-body \nstrongly-correlated systems, under field-free conditions, as well as under the \ninfluence of an applied magnetic field. We remark here, that the relatively \nlarge (spatial) size of QDs (resulting from materials' characteristics, e.g., a \nsmall electron effective mass and large dielectric constant), allows the full \nrange of orbital magnetic effects to be covered for magnetic fields that are \nreadily attained in the laboratory (less then 40 T). In contrast, for natural \natoms and molecules, magnetic fields of sufficient strength (i.e., larger than \n$10^5$ T) to produce novel phenomena related to orbital magnetism (beyond the \nperturbative regime), are known to occur only in astrophysical environments \n(e.g., on the surface of neutron stars).\n\nThe 2D hamiltonian of the problem under consideration is given by \n\\begin{equation}\nH=\\sum_{i=1}^N \\frac{1}{2m^*} \\left( {\\bf p}_i -\\frac{e}{c} {\\bf A}_i \\right)^2\n+ \\sum_{i=1}^N \\frac{m^*}{2} \\omega_0^2 {\\bf r}_i^2 +\n\\sum_{i=1}^N \\sum_{j>i}^N \\frac{e^2}{\\kappa r_{ij}},\n\\label{mbhn} \n\\end{equation}\ndescribing $N$ electrons (interacting via a Coulomb repulsion) confined by a \nparabolic potential of frequency $\\omega_0$ and subjected to a perpendicular \nmagnetic field $ {\\bf B}$, whose vector potential is given in the symmetric \ngauge by ${\\bf A} ({\\bf r}) = {\\bf B} \\times {\\bf r}\/2 = (-By,Bx,0)\/2$.\nFor sufficiently high magnetic field values (i.e., in the fractional \nquantum Hall effect, or FQHE, regime), the electrons are fully spin-polarized \nand the Zeeman term (not shown here) does not need to be considered. \nIn the $B \\rightarrow \\infty$ limit, the external confinement $\\omega_0$ can be \nneglected, and $H$ can be restricted to operate in the lowest Landau level \n(LLL), reducing to the form \\cite{yyl4,yyl5,yyl6,yyl7}\n\\begin{equation}\nH_{\\rm LLL} = N \\frac{\\hbar \\omega_c}{2} +\n\\sum_{i=1}^N \\sum_{j>i}^N \\frac{e^2}{\\kappa r_{ij}},\n\\label{hlll}\n\\end{equation}\nwhere $\\omega_c = eB\/(m^*c)$ is the cyclotron frequency. \n\nFor finite $N$, the solutions to the Schr\\\"{o}dinger equations corresponding to \nthe hamiltonians given by Eq\\. (\\ref{mbhn}) (with or without a magnetic field), \nor by Eq\\. (\\ref{hlll}) (in the $B \\rightarrow \\infty$ limit), must have a good \nangular momentum, $L$, and good spin quantum numbers (the latter is guaranteed \nin the high $B$, fully spin-polarized case). As described in detail elsewhere \n\\cite{yyl4,yyl5,yyl6,yyl7,yyl8,yyl9}, \nthese solutions can be well approximated by a two-step method \nconsisting of symmetry breaking at the spin-and-space unrestricted Hartree-Fock \nlevel and subsequent symmetry restoration via post-Hartree-Fock projection \ntechniques. We recall that the sS-UHF method relaxes both the double-occupancy \nrequirement (namely, different spatial orbitals are employed for different spin \ndirections), as well as relaxing the requirement that the electron (spatial) \norbitals be constrained by the symmetry of the external confining potential.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=14.5cm]{tnt_2005_fig2.eps}\\\\\n~~~~~\\\\\n\\caption{Various approximation levels for a field-free two-electron QD with \n$R_W=2.40$. (a) Electron density of the RHF solution, exhibiting circular \nsymmetry (due to the imposed symmetry restriction). The correlation energy \n$\\varepsilon_c = 2.94$ meV, is defined as the difference between the energy of \nthis state and the exact solution [shown in frame (e)]. (b1) and (b2) The two \noccupied orbitals (modulus square) of the symmetry-broken ``singlet'' sS-UHF \nsolution (b1), with the corresponding total electron density exhibiting \nnon-circular shape (b2). The energy of the sS-UHF solution shows a gain of \n44.3\\% of the correlation energy. (c) Electron density of the spin-projected \n(SP) singlet, showing broken spatial symmetry, but with an additional gain of \ncorrelation energy. (d) the spin-and-angular-momentum projected state (S\\&AMP) \nexhibiting restored circular symmetry with a 73.1\\% gain of the correlation \nenergy. The choice of parameters is: dielectric constant $\\kappa = 8$, parabolic\nconfinement $\\omega_0 = 5$ meV, and effective mass $m^* = 0.067m_e$. \nDistances are in nanometers and the densities in $10^{-4}$ nm$^{-2}$.\n}\n\\end{figure}\n\nResults obtained for various approximation levels for a two-electron QD with \n$B=0$ and $R_W=2.40$ (that is, in the Wigner-molecule regime) are displayed in \nFig.\\ 2. In these calculations \\cite{yyl9}, the spin projection was performed \nfollowing reference \\cite{lowd}, i.e., one constructs the wave function\n\\begin{equation}\n\\Psi_{\\rm SP}(s)={\\cal P}_{\\rm spin}(s) \\Psi_{\\rm UHF},\n\\label{psis}\n\\end{equation}\nwhere $\\Psi_{\\rm UHF}$ is the original symmetry-broken UHF determinant.\nIn Eq. (\\ref{psis}), the spin projection operator is given by\n\\begin{equation}\n{\\cal P}_{\\rm spin}(s) \\equiv \\prod_{s^\\prime \\neq s}\n\\frac{\\hat{S}^2 - s^\\prime(s^\\prime + 1) \\hbar^2}\n{[s(s+1) - s^\\prime(s^\\prime + 1)] \\hbar^2}~,\n\\label{prjp}\n\\end{equation}\nwhere the index $s^\\prime$ runs over the quantum numbers of $\\hat{S}^2$,\nwith $\\hat{S}$ being the total spin.\n\nThe angular momentum projector is given by\n\\begin{equation}  \n2\\pi {\\cal P}_L \\equiv \\int_0^{2\\pi} d\\gamma \\exp[-i\\gamma (\\hat{L}-L)],\n\\label{amp}\n\\end{equation} \nwhere $\\hat{L}=\\hat{l}_1+\\hat{l}_2$ is the total angular momentum \noperator. As seen from Eq.\\ (\\ref{amp}), application of the projection\noperator ${\\cal P}_L$ to the spin-restored state $\\Psi_{\\rm SP}(s)$\ncorresponds to a continuous configuration interaction (CCI) formalism. \n\nIn the following we focus on the ground state of the system, i.e., \n$L = 0$. The energy of the projected state is given by \n\\begin{equation}\nE_{\\rm PRJ}(L) = \\left. \\int_0^{2\\pi} h(\\gamma) e^{i\\gamma L} d\\gamma \\right\/ \n \\int_0^{2\\pi} n(\\gamma) e^{i\\gamma L}d\\gamma,\n\\label{eprj}\n\\end{equation}\nwith $h(\\gamma)=\\langle \\Psi_{\\rm SP}(s;0)|H|\\Psi_{\\rm SP}(s;\\gamma)\\rangle$\nand $n(\\gamma)=\\langle \\Psi_{\\rm SP}(s;0)|\\Psi_{\\rm SP}(s;\\gamma)\\rangle$,\nwhere $\\Psi_{\\rm SP}(s;\\gamma)$ is the spin-restored wave function \nrotated by an azimuthal angle $\\gamma$ and $H$ is the many-body hamiltonian.\nWe note that the UHF energies are simply given by $E_{\\rm UHF}=h(0)\/n(0)$.\n\nThe electron densities corresponding to the initial RHF approximation [shown in \nFig.\\ 2(a)] and the final spin-and-angular-momentum projection (S\\&AMP) [shown \nin Fig.\\ 2(d)], are circularly symmetric, while those corresponding to the two \nintermediate approximations, i.e., the sS-UHF and spin-projected (SP) solutions \n[Figs.\\ 2(b2) and 2(c), respectively] break the circular symmetry. This \nbehavior illustrates graphically the meaning of the term restoration of \nsymmetry, and the interpretation that the sS-UHF broken-symmetry solution refers\nto the intrinsic (rotating) frame of reference of the electron molecule. In \nlight of this discussion the final projected state is called a {\\it rotating \nWigner molecule}, or RWM.\n\n\\subsection{Electrons in circular quantum dots under high magnetic fields}\n\nTo illustrate the emergence of RWMs in parabolically confined QDs under high $B$,\nwe show in Fig.\\ 3 results obtained \\cite{yyl7} \nthrough the aforementioned two-step \ncomputational technique for $N=7$, 8, and 9 electrons, and compare them with the\nresults derived from exact diagonalization of the Hamiltonian [see Eq\\. \n(\\ref{mbhn})]. Systematic investigations of QDs under high $B$ revealed \nelectronic states of crystalline character. These states are found for \nparticular ``magic'' angular momentum values ($L$) that exhibit enhanced \nstability and are called cusp states. For a given value of $B$, one of these \n$L$'s corresponds to the global minimum, i.e., the ground state, and varying $B$ \ncauses the ground state and its angular momentum to change. The cusp states have\nbeen long recognized \\cite{laug1} as the finite-$N$ precursors of the fractional \nquantum Hall states in extended systems. In particular, the fractional fillings \n$\\nu$ (defined in the thermodynamic limit) are related to the magic angular \nmomenta of the finite-$N$ system as follows \\cite{girv}\n\\begin{equation}\n\\nu=\\frac{N(N-1)}{2L}.\n\\label{nu}\n\\end{equation}\n\nIn the literature of the fractional quantum Hall effect (FQHE), ever since the \ncelebrated paper \\cite{laug2} by Laughlin in 1983, the cusp states have been \nconsidered to be the antithesis of the Wigner crystal and to be described \naccurately by liquid-like wave functions, such as the Jastrow- Laughlin (JL) \n\\cite{laug2,laug3} and composite-fermion (CF) \\cite{jain,jain2} ones. This view,\nhowever, has been challenged recently \\cite{yyl4,yyl5} by the explicit \nderivation of trial wave functions for the cusp states that are associated with \na rotating Wigner molecule. As discussed elsewhere \\cite{yyl4,yyl5}, these \nparameter-free wave functions, which are by construction crystalline in \ncharacter, have been shown to provide a simpler and improved description of the \ncusp states, in particular for high angular momenta (corresponding to low \nfractional fillings).\n\nThe crystalline arrangements that were found \\cite{yyl4,yyl5,yyl6,yyl7} consist \nof concentric polygonal rings [see the conditional probabilities displayed in \nFig.\\ 3, with the reference (observation) point denoted by a filled dot]. These \nrings rotate independently of each other (see in particular the two cases shown \nfor $N=9$), with the electrons on each ring rotating coherently \\cite{yyl7}. The\nrotations stabilize the RWM relative to the static one -- namely, the projected \n(symmetry restored) states are lower in energy compared to the broken-symmetry \nones (the unrestricted Hartree-Fock solutions).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=12.0cm]{tnt_2005_fig3.ps}\n\\caption{Conditional probability distributions (CPDs) at high $B$, evaluated \nfor parabolic quantum dots through: (i) the two-step procedure of symmetry \nbreaking and subsequent restoration, resulting in rotating Wigner molecules \n(RWM) (shown in the left columns for given $N$), and (ii) exact diagonalization \n(EXD, shown in the right columns for given $N$). \nThe angular momentum values, and corresponding values of the \nfractional filling [see Eq\\. (\\ref{nu})], are given on the left. The optimal \npolygonal structure for a given $N$ is given by $(n_1,n_2)$ with $n_1+n_2=N$. \nFor $N= 7$, 8, and 9, these arrangements are (1,6), (1,7), and (2,7), \nrespectively. The reference point for the calculation of the CPD is denoted by a\nfilled dot. Note in particular the two CPDs shown for $N=9$, illustrating that \nfor a reference point located on the outer ring, the inner ring appears uniform,\nand vice versa for a reference point located on the inner ring (bottom row on\nthe right). These results illustrate that the rings rotate independently of each\nother. \n}\n\\end{figure}\n\n\\subsection{Symmetry breaking of trapped bosons}\n\nIn closing this section, we remark that the emergence of crystalline geometric \narrangements, discussed above for electrons confined in 2D quantum dots, appears\nto be a general phenomenon that is predicted \\cite{roma} to occur also for \ntrapped bosonic atomic systems (neutral or charged) when the interatomic \nrepulsion is tuned to exceed the characteristic energy of the harmonic trap. \nIndeed, application of the aforementioned two-step method, allowed the \nevaluation \\cite{roma} of solutions to the many-body Hamiltonians describing \nsuch bosonic systems, going beyond the mean-field (Gross-Pitaevskii) approach. \nThese highly-correlated trapped 2D states exhibit localization of the bosons \ninto polygonal-ringlike crystalline patterns, thus extending earlier work that \npredicted localization of strongly repelling 1D bosons (often referred to as the\nTonk-Girardeau regime \\cite{gira1,gira2}), to a higher dimension. \nIt is expected that these theoretical \nfindings will be the subject of experimental explorations, using methodologies \nthat have led recently to observations of localization transitions in 1D boson \nsystems \\cite{pare,weis}.\n\n\\section{Localization and entanglement in a two-electron elliptic \nquantum dot}\n\nAs discussed in the previous section, electron localization leading to formation\nof molecular-like structures [the aforementioned Wigner molecules] within a \n{\\it single circular\\\/} two-dimensional (2D) quantum dot at zero magnetic field \n($B$) has been theoretically predicted to occur \\cite{yyl1,yyl2,yyl3,yl1,mikh}, \nas the strength of the $e$-$e$ repulsive interaction relative to the zero-point \nenergy increases. Of particular interest is a two-electron $(2e)$ WM, in light \nof the current experimental effort \\cite{taru,eng} aiming at implementation of a\nspin-based \\cite{burk} solid-state quantum logic gate that employs two coupled \none-electron QDs (double dot). \n\nHere, we present an exact diagonalization (EXD) and an approximate\n(generalized Heitler-London, GHL) microscopic treatment for two electrons in a\n{\\it single\\\/} elliptic QD specified by the \nparameters of a recently investigated experimental device \\cite{marc}. While \nformation of Wigner molecules in circular QDs requires weak confinement, \nand thus large dots of lower densities (so that the interelectron repulsion \ndominates), we show that formation of such WMs is markedly enhanced in highly \ndeformed (e.g., elliptic) dots due to their lower symmetry.\nThe calculations provide a good description\nof the measured $J(B)$ curve (the singlet-triplet splitting) when screening \n\\cite{kou2,hall} due to the metal gates and leads is included (in addition to \nthe weakening of the effective inter-electron repulsion due to the\ndielectric constant of the semiconductor, GaAs). In particular, our results\nreproduce the salient experimental findings pertaining to the vanishing of \n$J(B)$ for a finite value of $B \\sim 1.3 $ T [associated with a change in sign \nof $J(B)$ indicating a singlet-triplet (ST) transition], \nas well as the flattening of the $J(B)$ curve after the ST crossing.\nThese properties, and in particular the latter one, are related \ndirectly to the formation of an electron molecular dimer and its effective \ndissociation for large magnetic fields. \nThe effective dissociation of the electron dimer is most naturally described \nthrough the GHL approximation, and it is fully supported by the more accurate,\nbut physically less transparent, EXD.\n\nOf special interest for quantum computing is the degree of entanglement \nexhibited by the two-electron molecule in its singlet state \\cite{burk}. \nHere, in relation to the microscopic calculations, we investigate two different \nmeasures of entanglement \\cite{note27}. The first, known as the concurrence \n(${\\cal C}$) for two {\\it indistinguishable\\\/} fermions \\cite{schl,loss}, \nhas been used in the analysis of the experiment in Ref.\\ \\cite{marc}\n(this measure is related to the operational cycle of a two-spin-qubit\nquantum logic gate \\cite{schl,loss}).\nThe second measure, referred to as the von Neumann entropy (${\\cal S}$) for \n{\\it indistinguishable\\\/} particles, has been developed in Ref.\\ \\cite{you}\nand used in Ref.\\ \\cite{zung}.\nWe show that the present {\\it wave-function-based\\\/} methods, in conjunction with\nthe knowledge of the dot shape and the $J(B)$ curve, enable theoretical \ndetermination of the degree of entanglement, in particular for the elliptic QD \nof Ref.\\ \\cite{marc}. The increase in the degree of entanglement (for both\nmeasures) with stronger magnetic fields correlates with the dissociation\nof the $2e$ molecule. This supports the experimental assertion \\cite{marc} that \ncotunneling spectroscopy can probe properties of the electronic wave function of\nthe QD, and not merely its low-energy spectrum. Our methodology can be \nstraightforwardly applied to other cases of strongly-interacting devices, e.g., \ndouble dots with strong interdot-tunneling. \n\n\\subsection{Microscopic treatment}\n\nThe Hamiltonian for two 2D interacting electrons is [see Eq.\\ (\\ref{mbhn})]\n\\begin{equation}\n{\\cal H} = H({\\bf r}_1)+H({\\bf r}_2)+e^2\/(\\kappa r_{12}),\n\\label{ham}\n\\end{equation}\nwhere the last term is the Coulomb repulsion, $\\kappa$ is the\ndielectric constant, and \n$r_{12} = |{\\bf r}_1 - {\\bf r}_2|$. $H({\\bf r})$ is the\nsingle-particle Hamiltonian for an electron in an external perpendicular\nmagnetic field ${\\bf B}$ and an appropriate confinement potential.\nWhen position-dependent screening is included, the last term in Eq.\\\n(\\ref{ham}) is modified by a function of $r_{12}$ (see below). \nFor an elliptic QD, the single-particle Hamiltonian is written as\n\\begin{equation}\nH({\\bf r}) = T + \\frac{1}{2} m^* (\\omega^2_{x} x^2 + \\omega^2_{y} y^2)\n    + \\frac{g^* \\mu_B}{\\hbar} {\\bf B \\cdot s},\n\\label{hsp}\n\\end{equation}\nwhere $T=({\\bf p}-e{\\bf A}\/c)^2\/2m^*$, with ${\\bf A}=0.5(-By,Bx,0)$ being the\nvector potential in the symmetric gauge. $m^*$ is the effective mass and\n${\\bf p}$ is the linear momentum of the electron. The second term is the\nexternal confining potential; the last term is the Zeeman interaction with \n$g^*$ being the effective $g$ factor, $\\mu_B$ the Bohr magneton, and ${\\bf s}$ \nthe spin of an individual electron. \n\nThe GHL method for solving the Hamiltoninian (\\ref{ham}) consists of two steps. \nIn the first step, we solve selfconsistently the ensuing \nunrestricted Hartree-Fock (UHF) equations allowing for lifting of the \ndouble-occupancy requirement (imposing this requirement gives the \n{\\it restricted\\\/} HF method, RHF).\nFor the $S_z=0$ solution, this step produces two single-electron \norbitals $u_{L,R}({\\bf r})$ that are localized left $(L)$ and right $(R)$ of the\ncenter of the QD [unlike the RHF method that gives a single doubly-occupied \nelliptic (and symmetric about the origin) orbital]. \nAt this step, the many-body wave function is a single Slater \ndeterminant $\\Psi_{\\text{UHF}} (1\\uparrow,2\\downarrow) \\equiv \n| u_L(1\\uparrow)u_R(2\\downarrow) \\rangle$ made out of the two occupied UHF \nspin-orbitals $u_L(1\\uparrow) \\equiv u_L({\\bf r}_1)\\alpha(1)$ and \n$u_R(2\\downarrow) \\equiv u_R({\\bf r}_2) \\beta(2)$, where \n$\\alpha (\\beta)$ denotes the up (down) [$\\uparrow (\\downarrow)$] spin. \nThis UHF determinant is an eigenfunction of the projection $S_z$ of the total \nspin $\\hat{S} = \\hat{s}_1 + \\hat{s}_2$, but not of $\\hat{S}^2$ (or the parity\nspace-reflection operator). \n\nIn the second step, we restore the broken parity and total-spin symmetries by \napplying to the UHF determinant the projection operator \\cite{yyl8,yl3} \n${\\cal P}_{\\rm spin}^{s,t}=1 \\mp \\varpi_{12}$, where the \noperator $\\varpi_{12}$ interchanges the spins of the two electrons\n[this is a special case of the operator given in Eq.\\ (\\ref{prjp})]; \nthe upper (minus) sign corresponds to the singlet. \nThe final result is a generalized Heitler-London (GHL) two-electron wave function\n$\\Psi^{s,t}_{\\text{GHL}} ({\\bf r}_1, {\\bf r}_2)$ for the ground-state singlet \n(index $s$) and first-excited triplet (index $t$), which uses\nthe UHF localized orbitals,\n\\begin{equation}\n\\Psi^{s,t}_{\\text{GHL}} ({\\bf r}_1, {\\bf r}_2) \\propto\n{\\bf (} u_L({\\bf r}_1) u_R({\\bf r}_2) \\pm u_L({\\bf r}_2) u_R({\\bf r}_1) {\\bf )}\n\\chi^{s,t},\n\\label{wfghl}\n\\end{equation}\nwhere $\\chi^{s,t} = (\\alpha(1) \\beta(2) \\mp \\alpha(2) \\beta(1))$ is the spin \nfunction for the 2$e$ singlet and triplet states.\nThe general formalism of the 2D UHF equations and of the subsequent restoration \nof broken spin symmetries can be found in Refs.\\ \\cite{yyl8,yyl9,yl1,yl3}.\n\nThe use of {\\it optimized\\\/} UHF orbitals in the GHL is suitable for treating \n{\\it single elongated\\\/} QDs. The GHL is equally applicable to double QDs with \narbitrary interdot-tunneling coupling \\cite{yyl8,yl3}. In contrast,\nthe Heitler-London (HL) treatment \\cite{hl} (known also as Valence bond), \nwhere non-optimized ``atomic'' orbitals of two isolated QDs are used, is \nappropriate only for the case of a double dot with small interdot-tunneling \ncoupling \\cite{burk}.\n\nThe orbitals $u_{L,R}({\\bf r})$ are expanded in a real Cartesian \nharmonic-oscillator basis, i.e.,\n\\begin{equation}\nu_{L,R}({\\bf r}) = \\sum_{j=1}^K C_j^{L,R} \\varphi_j ({\\bf r}),\n\\label{uexp}\n\\end{equation}\nwhere the index $j \\equiv (m,n)$ and $\\varphi_j ({\\bf r}) = X_m(x) Y_n(y)$,\nwith $X_m(Y_n)$ being the eigenfunctions of the one-dimensional oscillator in the\n$x$($y$) direction with frequency $\\omega_x$($\\omega_y$). The parity operator\n${\\cal P}$ yields ${\\cal P} X_m(x) = (-1)^m X_m(x)$, and similarly for $Y_n(y)$.\nThe expansion coefficients $C_j^{L,R}$ are real for $B=0$ and complex for finite\n$B$. In the calculations we use $K=79$, yielding convergent results.\n\nIn the EXD method, the many-body wave function is written as a linear\nsuperposition over the basis of non-interacting two-electron determinants, i.e.,\n\\begin{equation}\n\\Psi^{s,t}_{\\text{EXD}} ({\\bf r}_1, {\\bf r}_2) =\n\\sum_{i < j}^{2K} \\Omega_{ij}^{s,t} | \\psi(1;i) \\psi(2;j)\\rangle,\n\\label{wfexd}\n\\end{equation}\nwhere $\\psi(1;i) = \\varphi_i(1 \\uparrow)$ if $1 \\leq i \\leq K$ and\n$\\psi(1;i) = \\varphi_{i-K}(1 \\downarrow)$ if $K+1 \\leq i \\leq 2K$ [and\nsimilarly for $\\psi(2,j)$].\nThe total energies $E^{s,t}_{\\text{EXD}}$ and the coefficients\n$\\Omega_{ij}^{s,t}$ are obtained through a ``brute force'' diagonalization of\nthe matrix eigenvalue equation corresponding to the Hamiltonian in Eq.\\ \n(\\ref{ham}). The EXD wave function does not\nimmediately reveal any particular form, although, our calculations below\nshow that it can be approximated by a GHL wave function in the case of the\nelliptic dot under consideration. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=6.5cm]{tnt_2005_fig4.eps}   \n\\caption{\nThe singlet-triplet splitting $J=E^s-E^t$ as a function of the magnetic field\n$B$ for an elliptic QD with $\\hbar \\omega_x=1.2$ meV and $\\hbar \\omega_y=3.3$ meV\n(these values correspond to the device of Ref.\\ \\cite{marc}).\nSolid line: GHL (broken-symmetry UHF + restoration of symmetries) results\nwith a coordinate-independent screening ($\\kappa=22$).\nDashed line: EXD results with $\\kappa=12.9$ (GaAs), but including screening with\na coordinate dependence according to Ref.\\ \\cite{hall} and $d=18.0$ nm \n(see text). The rest of the material parameters used are: \n$m^*$(GaAs)$=0.067 m_e$, and $g^*=0$ (see text).\nThe experimental measurements \\cite{marc} are denoted by open squares.\nOur sign convention for $J$ is opposite to that in Ref.\\ \\cite{marc}.\n}\n\\end{figure}\n\nTo model the experimental elliptic QD device, we take, following Ref.\\ \n\\cite{marc}, $\\hbar \\omega_x=1.2$ meV and $\\hbar \\omega_y=3.3$ meV. \nThe effective mass of the electron is taken as $m^*=0.067 m_e$ (GaAs). Since \nthe experiment did not resolve the lifting of the triplet degeneracy caused by \nthe Zeeman term, we take $g^*=0$. Using the two step method, \nwe calculate the GHL singlet-triplet splitting \n$J_{\\text{GHL}}(B)=E^s_{\\text{GHL}}(B)-E^t_{\\text{GHL}}(B)$ \nas a function of the magnetic field in the range $0 \\leq B \\leq 2.5$ T.\nScreening of the $e$-$e$ interaction due to the metal gates and leads \nmust be considered in order to reproduce the experimental $J(B)$ \ncurve \\cite{note2}. This screening can be modeled, to first approximation, by a \nposition-independent adjustment of the dielectric constant \n$\\kappa$ \\cite{kyri}. Indeed, with $\\kappa=22.0$ (instead of the GaAs \ndielectric constant, i.e., $\\kappa = 12.9$), good agreement with\nthe experimental data is obtained [see Fig.\\ 4]. In particular, we note the \nsinglet-triplet crossing for $B \\approx 1.3$ T, and the flattening of the \n$J(B)$ curve beyond this crossing.\n\nWe have also explored, particularly in the context of the EXD treatment, \na position-dependent screening using the functional form,\n$(e^2\/\\kappa r_{12}) [1-(1+4d^2\/r_{12}^2)^{-1\/2}]$, \nproposed in Ref.\\ \\cite{hall}, with $d$ as a fitting parameter. \nThe $J_{\\text{EXD}}(B)$ result for $d=18.0$ nm is depicted in Fig.\\ 4 \n(dashed line), and it is in very good agreement with the experimental \nmeasurement.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=6.5cm]{tnt_2005_fig5.ps}   \n\\caption{ \nTotal electron densities (EDs) associated with the singlet state of the\nelliptic dot at $B=0$ and $B=2.5$ T.\n(a) The GHL densities. (b) The EXD densities.\nThe rest of the parameters and the screening of the Coulomb interaction \nare as in Fig.\\ 4.\nLengths in nm and densities in $10^{-4}$ nm$^{-2}$.\n}\n\\end{figure}\n\n\nThe singlet state electron densities from the GHL and the EXD treatments\nat $B=0$ and $B=2.5$ T are displayed in Fig.\\ 5. These densities illustrate the \ndissociation of the electron dimer with increasing magnetic field. \nThe asymptotic convergence (beyond the ST point) of the energies of the singlet \nand triplet states, i.e., [$J(B) \\rightarrow 0$ as $B \\rightarrow \\infty$] is a \nreflection of the dissociation of the 2$e$ molecule, since the ground-state \nenergy of two fully spatially separated electrons (zero overlap) does not \ndepend on the total spin \\cite{note67}. \n\n\\subsection{Measures of entanglement \\cite{note27}}\n\nTo calculate the concurrence ${\\cal C}$ \\cite{schl,loss}, one needs a \ndecomposition of the GHL wave function into a \nlinear superposition of {\\it orthogonal\\\/}\nSlater determinants. Thus one needs to expand the {\\it nonorthogonal\\\/} \n$u^{L,R}({\\bf r})$ orbitals as a superposition of two other {\\it orthogonal\\\/}\nones. To this effect, we write\n$u^{L,R}({\\bf r}) \\propto \\Phi^+({\\bf r}) \\pm  \\xi \\Phi^-({\\bf r})$,\nwhere $\\Phi^+({\\bf r})$ and $\\Phi^-({\\bf r})$ are the parity symmetric and \nantisymmetric (along the $x$-axis) components in their expansion given by \nEq.\\ (\\ref{uexp}). \nSubsequently, with the use of Eq.\\ (\\ref{wfghl}), the GHL \nsinglet can be rearranged as follows:\n\\begin{equation}\n\\Psi^{s}_{\\text{GHL}} \\propto\n| \\Phi^+(1\\uparrow) \\Phi^+(2\\downarrow) \\rangle - \n\\eta |\\Phi^-(1\\uparrow)\\Phi^-(2\\downarrow) \\rangle,\n\\label{rear}\n\\end{equation}\nwhere the so-called interaction parameter \\cite{loss}, $\\eta=\\xi^2$, is the \ncoefficient in front of the second determinant.\nKnowing $\\eta$ allows a direct evaluation of the concurrence of the singlet\nstate, since ${\\cal C}^s = 2\\eta\/(1+\\eta^2)$ \\cite{loss}. Note that \n$\\Phi^+({\\bf r})$ and $\\Phi^-({\\bf r})$ are properly normalized.\nIt is straightforward to show that $\\eta=(1-|S_{LR}|)\/(1+|S_{LR}|)$, where \n$S_{LR}$ (with $|S_{LR}| \\leq 1$) is the overlap of the original \n$u^{L,R}({\\bf r})$ orbitals.\n\nFor the GHL triplet, one obtains an expression independent of the \ninteraction parameter $\\eta$, i.e.,\n\\begin{equation}\n\\Psi^{t}_{\\text{GHL}} \\propto\n| \\Phi^+(1\\uparrow) \\Phi^-(2\\downarrow) \\rangle +\n|\\Phi^+(1\\downarrow)\\Phi^-(2\\uparrow) \\rangle,\n\\label{reart}\n\\end{equation}\nwhich is a maximally (${\\cal C}^t=1$) entangled state. Note that underlying the \nanalysis of the experiments in Ref.\\ \\cite{marc} is a {\\it conjecture\\\/} that \nwave functions of the form given in Eqs.\\ (\\ref{rear}) and (\\ref{reart}) \ndescribe the two electrons in the elliptic QD.\n\nFor the GHL singlet, using the overlaps of the left and right orbitals, we find\nthat starting with $\\eta=0.46$ $({\\cal C}^s=0.76)$ at $B=0$, the interaction\nparameter (singlet-state concurrence) increases monotonically to $\\eta=0.65$ \n$({\\cal C}^s=0.92)$ at $B=2.5$ T. At the intermediate value corresponding to the\nST transition ($B=1.3$ T), we find $\\eta=0.54$ $({\\cal C}^s=0.83)$ \n\\cite{note34}. Our $B=0$ theoretical results for \n$\\eta$ and ${\\cal C}^s$ are in remarkable agreement with \nthe experimental estimates \\cite{marc} of $\\eta=0.5 \\pm 0.1$ and \n${\\cal C}^s=0.8$, which were based solely on conductance measurements below the \nST transition (i.e., near $B=0$). \n\nTo compute the von Neumann entropy, one needs to bring both \nthe EXD and the GHL wave functions into a diagonal form (the socalled \n``canonical form'' \\cite{you,schl2}), i.e.,\n\\begin{equation}\n\\Psi^{s,t}_{\\text{EXD}} ({\\bf r}_1, {\\bf r}_2) =\n\\sum_{k=1}^M z^{s,t}_k | \\Phi(1;2k-1) \\Phi(2;2k) \\rangle,\n\\label{cano}\n\\end{equation}\nwith the $\\Phi(i)$'s being appropriate spin orbitals resulting from a unitary \ntransformation of the basis spin orbitals $\\psi(j)$'s [see Eq.\\ (\\ref{wfexd})]; \nonly terms with $z_k \\neq 0$ contribute. The upper bound $M$ can be \nsmaller (but not larger) than $K$ (the dimension of the \nsingle-particle basis); $M$ is referred to as the Slater rank.\nOne obtains the coefficients of the canonical expansion from the fact that \nthe $|z_k|^2$ are eigenvalues of the hermitian matrix $\\Omega^\\dagger \\Omega$\n[$\\Omega$, see Eq.\\ (\\ref{wfexd}), is antisymmetric]. The von Neumann \nentropy is given by ${\\cal S} = -\\sum_{k=1}^M |z_k|^2 \\log_2(|z_k|^2)$ with the\nnormalization $\\sum_{k=1}^M |z_k|^2 =1$.\nNote that the GHL wave functions in Eqs.\\ (\\ref{rear}) and (\\ref{reart}) \nare already in canonical form, which shows that they always have a Slater rank \nof $M=2$. One finds ${\\cal S}^s_{\\text{GHL}} =\n\\log_2(1+\\eta^2) - \\eta^2 \\log_2(\\eta^2)\/(1+\\eta^2)$, and\n${\\cal S}^t_{\\text{GHL}}=1$ for all $B$. For large $B$, the overlap \nbetween the two electrons of the dissociated dimer vanishes, and thus \n$\\eta \\rightarrow 1$ and ${\\cal S}^s_{\\text{GHL}} \\rightarrow 1$.\n\n\n\\begin{figure}[t]\n\\centering\\includegraphics[width=10cm]{tnt_2005_fig6.eps}\n\\caption{\nVon Neumann entropy for the singlet state of the elliptic dot as a function of \nthe magnetic field $B$. Solid line: GHL. Dashed line: EXD. \nThe rest of the parameters and the screening of the Coulomb interaction \nare as in Fig.\\ 4. On the left, we show histograms for the $|z_k|^2$\ncoefficients [see Eq.\\ (\\ref{cano})] of the singlet state at $B=1.3$ T, \nillustrating the dominance of two configurations. Note the small third\ncoefficient $|z_3|^2=0.023$ in the EXD case.\n}\n\\end{figure}\n\nSince the EXD singlet has obviously a Slater rank $M > 2$, the definition\nof concurrence is not applicable to it.  \nThe von Neumann entropy for the EXD singlet (${\\cal S}^s_{\\text{EXD}}$) is \ndisplayed in Fig.\\ 6, along with that (${\\cal S}^s_{\\text{GHL}}$) of the GHL \nsinglet. ${\\cal S}^s_{\\text{EXD}}$ and \n${\\cal S}^s_{\\text{GHL}}$ are rather close to each other for the entire\n$B$ range, and it is remarkable that both \nremain close to unity for large $B$, although the maximum allowed mathematical \nvalue is $\\log_2(K)$ [as aforementioned we use $K=79$, i.e., $\\log_2(79)=6.3$];\nthis maximal value applies for both the EXD and GHL approaches. The saturation \nof the entropy for large $B$ to a value close to unity reflects the dominant \n(and roughly equal at large $B$) weight of two configurations in the \ncanonical expansion [see Eq.\\ (\\ref{cano})] of the EXD wave function, which\nare related to the two terms ($M=2$) in the canonical expansion of the GHL\nsinglet [Eq.\\ (\\ref{rear})]. This is illustrated by the histograms of the\n$|z_k^s|^2$ coefficients for $B=1.3$ T in Fig.\\ 6 (left column).\nThese observations support the GHL approximation, which is\ncomputationally less demanding than the exact diagonalization, and can be used\neasily for larger $N$.\n\n\\section{Summary}\n\nWe discussed symmetry breaking in two-dimensional quantum dots resulting from\nstrong interelectron repulsion relative to the zero-point kinetic energy\nassociated with the confining potential. Such symmetry breaking leads to the\nemergence of crystalline arrangements of electrons in the dot. The so-called\nWigner molecules form already at field-free conditions. The appearance of\nrotating Wigner molecules in circular dots under high magnetic field, and their\nrelation to magic angular momenta and quantum-Hall-effect fractional fillings was\nalso discussed. \n\nFurthermore, we have shown, through exact and approximate microscopic\ntreatments, formation of an electron molecular dimer in an\nelliptic QD (Fig.\\ 5) for screened interelectron repulsion.\nThe formation and effective dissociation (in high magnetic fields) of the \nelectron dimer are reflected in the behavior of the computed singlet-triplet\nsplitting, $J(B)$, that agrees well (see Fig.\\ 4) with the measurements \n\\cite{marc}. Furthermore, we showed that, from a knowledge of the dot shape and \nof $J(B)$, theoretical analysis along the lines introduced here allows probing \nof the correlated ground-state wave function and determination of its degree of \nentanglement. This presents an alternative to the experimental study where\ndetermination of the concurrence utilized conductance data \\cite{marc}.\nWe have employed two measures of entanglement for {\\it\nindistinguishable fermions\\\/} (the concurrence and the von Neumann entropy) \nand have shown that their behavior correlates with the effective dissociation of\nthe electron dimer. Such information is of interest to the implementation of \nspin-based solid-state quantum logic gates.\n\nThis research is supported by the US D.O.E. (Grant No. FG05-86ER45234), and\nNSF (Grant No. DMR-0205328). We thank M. Pustilnik for comments on the\nmanuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:intro}\n\nA major impairment in orthogonal frequency-division multiple-access (OFDMA)\nnetworks is the remarkable sensitivity to timing errors and carrier\nfrequency offsets (CFOs) between the uplink signals and the base station\n(BS) local references. For this reason, the IEEE 802.16e-2005 standard for\nOFDMA-based wireless metropolitan area networks (WMANs) specifies a\nsynchronization procedure called Initial Ranging (IR) in which subscriber\nstations that intend to establish a link with the BS transmit pilot symbols\non dedicated subcarriers using specific ranging codes. Once the BS has\ndetected the presence of these pilots, it has to estimate some fundamental\nparameters of ranging subscriber stations (RSSs) such as timing errors, CFOs\nand power levels.\n\nInitial synchronization and power control in OFDMA was originally discussed\nin \\cite{Krinock2001} and \\cite{Minn2004} while similar solutions can be\nfound in \\cite{Mahmoud2006}-\\cite{Zhou2006}. A different IR approach has\nrecently been proposed in \\cite{Minn07}. Here, each RSS transmits pilot\nstreams over adjacent OFDMA blocks using orthogonal spreading codes. As long\nas channel variations are negligible over the ranging period, signals of\ndifferent RSSs can be easily separated at the BS as they remain orthogonal\nafter propagating through the channel. Timing information is eventually\nacquired in an iterative fashion by exploiting the autocorrelation\nproperties of the received samples induced by the use of the cyclic prefix\n(CP).\n\nAll the aforementioned schemes are derived under the assumption of perfect\nfrequency alignment between the received signals and the BS local reference.\nHowever, the occurrence of residual CFOs results into a loss of\northogonality among ranging codes and may compromise the estimation accuracy\nand detection capability of the IR process. Motivated by the above problem,\nin the present work we propose a novel ranging scheme for OFDMA networks\nwith increased robustness against frequency errors and lower computational\ncomplexity than the method in \\cite{Minn07}. To cope with the large number\nof parameters to be recovered, we adopt a three-step procedure. In the first\nstep the number of active codes is estimated by resorting to the minimum\ndescription length (MDL) principle \\cite{Wax85}. Then, the ESPRIT\n(Estimation of Signal Parameters by Rotational Invariance Techniques) \\cite%\n{Roy1989} algorithm is employed in the second and third steps to detect\nwhich codes are actually active and determine their corresponding timing\nerrors and CFOs.\n\n\n\\section{System description and signal model}\n\n\\subsection{System description}\n\nWe consider an OFDMA\\ system employing $N$ subcarriers with index set $%\n\\{0,1,\\ldots ,N-1\\}$. As in \\cite{Minn07}, we assume that a ranging\ntime-slot is composed by $M$ consecutive OFDMA\\ blocks where the $N$\navailable subcarriers are grouped into ranging subchannels and data\nsubchannels. The former are used by the active RSSs to complete their\nranging processes, while the latter are assigned to data subscriber stations\n(DSSs) for data transmission. We denote by $R$ the number of ranging\nsubchannels and assume that each of them is divided into $Q$ subbands. A\ngiven subband is composed of a set of $V$ adjacent subcarriers which is\ncalled a \\textit{tile}. The subcarrier indices of the $q$th tile $%\n(q=0,1,\\ldots ,Q-1)$ in the $r$th subchannel $(r=0,1,\\ldots ,R-1)$ are\ncollected into a set $\\mathcal{J}_{q}^{(r)}=\\{i_{q}^{(r)}+v\\}_{v=0}^{V-1}$,\nwhere the tile index $i_{q}^{(r)}$ can be chosen adaptively according to the\nactual channel conditions. The only constraint in the selection of $%\ni_{q}^{(r)}$ is that different tiles must be disjoint, i.e., $\\mathcal{J}%\n_{q_{1}}^{(r_{1})}\\cap \\mathcal{J}_{q_{2}}^{(r_{2})}=\\varnothing $ for $%\nq_{1}\\neq q_{2}$ or $r_{1}\\neq r_{2}$. The $r$th ranging subchannel is thus\ncomposed of $QV$ subcarriers with indices in the set $\\mathcal{J}^{(r)}=\\cup\n_{q=0}^{Q-1}{\\mathcal{J}_{q}^{(r)}}$, while a total of $N_{R}=QVR$ ranging\nsubcarriers is available in each OFDMA block.\n\nWe assume that each subchannel can be accessed by a maximum number of $K_{\\max }=\\min \\{V,M\\}-1$\nRSSs, which are separated by means of orthogonal codes in both the time and\nfrequency domains. The codes are selected in a pseudo-random fashion from a\npredefined set $\\{\\mathbf{C}_{0},\\mathbf{C}_{1},\\ldots ,\\mathbf{C}_{K_{\\max\n}-1}\\}$ with\n\\begin{equation}\n\\mathbf{[C}_{k}\\mathbf{]}_{v,m}=e^{j2\\pi k(\\frac{v}{V-1}+\\frac{m}{M-1})}\n\\label{eq1}\n\\end{equation}%\nwhere $v=0,1,\\ldots ,V-1$ counts the subcarriers within a tile and is used\nto perform spreading in the frequency domain, while $m=0,1,\\ldots ,M-1$\nis the block index by which spreading is done in the time domain across\nthe ranging time-slot. As in \\cite{Minn07}, we assume that different RSSs\nselect different codes. Also, we assume that a selected code is employed by\nthe corresponding RSS over all tiles in the considered subchannel. Without\nloss of generality, we concentrate on the $r$th subchannel and denote by $%\nK\\leq K_{\\max }$ the number of simultaneously active RSSs. To simplify the\nnotation, the subchannel index $^{(r)}$ is dropped henceforth.\n\nThe signal transmitted by the $k$th RSS propagates through a multipath\nchannel characterized by a channel impulse response (CIR) $\\mathbf{h}'%\n_{k}=[h'_{k}(0),h'_{k}(1),\\ldots ,h'_{k}(L-1)]^{T}$ of length $L$ (in sampling\nperiods). We denote by $\\theta _{k}$ the timing error of the \\textit{k}th\nRSS expressed in sampling intervals $T_{s}$, while $\\varepsilon _{k}$ is the\nfrequency offset normalized to the subcarrier spacing. As discussed in \\cite%\n{Morelli2004}, during IR the CFOs are only due to Doppler shifts and\/or\nto estimation errors and, in consequence, they are assumed to lie \\textit{%\nwithin a small fraction} of the subcarrier spacing. Timing offsets depend on\nthe distance of the RSSs from the BS and their maximum value is thus limited\nto the round trip delay from the cell boundary. In order to eliminate\ninterblock interference (IBI), we assume that during the ranging process the\nCP length comprises $N_{G}\\geq \\theta _{\\max }+L$ sampling periods, where $%\n\\theta _{\\max }$ is the maximum expected timing error. This assumption is\nnot restrictive as initialization blocks are usually preceded by long CPs in\nmany standardized OFDMA systems.\n\n\\subsection{System model}\n\nWe denote by $\\mathbf{X}_{m}(q)=[X_{m}(i_{q}),X_{m}(i_{q}+1),\\ldots\n,X_{m}(i_{q}+V-1)]^{T}$ the discrete Fourier transform (DFT) outputs\ncorresponding to the $q$th tile in the $m$th OFDMA block. Since DSSs have\nsuccessfully completed their IR processes, they are perfectly aligned to the\nBS references and their signals do not contribute to $\\mathbf{X}_{m}(q)$. In\ncontrast, the presence of uncompensated CFOs destroys orthogonality among\nranging signals and gives rise to interchannel interference (ICI). The\nlatter results in a disturbance term plus an attenuation of the useful\nsignal component. To simplify the analysis, in the ensuing discussion the\ndisturbance term is treated as a zero-mean Gaussian random variable while\nthe signal attenuation is considered as part of the channel impulse\nresponse. Under the above assumptions, we may write\n\\begin{equation}\nX_{m}(i_{q}+v)=\\sum\\limits_{k=1}^{K}\\mathbf{[C}_{\\ell _{k}}\\mathbf{]}_{v,m}{%\ne^{j{m\\omega _{k}N_{T}}}}H_{k}(\\theta _{k},{\\varepsilon _{k},}%\ni_{q}+v)+w_{m}(i_{q}+v)  \\label{eq2}\n\\end{equation}%\nwhere $\\omega _{k}=2\\pi \\varepsilon _{k}\/N$, $N_{T}=N+N_{G}$ denotes the\nduration of the cyclically extended block and $\\mathbf{C}_{\\ell _{k}}$ is\nthe code matrix selected by the \\textit{k}th RSS. The quantity $H_{k}(\\theta\n_{k},{\\varepsilon _{k},}n)$ is the $k$th \\textit{equivalent} channel\nfrequency response over the $n$th subcarrier and is given by\n\\begin{equation}\nH_{k}(\\theta _{k},{\\varepsilon _{k},}n)=\\gamma _{N}({\\varepsilon _{k})}%\nH_{k}^{\\prime }(n)e^{-j{2\\pi n\\theta _{k}}\/N}  \\label{eq3}\n\\end{equation}%\nwhere $H_{k}^{\\prime }(n)=\\sum_{\\ell =0}^{L-1}h_{k}^{\\prime }(\\ell )e^{-j{%\n2\\pi n\\ell }\/N}$ is the true channel frequency response, while\n\\begin{equation}\n\\gamma _{N}({\\varepsilon })=\\frac{\\sin (\\pi {\\varepsilon })}{N\\sin (\\pi {%\n\\varepsilon }\/N)}e^{j\\pi {\\varepsilon }(N-1)\/N}  \\label{eq5}\n\\end{equation}%\nis the attenuation factor induced by the CFO. The last term in (\\ref{eq2})\naccounts for background noise plus interference and is modeled as a circularly symmetric\ncomplex Gaussian random variable with zero-mean and variance $\\sigma\n_{w}^{2}=\\sigma _{n}^{2}+\\sigma _{ICI}^{2}$, where $\\sigma _{n}^{2}$ and $%\n\\sigma _{ICI}^{2}$ are the average noise and ICI powers, respectively. From (%\n\\ref{eq3}) we see that $\\theta _{k}$ appears only as a phase shift across\nthe DFT outputs. The reason is that the CP duration is longer than the\nmaximum expected propagation delay.\n\nTo proceed further, we assume that the tile width is much smaller than the\nchannel coherence bandwidth. In this case, the channel response is nearly\nflat over each tile and we may reasonably replace the quantities $%\n\\{H_{k}^{\\prime }(i_{q}+v)\\}_{v=0}^{V-1}$ with an average frequency response\n\\begin{equation}\n\\overline{H}_{k}^{\\prime }(q)=\\frac{1}{V}\\sum_{v=0}^{V-1}H_{k}^{\\prime\n}(i_{q}+v).  \\label{eq6}\n\\end{equation}%\nSubstituting (\\ref{eq1}) and (\\ref{eq3}) into (\\ref{eq2}) and bearing in\nmind (\\ref{eq6}), yields\n\\begin{equation}\nX_{m}(i_{q}+v)=\\sum\\limits_{k=1}^{K}{e^{j2\\pi ({m\\xi _{k}+v\\eta }_{k}{)}}}%\nS_{k}(q)+w_{m}(i_{q}+v)  \\label{eq7}\n\\end{equation}%\nwhere $S_{k}(q)=\\gamma _{N}({\\varepsilon _{k})}\\overline{H}_{k}^{\\prime\n}(q)e^{-j{2\\pi i}_{q}{\\theta _{k}}\/N}$ and we have defined the quantities\n\\begin{equation}\n\\xi _{k}=\\frac{\\ell _{k}}{M-1}+\\frac{\\varepsilon _{k}N_{T}}{N}  \\label{eq8}\n\\end{equation}\nand\n\\begin{equation}\n\\eta _{k}=\\frac{\\ell _{k}}{V-1}-\\frac{\\theta _{k}}{N}  \\label{eq9}\n\\end{equation}%\nwhich are referred to as the \\textit{effective }CFOs and timing errors,\nrespectively.\n\nIn the following sections we show how the DFT outputs $\\{X_{m}(i_{q}+v)\\}$\ncan be exploited to identify the active codes and to estimate the\ncorresponding timing errors and CFOs.\n\n\\section{ESPRIT-based estimation}\n\n\\subsection{Determination of the number of active codes}\n\nThe first problem to solve is to determine the number $K$ of active codes\nover the considered ranging subchannel. For this purpose, we collect the $%\n(i_{q}+v)$th DFT outputs across all ranging blocks into an \\textit{M}%\n-dimensional vector $\\mathbf{Y}(i_{q,v})=[X_{0}(i_{q}+v),X_{1}(i_{q}+v),%\n\\ldots ,X_{M-1}(i_{q}+v)]^{T}$ given by\n\\begin{equation}\n\\mathbf{Y}(i_{q,v})=\\sum_{k=1}^{K}e^{j2\\pi v\\eta _{k}}S_{k}(q)\\mathbf{e}%\n_{M}(\\xi _{k})+\\mathbf{w}(i_{q,v})  \\label{eq30}\n\\end{equation}%\nwhere $\\mathbf{w}(i_{q,v})=[w_{0}(i_{q}+v),w_{1}(i_{q}+v),\\ldots\n,w_{M-1}(i_{q}+v)]^{T}$ is Gaussian distributed with zero mean and\ncovariance matrix $\\sigma _{w}^{2}\\mathbf{I}_{M}$ while $\\mathbf{e}_{M}(\\xi\n)=[1,e^{j2\\pi \\xi },e^{j4\\pi \\xi },\\ldots ,e^{j2\\pi (M-1)\\xi }]^{T}$.\n\nFrom the above equation, we observe that $\\mathbf{Y}(i_{q,v})$\nhas the same structure as measurements of multiple uncorrelated sources from\nan array of sensors. Hence, an estimate of $K$ can be obtained by performing\nan eigendecomposition (EVD) of the correlation matrix $\\mathbf{R}_{Y}=%\n\\mathrm{E}\\{\\mathbf{Y}(i_{q,v})\\mathbf{Y}^{H}(i_{q,v})\\}$. In practice,\nhowever, $\\mathbf{R}_{Y}$ is not available at the receiver and must be\nreplaced by some suitable estimate. One popular strategy to get an estimate\nof $\\mathbf{R}_{Y}$ is based on the forward-backward (FB) principle.\nFollowing this approach, $\\mathbf{R}_{Y}$ is replaced by\n$\n\\hat{\\mathbf{R}}_{Y}=\\frac{1}{2}(\\tilde{\\mathbf{R}}_{Y}+\\mathbf{J}\\tilde{%\n\\mathbf{R}}_{Y}^{T}\\mathbf{J})$, where $\\tilde{\\mathbf{R}}_{Y}$ is the sample correlation matrix\n\\begin{equation}\n\\tilde{\\mathbf{R}}_{Y}=\\frac{1}{QV}\\sum_{v=0}^{V-1}\\sum_{q=0}^{Q-1}\\mathbf{Y}%\n(i_{q,v})\\mathbf{Y}^{H}(i_{q,v})  \\label{eq31}\n\\end{equation}%\nwhile $\\mathbf{J}$ is the exchange matrix with 1's on the anti-diagonal and\n0's elsewhere. Arranging the eigenvalues $\\hat{\\lambda}_{1}\\geq \\hat{\\lambda}%\n_{2}\\geq \\cdots \\geq \\hat{\\lambda}_{M}$ of $\\hat{\\mathbf{R}}_{Y}$ in\nnon-increasing order, we can find an estimate $\\hat{K}$ of the number of\nactive codes by applying the MDL information-theoretic criterion. This\namounts to looking for the minimum of the following objective function \\cite%\n{Wax85}\n\\begin{equation}\n\\mathcal{F(}\\tilde{K})=\\frac{1}{2}\\tilde{K}(2V-\\tilde{K})\\ln (MQ)-MQ(V-%\n\\tilde{K})\\ln \\rho (\\tilde{K})  \\label{eq22}\n\\end{equation}%\nwhere $\\rho (\\tilde{K})$ is the ratio between the geometric and arithmetic\nmeans of $\\{\\hat{\\lambda}_{\\tilde{K}+1},\\hat{\\lambda}_{\\tilde{K}+2},\\ldots ,%\n\\hat{\\lambda}_{M}\\}$.\n\n\\subsection{Frequency estimation}\n\nFor simplicity, we assume that the number of active codes has been perfectly\nestimated. An estimate of ${\\boldsymbol{\\xi }}{=[}\\xi _{1},\\xi\n_{2},\\ldots ,\\xi _{K}\\mathbf{]}^{T}$ can be found by applying the ESPRIT\nalgorithm to the model (\\ref{eq30}). To elaborate on this, we arrange the\neigenvectors of $\\hat{\\mathbf{R}}_{Y}$ associated to the $K$ largest\neigenvalues $\\hat{\\lambda}_{1}\\geq \\hat{\\lambda}_{2}\\geq \\cdots \\geq \\hat{%\n\\lambda}_{K}$ into an $M\\times K$ matrix $\\mathbf{Z}=[\\mathbf{z}_{1}~\\mathbf{%\nz}_{2}~\\cdots \\mathbf{z}_{K}]$. Next, we consider the matrices $\\mathbf{Z}%\n_{1}$ and $\\mathbf{Z}_{2}$ that are obtained by collecting the first $M-1$\nrows and the last $M-1$ rows of $\\mathbf{Z}$, respectively. The entries of ${%\n\\boldsymbol{\\xi }}$ are finally estimated in a decoupled fashion as\n\\begin{equation}\n\\hat{\\xi}_{k}=\\frac{1}{2\\pi }\\arg \\{\\rho _{y}(k)\\}\\text{ \\ \\ \\ \\ }%\nk=1,2,\\ldots ,K  \\label{eq25}\n\\end{equation}%\nwhere $\\{\\rho _{y}(1),\\rho _{y}(2),\\ldots ,\\rho _{y}(K)\\}$ are the\neigenvalues of\n\\begin{equation}\n\\mathbf{Z}_{Y}=(\\mathbf{Z}_{1}^{H}\\mathbf{Z}_{1})^{-1}\\mathbf{Z}_{1}^{H}%\n\\mathbf{Z}_{2}  \\label{eq26}\n\\end{equation}%\nand $\\arg \\{\\rho _{y}(k)\\}$ denotes the phase angle of $\\rho _{y}(k)$ taking\nvalues in the interval $[-\\pi ,\\pi )$.\n\nAfter computing estimates of the effective CFOs through (\\ref{eq25}), the\nproblem arises of matching each $\\hat{\\xi}_{k}$ to the corresponding code $%\n\\mathbf{C}_{\\ell _{k}}$. This amounts to finding an estimate of $\\ell _{k}$\nstarting from $\\hat{\\xi}_{k}$. For this purpose, we denote by $\\left\\vert\n\\varepsilon _{\\max }\\right\\vert $ the magnitude of the maximum expected CFO\nand observe from (\\ref{eq8}) that $(M-1)\\xi _{k}$ belongs to the interval $%\nI_{\\ell _{k}}=[\\ell _{k}-\\beta ;\\ell _{k}+\\beta ]$, with $\\beta =\\left\\vert\n\\varepsilon _{\\max }\\right\\vert N_{T}(M-1)\/N$. It follows that the effective\nCFOs can be univocally mapped to their corresponding codes as long as $\\beta\n<1\/2$ since only in that case intervals $\\{I_{\\ell _{k}}\\}_{k=1}^{K}$ are\ndisjoint. The acquisition range of the frequency estimator is thus limited\nto $\\left\\vert \\varepsilon _{\\max }\\right\\vert <N\/(2N_{T}(M-1))$ and an estimate\nof the pair $(\\ell _{k},\\varepsilon _{k})$ is computed as\n\\begin{equation}\n\\hat{\\ell}_{k}=\\text{round}\\left((M-1)\\hat{\\xi}_{k}\\right)  \\label{eq35}\n\\end{equation}%\nand\n\\begin{equation}\n\\hat{\\varepsilon}_{k}=\\frac{N}{N_{T}}\\left(\\hat{\\xi}_{k}-\\frac{\\hat{\\ell}_{k}}{M-1}\\right).\n\\label{eq36}\n\\end{equation}%\nIt is worth noting that the $\\arg \\{\\cdot \\}$ function in (\\ref{eq25}) has\nan inherent ambiguity of multiples of $2\\pi $, which translates into a\ncorresponding ambiguity of the quantity $\\hat{\\ell}_{k}$ by multiples of $M-1$%\n. Hence, recalling that $\\ell _{k}\\in \\{0,1,\\ldots ,K_{\\max }-1\\}$ with $%\nK_{\\max } < M$, a refined estimate of $\\ell _{k}$ can be found as\n\\begin{equation}\n\\hat{\\ell}_{k}^{(F)}=[\\hat{\\ell}_{k}]_{M-1}  \\label{eq29}\n\\end{equation}%\nwhere $[x]_{M-1}$ is the value of $x$ reduced to the interval $[0,M-2]$. In\nthe sequel, we refer to (\\ref{eq36}) as the ESPRIT-based frequency estimator\n(EFE).\n\n\\subsection{Timing estimation}\n\nWe call $\\mathbf{X}_{m}(q)=[X_{m}(i_{q}),X_{m}(i_{q}+1),\\ldots\n,X_{m}(i_{q}+V-1)]^{T}$ the $V$-dimensional vector of the DFT outputs\ncorresponding to the \\textit{q}th tile in the \\textit{m}th OFDMA\\ block.\nThen, from (\\ref{eq7}) we have\n\\begin{equation}\n\\mathbf{X}_{m}(q)=\\sum\\limits_{k=1}^{K}{e^{j2\\pi {m\\xi _{k}}}}S_{k}(q)%\n\\mathbf{e}_{V}(\\eta _{k})+\\mathbf{w}_{m}(q)  \\label{eq18}\n\\end{equation}%\nwhere $\\mathbf{w}_{m}(q)=[w_{m}(i_{q}),w_{m}(i_{q}+1),\\ldots\n,w_{m}(i_{q}+V-1)]^{T}$ is Gaussian distributed with zero mean and\ncovariance matrix $\\sigma _{w}^{2}\\mathbf{I}_{V}$ while $\\mathbf{e}_{V}(\\eta\n)=[1,e^{j2\\pi \\eta },e^{j4\\pi \\eta },\\ldots ,e^{j2\\pi (V-1)\\eta }]^{T}$.\nSince $\\mathbf{X}_{m}(q)$ is a superposition of complex sinusoidal signals\nwith random amplitudes embedded in white Gaussian noise, an estimate of ${%\n\\boldsymbol{\\eta }}{=[}\\eta _{1},\\eta _{2},\\ldots ,\\eta _{K}\\mathbf{]}^{T}$\ncan still be obtained by resorting to the ESPRIT algorithm. Following the\nprevious steps, we first compute $\\hat{\\mathbf{R}}_{X}=\\frac{1}{2}(\\tilde{%\n\\mathbf{R}}_{X}+\\mathbf{J}\\tilde{\\mathbf{R}}_{X}^{T}\\mathbf{J})$ with\n\\begin{equation}\n\\tilde{\\mathbf{R}}_{X}=\\frac{1}{MQ}\\sum_{m=0}^{M-1}\\sum_{q=0}^{Q-1}\\mathbf{X}%\n_{m}(q)\\mathbf{X}_{m}^{H}(q).  \\label{eq32}\n\\end{equation}%\nThen, we define a $V\\times K$ matrix $\\mathbf{U}=[\\mathbf{u}_{1}~\\mathbf{u}%\n_{2}~\\cdots \\mathbf{u}_{K}]$ whose columns are the eigenvectors of $\\hat{%\n\\mathbf{R}}_{X}$ associated to the $K$ largest eigenvalues. The effective\ntiming errors are eventually estimated as\n\\begin{equation}\n\\hat{\\eta}_{k}=\\frac{1}{2\\pi }\\arg \\{\\rho _{x}(k)\\},\\text{ \\ \\ \\ \\ }%\nk=1,2,\\ldots ,K  \\label{eq33}\n\\end{equation}%\nwhere $\\{\\rho _{x}(1),\\rho _{x}(2),\\ldots ,\\rho _{x}(K)\\}$ are the\neigenvalues of\n\\begin{equation}\n\\mathbf{U}_{X}=(\\mathbf{U}_{1}^{H}\\mathbf{U}_{1})^{-1}\\mathbf{U}_{1}^{H}%\n\\mathbf{U}_{2}  \\label{eq34}\n\\end{equation}%\nwhile the matrices $\\mathbf{U}_{1}$ and $\\mathbf{U}_{2}$ are obtained by\ncollecting the first $V-1$ rows and the last $V-1$ rows of $\\mathbf{U}$,\nrespectively.\n\nThe quantities $\\{\\hat{\\eta}_{k}\\}_{k=1}^{K}$ are eventually used to find\nestimates $(\\hat{\\ell}_{k},\\hat{\\theta}_{k})$ of the associated ranging code\nand timing error. To accomplish this task, we let $\\alpha =\\theta _{\\max\n}(V-1)\/(2N)$. Then, recalling that $0\\leq \\theta _{k}\\leq \\theta _{\\max }$, from\n(\\ref{eq9}) we see that $(V-1)\\eta _{k}+\\alpha $ falls into the range $I_{\\ell\n_{k}}=[\\ell _{k}-\\alpha ;\\ell _{k}+\\alpha ]$. If $\\theta _{\\max }<N\/(V-1)$, the\nquantity $\\alpha $ is smaller than 1\/2 and, in consequence, intervals $%\n\\{I_{\\ell _{k}}\\}_{k=1}^{K}$ are disjoint. In this case, there is only one\npair $(\\ell _{k},\\theta _{k})$ that results into a given $\\eta _{k}$ and an\nestimate of $(\\ell _{k},\\theta _{k})$ is found as\n\\begin{equation}\n\\hat{\\ell}_{k}=\\text{round}\\left((V-1)\\hat{\\eta}_{k}+\\alpha \\right)  \\label{eq27}\n\\end{equation}%\nand\n\\begin{equation}\n\\hat{\\theta}_{k}=N\\left(\\frac{\\hat{\\ell}_{k}}{V-1}-\\hat{\\eta}_{k}\\right).  \\label{eq28}\n\\end{equation}%\nAs done in Sect. 3.2, a refined estimate of $\\ell _{k}$ is obtained in the\nform\n\n\\begin{equation}\n\\hat{\\ell}_{k}^{(f)}=[\\hat{\\ell}_{k}]_{V-1}.  \\label{eq37}\n\\end{equation}%\nIn the sequel, we refer to (\\ref{eq28}) as the ESPRIT-based timing estimator\n(ETE).\n\n\\subsection{Code detection}\n\nFrom (\\ref{eq29}) and (\\ref{eq37}) we see that two distinct estimates\\ $\\hat{%\n\\ell}_{k}^{(F)}$ and $\\hat{\\ell}_{k}^{(f)}$ are available at the receiver\nfor each code index $\\ell _{k}$. These estimates are now used to decide\nwhich codes are actually active in the considered ranging subchannel. For\nthis purpose, we define two sets $I^{(f)}=\\{\\hat{\\ell}_{k}^{(f)}\\}_{k=1}^{K}$\nand $I^{(F)}=\\{\\hat{\\ell}_{k}^{(F)}\\}_{k=1}^{K}$ and observe that, in the\nabsence of any detection error, it should be $I^{(f)}=I^{(F)}=\\{\\ell\n_{k}\\}_{k=1}^{K}$. Hence, for any code matrix $\\mathbf{C}_{m}$\nwe suggest the following detection strategy\n\\begin{equation}\n\\begin{array}{c}\n\\text{if }m\\in I^{(f)}\\cap I^{(F)}\\Longrightarrow \\mathbf{C}_{m}\\text{ is\ndeclared \\textit{detected }} \\\\\n\\text{if }m\\notin I^{(f)}\\cap I^{(F)}\\Longrightarrow \\mathbf{C}_{m}\\text{ is\ndeclared \\textit{undetected}}.%\n\\end{array}\n\\label{eq37bis}\n\\end{equation}%\nIn the downlink response message, the BS will indicate only\nthe detected codes while undetected RSSs must restart their ranging process.\nIn the sequel, we refer to (\\ref{eq37bis}) as the ESPRIT-based code detector\n(ECD).\n\n\\section{Numerical results}\n\nThe investigated system has a total of $N=1024$ subcarriers over an uplink\nbandwidth of 3 MHz. The sampling period is $T_{s}=0.33$ $\\mu s$,\ncorresponding to a subcarrier distance of $1\/(NT_{s})=2960$ Hz. We assume\nthat $R=4$ subchannels are available for IR. Each subchannel is divided into\n$Q=16$ tiles uniformly spaced over the signal spectrum at a distance of $%\nN\/Q=64$ subcarriers. The number of subcarriers in any tile is $V=4$ while $%\nM=4$. The discrete-time CIRs have $L=12$ taps which are modeled as\nindependent and circularly symmetric Gaussian random variables with zero\nmeans and an exponential power delay profile, i.e., $E\\{\\left\\vert\nh_{k}(\\ell )\\right\\vert ^{2}\\}=\\sigma _{h}^{2}\\cdot \\exp (-\\ell \/12)$ for $%\n\\ell =0,1,\\ldots ,11$, where $\\sigma _{h}^{2}$ is chosen such that $%\nE\\{\\left\\Vert \\mathbf{h}_{k}\\right\\Vert ^{2}\\}=1$. Channels of different\nusers are statistically independent of each other and are kept fixed over an\nentire time-slot. We consider a maximum propagation delay of $\\theta _{\\max\n}=204$ sampling periods. Ranging blocks are preceded by a CP of length $%\nN_{G}=256$. The normalized CFOs are uniformly distributed over the interval $%\n[-\\Omega ,\\Omega ]$ and vary at each run. Recalling that the estimation\nrange of EFE is $\\left\\vert \\varepsilon _{k}\\right\\vert <N\/(2N_{T}(M-1))$, we\nset $\\Omega \\leq 0.1$.\n\n\\begin{figure}[htb]\n\\centering\n\\centerline{\\epsfig{figure=picture1.eps,width=7.4cm}}\n\\caption{$P_{f}$ vs. SNR for $K$ = 3 when $\\Omega$ is 0.05 or 0.1.}\n\\label{fig:res}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\centering\n\\centerline{\\epsfig{figure=picture5.eps,width=7.4cm}}\n\\caption{RMSE vs. SNR for $K$ = 2 or 3 when $\\Omega$ is 0.05 or 0.1.}\n\\label{fig:res}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\centerline{\\epsfig{figure=picture7.eps,width=7.4cm}}\n\\caption{$P(\\protect\\epsilon)$ vs. SNR for $K$ = 2 or 3 when $\\Omega$ is\n0.05 or 0.1.}\n\\label{fig:res}\n\\end{figure}\nWe begin by investigating the performance of ECD in terms of probability of making an incorrect detection, say\n$P_{f}$. Fig. 1 illustrates $P_{f}$ as a function of $%\n\\mathrm{SNR}=1\/\\sigma _{n}^{2}$. The number of active RSSs is 3 while\nthe maximum CFO is either $\\Omega =0.05$ or 0.1. Comparisons are made with\nthe ranging scheme proposed by Fu, Li and Minn (FLM) in \\cite{Minn07}. The\nresults of Fig. 1 indicate that ECD performs remarkably better than FLM.\n\nFig. 2 illustrates the root mean-square error (RMSE) of the frequency\nestimates obtained with EFE vs. SNR for $K=2$ or 3 and $\\Omega =0.05$ or\n0.1. We see that the accuracy of EFE is satisfactory at SNR values\nof practical interest. Moreover, EFE has virtually the same performance\nas $K$ or $\\Omega$ increase.\n\nThe performance of the timing estimators is measured in terms of probability\nof making a timing error, say $P(\\epsilon )$, as defined in \\cite%\n{Morelli2004}. An error event is declared to occur whenever the estimate $%\n\\hat{\\theta}_{k}$ gives rise to IBI during the data section of the frame. This\nis tantamount to saying that the timing\nerror $\\hat{\\theta}_{k}-\\theta _{k}+(-N_{G,D}+L)\/2$ is larger than zero or\nsmaller than $-N_{G,D}+L-1$, where $N_{G,D}$ is the CP length during the data transmission phase. In the sequel, we set $N_{G,D}=32$.\nFig. 3 illustrates $P(\\epsilon )$ vs. SNR as\nobtained with ETE and FLM. The number of active codes is $K=2$ or 3 while $%\n\\Omega =0.05$ or 0.1. We see that ETE provides much better results than FLM.\n\n\\section{Conclusions}\n\nWe have presented a new ranging method for OFDMA systems in which uplink\nsignals arriving at the BS are impaired by frequency errors in addition to\ntiming misalignments. The synchronization parameters of all ranging users\nare estimated with affordable complexity through an ESPRIT-based approach.\nCompared to previous works, the proposed scheme exhibits increased\nrobustness against residual frequency errors and can cope with situations\nwhere the CFOs are as large as 10\\% of the subcarrier spacing.\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOverlapping speech is quite common for many scenarios, as the meetings recorded in the AMI corpus \\cite{McCowan2005_AMIMeetingCorpus} show overlap in about \\SIrange{5}{10}{\\%} of the time, and in informal situations, as recorded in the CHiME-5 \\cite{Barker2018_FifthCHiMESpeech} challenge data, it can exceed \\SI{20}{\\%}.\nA typical task for analysis of such recordings is \\gls{ASR}.\nIn the case of overlapping speech, this is called multi-talker speech recognition where the speech of multiple concurrently active talkers is to be recognized.\nConventional speech recognizers are limited to handling a single talker at a time which makes them inapplicable in those scenarios.\n\nMany efforts have already been put into the field of multi-talker speech recognition \\cite{Seki2018_PurelyEndtoEndSystem,Chang2019_EndtoendMonauralMultispeaker,Kanda2020_SerializedOutputTraining,Settle2018_EndtoendMultispeakerSpeech,Menne2019_AnalysisDeepClustering,vonNeumann2020_EndtoEndTrainingTime,Yu2017_RecognizingMultiTalkerSpeech,Qian2018_SinglechannelMultitalkerSpeech,Bahmaninezhad2019_ComprehensiveStudySpeecha}.\nThe approaches can basically be divided into monolithic and cascade systems.\nThe monolithic ones form one large neural network that is optimized as a whole, e.g., by extending a single-talker CTC\/attention ASR system so that its encoder outputs one latent representation for each speaker in the mixture and one or multiple attention decoders reconstruct one transcription for each representation \\cite{Seki2018_PurelyEndtoEndSystem,Chang2019_EndtoendMonauralMultispeaker}.\nAnother monolithic approach called Serialized Output Training modifies the target label sequence of a single-speaker recognizer to output the transcriptions for all talkers serially delimited by a special \\enquote{speaker change} token \\cite{Kanda2020_SerializedOutputTraining}.\nThese systems have the disadvantage that they do not provide interpretable signals as, e.g., separated speech signals.\n\n\\begin{figure}[t]\n    \\centering\n    \\include{tikz\/architecture}\n    \\vspace{-\\baselineskip}\n    \\caption{Example of the proposed iterative multi-speaker speech recognition system processing a three-speaker mixture.}\n    \\label{fig:architecture}\n   \n\\end{figure}\n\nThe cascade systems use source separation techniques followed by single-talker speech recognizers.\nThese systems have the advantage that intermediate signals are interpretable and individual system parts can be trained and tested on their own.\nAs separation front-ends, \\gls{DPCL} \\cite{Isik2016_SingleChannelMultiSpeakerSeparation}, \\gls{PIT} and the currently most promising \\gls{TasNet} architecture \\cite{Luo2018_TaSNetTimeDomainAudio} have been combined with different speech recognizers \\cite{Settle2018_EndtoendMultispeakerSpeech,Menne2019_AnalysisDeepClustering,Yu2017_RecognizingMultiTalkerSpeech,Isik2016_SingleChannelMultiSpeakerSeparation,Bahmaninezhad2019_ComprehensiveStudySpeecha,vonNeumann2020_EndtoEndTrainingTime}.\nThe experiments in general suggest that having both, dedicated parts for the separation and the recognition part, and joint end-to-end fine-tuning is beneficial.\n\nMost separation and multi-talker ASR approaches assume that the number of talkers is known \\cite{Kolbaek2017_MultitalkerSpeechSeparationa,Luo2018_TaSNetTimeDomainAudio,Luo2020_DualPathRNNEfficient,Seki2018_PurelyEndtoEndSystem,Chang2019_EndtoendMonauralMultispeaker,Settle2018_EndtoendMultispeakerSpeech,Qian2018_SinglechannelMultitalkerSpeech}, although this is not the case in realistic situations.\nSource number counting has been combined with source separation in iterative speech extraction  \\cite{Kinoshita2018_ListeningEachSpeaker,Takahashi2019_RecursiveSpeechSeparation} and model selection schemes \\cite{Nachmani2020_VoiceSeparationUnknown}, but not yet with speech recognition.\n\nWe propose the first jointly optimized multi-talker ASR system for an unknown number of speakers by combining source separation and number counting techniques from \\cite{Luo2020_DualPathRNNEfficient,Kinoshita2018_ListeningEachSpeaker,Takahashi2019_RecursiveSpeechSeparation} with a single-speaker CTC\/attention speech recognizer \\cite{Watanabe2017_HybridCTCAttention,Kim2017_JointCTCattentionBased,Watanabe2018_EspnetEndtoendSpeech,Xiao2018_HybridCTCattentionBased}.\nTo do that, we first investigate a \\gls{DPRNN}-TasNet separator for fixed numbers of speakers as a front-end for ASR.\nWe jointly fine-tune the \\gls{DPRNN} with a pre-trained ASR system, which was shown to be effective in other scenarios \\cite{Heymann2017_BeamnetEndtoendTraining} and multi-talker ASR \\cite{vonNeumann2020_EndtoEndTrainingTime}, to optimize the overall model performance.\nBy doing so, we achieve new state-of-the-art performance of \\SI{7.5}{\\%} \\gls{WER}.\nWe then integrate the \\gls{DPRNN} into the \\gls{OR-PIT} architecture and extend it with elegant mechanisms for source number counting.\nThe counting mechanisms show a promising performance.\nFinally, we combine the \\gls{OR-PIT} with an ASR system to form our new multi-speaker ASR system for unknown numbers of speakers.\nExperiments show that our system generalizes to a larger number of speakers than it saw during training.\n\n\\section{Source separation and counting}\nIn the following two subsections we first introduce the baseline approach to source separation, where the maximum number of talkers is assumed to be known.\nThen, we generalize it to separate an a priori unknown number of concurrent talkers.\n\n\\subsection{Known number of talkers: Dual-Path RNN-TasNet}\n\\defx(t){x(t)}\n\\defs{s}\n\\def\\sourceletter(t){s(t)}\n\\defk{k}\n\\def\\sourceindexed#1{s_{#1}(t)}\n\\def\\hat{\\sourceletter}(t){\\hat{s}(t)}\n\\def\\sourceletter_\\spkidx(t){s_k(t)}\n\\def\\hat{\\sourceletter}_\\spkidx(t){\\hat{s}_k(t)}\n\\defK{K}\n\\defL{L}\n\\def\\alpha{\\alpha}\n\nA single channel discrete-time speech mixture signal $x(t)$ is modeled as a sum of $K$ single-talker speech signals $\\sourceletter_\\spkidx(t)$:\n\\begin{equation}\n    x(t) = \\sum_{k = 1}^{K} \\sourceletter_\\spkidx(t),\n\\end{equation}\nwhere $k$ is the talker index.\nSource separation is concerned with extracting the source signals $\\sourceletter_\\spkidx(t)$ from the mixture $x(t)$.\n\nWe use the \\gls{TasNet} \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2019_ConvTasNetSurpassingIdeal} with a \\gls{DPRNN} \\cite{Luo2020_DualPathRNNEfficient} as the separation network to obtain a fixed number of estimations $z_k(t)$ for the sources $\\sourceletter_\\spkidx(t)$, where the order of $z_k(t)$ can be permuted to the actual source signals:\n\\begin{equation}\n    [z_1 (t), ..., z_K (t)] = \\text{DPRNN-TasNet}\\big(x(t)\\big).\n\\end{equation}\nIt works by encoding the time-domain signal into a latent domain with a convolutional encoder, separating this representation with the \\gls{DPRNN}, and transforming it back to time domain with a de-convolutional decoder.\nThe \\gls{DPRNN} models short- and long-term dependencies in an alternating manner by segmenting the input and skipping different numbers of time-steps in adjacent layers.\nIt was shown in \\cite{Luo2020_DualPathRNNEfficient} to be superior to a separator based on 1-D convolutions used in the Conv-\\gls{TasNet} \\cite{Luo2019_ConvTasNetSurpassingIdeal}.\n\n\n\n\\subsubsection{Time-domain training objective}\n\nIn recent work, the training objective often is to maximize the scale-invariant source-to-distortion ratio (SI-SDR)\\footnote{Sometimes called SI-SNR \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2019_ConvTasNetSurpassingIdeal}} \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2019_ConvTasNetSurpassingIdeal} by minimizing the negative SI-SDR.\nBy setting $\\alpha=1$ and removing terms that do not depend on $\\hat{\\sourceletter}(t)$, the time-domain logarithmic MSE loss can be obtained \\cite{Heitkaemper2020_DemystifyingTasNetDissecting}:\n\\begin{equation}\n    \\loss{T-LMSE}\\big(\\sourceletter(t), \\hat{\\sourceletter}(t)\\big) = 10 \\log_{10} \\sum_t |\\sourceletter(t) - \\hat{\\sourceletter}(t)|^2.\n\\end{equation}\nThe missing scale-invariance did not show negative effects on the separation performance \\cite{Heitkaemper2020_DemystifyingTasNetDissecting}.\nTo be able to handle different numbers of speakers with such a model, it is common for frequency-domain separators to set the missing outputs to silence \\cite{Kolbaek2017_MultitalkerSpeechSeparationa}, i.e., $\\sourceletter(t)=0$.\nTo use silent targets here, the loss has to be prevented from going to negative infinity for perfect reconstruction and masking any loss terms from other target signals by, e.g., adding $1$ to the argument of the logarithm:\n\\begin{equation}\n    \\loss{T-L1PMSE}\\big(\\sourceletter(t), \\hat{\\sourceletter}(t)\\big) = 10 \\log_{10}\\big(1 + \\sum_t |\\sourceletter(t) - \\hat{\\sourceletter}(t)|^2\\big).\n\\end{equation}\nThe total loss is calculated in a permutation-invariant manner with the set $\\mathcal{P}$ of all permutations of length $K$:\n\\begin{equation}\n    \\loss{reconstruction} = \\min_{\\phi\\in\\mathcal{P}} \\frac{1}{K} \\sum_{k=1}^{K}\\mathcal{L}(\\sourceindexed{\\phi(k)}, z_k (t)).\n\\end{equation}\n\n\\def{n\/a}{{n\/a}}\n\n\\subsection{Unknown number of talkers: OR-PIT}\n\nConventional source separators are limited to a fixed number of talkers.\nAn arbitrary number of talkers can be handled by iterative source extraction approaches \\cite{Kinoshita2018_ListeningEachSpeaker,Takahashi2019_RecursiveSpeechSeparation}.\nInstead of directly separating the mixture into one stream for each talker, they apply a network repeatedly to extract one talker at a time.\n\n\\defz_1(t){z_1(t)}\n\\defz_2(t){z_2(t)}\nFollowing the \\gls{OR-PIT} \\cite{Takahashi2019_RecursiveSpeechSeparation} scheme, an iterative source extractor is a two-output separator, in our case a DPRNN-TasNet, trained to output one talker at its primary output $z_1(t)$ and the sum of all remaining talkers at its secondary output $z_2(t)$ so that $z_2(t)$ can be fed back to extract the next talker, as visualized in the left part of \\cref{fig:architecture}.\nIt is first trained with\n\\begin{equation}\n  \\label{eq:or-pit}\n    \\loss{OR-PIT} = \\min_k \\mathcal{L} \\big(z_1(t), \\sourceletter_\\spkidx(t)\\big) + \\mathcal{L}\\big(z_2(t), \\sum_{i\\neqk} \\sourceindexed{i}\\big)\n\\end{equation}\non clean mixtures, where $\\mathcal{L}$ is a time-domain loss.\nIt is then fine-tuned by feeding $z_2(t)$ as additional training data.\n\nThe number of talkers can be determined by counting the number of iterations required, until $z_2(t)$ does not contain speech.\nThe authors of \\cite{Takahashi2019_RecursiveSpeechSeparation} use Alexnet \\cite{Sutskever2012_ImagenetClassificationDeep} as an external classifier to detect the absence of speech.\nA stopping criterion can, however, be integrated into the separator by using thresholding or an additional output flag, as inspired by \\cite{Kinoshita2018_ListeningEachSpeaker}.\n\n\n\nAssuming that a speech signal contains a certain minimal amount of average energy and the network is able to suppress speech well enough, absence of speech can be detected by:\n\\def\\gamma{\\gamma}\n\\begin{equation}\n    \\frac{1}{T} \\sum_t |z_2(t)|^2 < \\gamma,\n\\end{equation}\nwith $T$ being the length of the signal and the threshold value $\\gamma$ being determined manually.\nModels that use this stopping criterion are called \\enquote{threshold} models.\n\n\\def\\hat{f}{\\hat{f}}\n\\deff{f}\nAn elegant way to integrate the classifier into the separator is to let the separation network predict a stop flag.\nThis is done by adding an additional scalar output $f$ to the network.\nThe output size of the \\gls{DPRNN} is increased. \nThe newly added \\gls{DPRNN} outputs are transformed by a fully connected layer to a scalar for each time step.\nThe stop flag per utterance is obtained with, e.g., an average pooling over time followed by a sigmoidal function.\nThe estimated flag $\\hat{f}$ is trained to indicate when to stop iterating using a binary cross-entropy objective\n\\begin{equation}\n    \\loss{flag} = - f \\log(\\hat{f}) - (1 - f) \\log (1 - \\hat{f}).\n\\end{equation}\nThe target flag $f$ is set to $1$ when the second output should be empty and $0$ otherwise.\n\n\n\n\n\\section{Joint optimization of source separation and speech recognition}\n\nWe propose to jointly fine-tune a source separation front-end (FE) with source number counting and a speech recognition back-end.\nThe FE is either a TasNet or an OR-PIT and the back-end is a CTC\/attention speech recognizer similar to \\cite{Watanabe2017_HybridCTCAttention,Seki2018_PurelyEndtoEndSystem} taken from the ESPnet toolkit \\cite{Watanabe2018_EspnetEndtoendSpeech}.\nWe replace the original feature extraction with differentiable STFT-based features.\nIt is trained with a multi-target loss function with the factor $\\lambda=0.2$:\n\\begin{equation}\n    \\loss{ASR} = \\lambda \\loss{CTC} + (1 - \\lambda)\\loss{att}.\n\\end{equation}\n\nJoint fine-tuning is straight forward for the TasNet-based system without counting.\nThe gradients are propagated from the back-end into the FE and their losses are combined like\n\\begin{equation}\n    \\mathcal{L} = \\loss{ASR} + \\loss{FE}.\n\\end{equation}\nWe solve the permutation problem with the FE signal-level loss.\n\nFor the iterative OR-PIT system, there are different options that arise from the fact that $z_2(t)$ can contain more than one talker and is, thus, unusable for training the back-end.\nWe formulate two training schemes, namely the single- and multi-iteration schemes.\nThe single-iteration scheme unrolls a single iteration of the OR-PIT FE and uses $z_1(t)$ to train the back-end.\n$z_2(t)$ is optimized using only the signal-level loss.\nHere, the OR-PIT always sees unprocessed data, so there is a mismatch to evaluation where its own output is fed back.\nTo mitigate this, the model can be unrolled in the multi-iteration scheme, where the secondary output is used as the input for the following iteration and all primary outputs are used to train the ASR back-end.\n\n\n\\section{Experiments}\n\n\nWe evaluate our systems in terms of average improvement of signal-to-distortion ratio (SDRi)\\footnote{Provided by the mir\\_eval toolbox \\cite{Raffel2014_MirEvalTransparent}.} \\cite{Vincent2006_PerformanceMeasurementBlind,Fevotte2005_BSSEVALToolbox}, \\gls{CER}, \\gls{WER} and counting accuracy on the WSJ, WSJ0-2mix, WSJ0-3mix and WSJ0-4mix databases \\cite{Paul1992_DesignWallStreet,Isik2016_SingleChannelMultiSpeakerSeparation,Kolbaek2017_MultitalkerSpeechSeparationa} with a sample rate of \\SI{8}{kHz}.\nWe use WSJ0-4mix as created in \\cite{Takahashi2019_RecursiveSpeechSeparation}.\nThe experiments on source separation and number counting are conducted on the \\emph{min} subsets of the WSJ0-mix databases that contain full overlap only.\nSpeech recognition is evaluated on the \\emph{max} subset where no utterances are truncated.\n\n\\subsection{Source Separation}\n\\begin{table}[t]\n    \\centering\n    \\caption{\n    Source separation performance in SDRi in dB for varying numbers of talkers given the oracle number of sources. The mixtures contain full overlap only (min subset).\n    }\\label{tab:separation}\n    \\setlength{\\tabcolsep}{3pt}\n   \n    \\begin{tabular}{ll SSSS}\n        \\toprule\n        \\multicolumn{2}{l}{\\mrow{Method}} & \\multicolumn{4}{c}{number of talkers} \\\\\n        \\cmidrule{3-6}\n        && {1} & {2} & {3} & {4} \\\\\n        \\midrule\n        \n        \\multirow{5}{*}{\\rotatebox[origin=c]{90}{DPRNN-}}&TasNet 2 talker & 10.8 & 17.0 & {---} & {---} \\\\\n        &TasNet 3 talker & 8.2 & 11.7 & 11.3 & {---} \\\\\n        &TasNet 2+3 talker & \\fontseries{b}\\selectfont 39.4 & 17.0 & 10.9 & {---} \\\\\n       \n        \\cmidrule{2-6}\n        &OR-PIT stop-flag & 30.6 & 17.2 & 12.5 & 8.7 \\\\\n        &OR-PIT threshold & 30.9 & \\fontseries{b}\\selectfont 17.5 & 12.8 & 8.4 \\\\\n        \\midrule\n        \\multicolumn{2}{l}{RSAN stop-flag \\cite{Kinoshita2018_ListeningEachSpeaker}} & {---} & 8.6 & {---} & {---}\\\\\n       \n        \\multicolumn{2}{l}{Conv-TasNet-OR-PIT \\cite{Takahashi2019_RecursiveSpeechSeparation}} & {---} & 15.0 & \\fontseries{b}\\selectfont 12.9 & \\fontseries{b}\\selectfont 10.6 \\\\\n        \\multicolumn{2}{l}{Original DPRNN-TasNet \\cite{Luo2020_DualPathRNNEfficient}} & {---} & 16.1 & {---} & {---} \\\\\n        \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\\noindent\nWe first assume that the number of talkers is known.\nWe train all source separators on \\SI{4}{s} long signal segments to comply with \\cite{Luo2018_TaSNetTimeDomainAudio,Luo2020_DualPathRNNEfficient}.\nWe choose the DPRNN parameters according to the original paper \\cite{Luo2020_DualPathRNNEfficient}%\n, i.e., six blocks with two \\glspl{BLSTM} with 128 units, an encoder window size of 16 and a chunk size of 100.\nWe train two \\gls{DPRNN}-\\glspl{TasNet} for two or three speakers only, respectively, and one with three outputs for two and three speakers where the last target is set to silence for two-speaker mixtures.\nOur OR-PIT is trained on single-, two- and three-speaker recordings.\nWe use $\\loss{T-L1PMSE}$ for \\enquote{DPRNN-TasNet 2+3 talker} to cope with silent output, and $\\loss{T-LMSE}$ for all other models.\n\nThe separation performance displayed in \\cref{tab:separation} is evaluated given the oracle number of talkers, so that counting issues are not reflected in the SDRi.\nThe OR-PIT is forced to the correct number of iterations.\nFor the TasNet, only the $K$ outputs with the largest energy are considered for evaluation.\n\nOur two-talker TasNet (DPRNN-TasNet 2 talker) achieves a separation performance slightly better than \\cite{Luo2020_DualPathRNNEfficient} on two talkers and the same architecture works well on three talkers (DPRNN-TasNet 3 talker).\nBoth do not generalize well to smaller numbers of talkers.\nThe models specialized to one specific number are not able to produce a better reconstruction quality than what they learned during training.\nThe model trained on both, two and three speakers, handles both types of mixtures well and generalizes even to single-speaker recordings.\n\nThe OR-PIT can handle one to three speakers slightly better than the TasNet.\nIt especially has advantages in the three-speaker case, as also observed in \\cite{Takahashi2019_RecursiveSpeechSeparation}, where the OR-PIT network extracts one speaker at a time while the TasNet has to solve the more complex task of separating three speakers simultaneously.\nIts iterative structure allows the model to generalize to some extent to a larger number of talkers.\nThe performance, however, drops compared to the original Conv-TasNet-OR-PIT.\nThis might be caused by slight differences in training scheme.\n\n\n\n\\subsection{Source Number Counting}\n\n\\begin{table}[t]\n    \\centering\n    \\caption{\n    Source number counting accuracy in \\% for varying numbers of talkers. The mixtures contain full overlap only (min subset). ($^*$: Threshold not optimized for this number of sources)\n    }\\label{tab:counting}\n    \\setlength{\\tabcolsep}{2pt}\n    \\begin{tabular}{ll SSSS}\n        \\toprule\n        \\multicolumn{2}{l}{\\mrow{Method}} & \\multicolumn{4}{c}{number of talkers} \\\\\n        \\cmidrule{3-6}\n         && {1} & {2} & {3} & {4} \\\\\n        \\midrule\n        \\multirow{3}{*}{\\rotatebox[origin=c]{90}{DPRNN-}}\n        & TasNet 2+3 talker & 0.0{$^*$} & 99.9 & \\fontseries{b}\\selectfont 99.4 & {---} \\\\\n       \n        \\cmidrule{2-6}\n        & OR-PIT stop-flag & \\fontseries{b}\\selectfont 100.0 & \\fontseries{b}\\selectfont 100.0 & 96.7 & \\fontseries{b}\\selectfont 91.9 \\\\\n        & OR-PIT threshold & \\fontseries{b}\\selectfont 100.0 & \\fontseries{b}\\selectfont 100.0 & 98.2 & 0.0{$^*$}\\\\\n        \\midrule\n        \\multicolumn{2}{l}{RSAN stop-flag \\cite{Kinoshita2018_ListeningEachSpeaker}} & \\fontseries{b}\\selectfont 100.0 & 99.9 & {---} & {---} \\\\\n        \\multicolumn{2}{l}{Model selection \\cite{Nachmani2020_VoiceSeparationUnknown}} & {---} & 37.0 & 41.0 & 47.0 \\\\\n        \\bottomrule\n    \\end{tabular}\n   \n\\end{table}\n\n\\begin{table*}[ht]\n\\centering\n\\caption{\nRecognition performance of the multi-talker ASR systems for varying numbers of speakers given the oracle number of sources. The mixture signals contain full utterances (max subset). (\\enquote{FE}: front-end, \\enquote{ASR}: speech recognition part.)}\n\\label{tab:or-pit-recog}\n\\setlength{\\tabcolsep}{4.5pt}\n\\newcommand*{\\rn}[1]{(#1)}\n\\begin{tabular}{llSSS c SSS c SSS}\n    \\toprule\n    \\multicolumn{2}{l}{\\mrow{Experiment}} & \\multicolumn{3}{c}{WSJ0-2mix} && \\multicolumn{3}{c}{WSJ0-3mix} && \\multicolumn{3}{c}{WSJ0-4mix} \\\\\n    \\cmidrule{3-5}\\cmidrule{7-9}\\cmidrule{11-13}\n    && {CER} & {WER} & {SDRi} && {CER} & {WER} & {SDRi} && {CER} & {WER} & {SDRi} \\\\\n    \\midrule\n    \\rn{1} &DPRNN-TasNet (2 talker) + ASR & 9.6 & 15.8 & 16.5 && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{2} &\\phantom{+}+ fine-tune ASR & 5.1 & 8.9 & 16.5 && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{3} &\\phantom{+}+ fine-tune FE + ASR & \\fontseries{b}\\selectfont 4.0 & \\fontseries{b}\\selectfont 7.5 & 11.9 && {---} & {---} & {---} && {---} & {---} & {---} \\\\ \n    \\rn{4} &DPRNN-TasNet (3 talker) + ASR & {---} & {---} & {---} && 31.1 & 49.2 & 10.4 && {---} & {---} & {---} \\\\\n    \\rn{5} &\\phantom{+}+ fine-tune ASR & {---} & {---} & {---} && 25.5 & 35.3 & 10.4 && {---} & {---} & {---} \\\\\n    \\rn{6} &\\phantom{+}+ fine-tune FE + ASR & {---} & {---} & {---} && 29.0 & 39.7 & 7.5 && {---} & {---} & {---} \\\\\n    \\rn{7} &DPRNN-TasNet (2+3 talker) + ASR & 17.5 & 28.7 & 14.1 && 41.1 & 61.2 & 9.9 && {---} & {---} & {---} \\\\\n    \\rn{8} &\\phantom{+}+ fine-tune ASR & 7.3 & 11.9 & 14.1 && 19.0 & 27.1 & 9.9 && {---} & {---} & {---} \\\\\n    \\rn{9} &\\phantom{+}+ fine-tune FE + ASR & 5.8 & 10.5 & 14.4 && 24.9 & 34.9 & 7.8 && {---} & {---} & {---} \\\\\n    \\midrule\n    \\rn{10} &DPRNN-OR-PIT + ASR & 10.8 & 18.3 & \\fontseries{b}\\selectfont 16.7 && 25.2 & 40.7 & \\fontseries{b}\\selectfont 11.9 && 50.5 & 79.9 & \\fontseries{b}\\selectfont 7.8 \\\\\n   \n   \n   \n   \n   \n   \n   \n    \\rn{11}&\\phantom{+}+ VAD & 5.8 & 11.5 & \\fontseries{b}\\selectfont 16.7 && 14.8 & 26.2 & \\fontseries{b}\\selectfont 11.9 && 31.6 & 51.8 & \\fontseries{b}\\selectfont 7.8\\\\\n    \\rn{12}&\\phantom{++}+ single-iteration fine-tune ASR & 4.3 & 8.7 & \\fontseries{b}\\selectfont 16.7 && \\fontseries{b}\\selectfont 9.3 & \\fontseries{b}\\selectfont 16.3 & \\fontseries{b}\\selectfont 11.9 && 24.6 & 41.1 & \\fontseries{b}\\selectfont 7.8 \\\\\n    \\rn{13}&\\phantom{++}+ single-iteration fine-tune FE + ASR & 5.2 & 10.0 & 11.9 && 12.2 & 21.2 & 8.9 && 26.2 & 42.6 & 5.1 \\\\\n    \\rn{14}&\\phantom{++}+ multi-iteration fine-tune ASR & 4.7 & 9.0 & \\fontseries{b}\\selectfont 16.7 && 9.9 & 17.0 & \\fontseries{b}\\selectfont 11.9 && \\fontseries{b}\\selectfont 21.0 & \\fontseries{b}\\selectfont 33.4 & \\fontseries{b}\\selectfont 7.8 \\\\\n    \\rn{15}&\\phantom{++}+ multi-iteration fine-tune FE + ASR  & 4.6 & 8.7 & 12.9 && 11.8 & 20.5 & 9.0 && 27.8 & 45.2 & 4.8 \\\\\n    \\midrule\n    \\rn{16} &End-to-end ASR \\cite{Chang2019_EndtoendMonauralMultispeaker} & {---} & 25.4 & {---} && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{17} &Conv-TasNet + ASR \\cite{vonNeumann2020_EndtoEndTrainingTime} & 6.0 & 11.1 & 13.8 && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\rn{18} &End-to-end ASR \\cite{Zhang2020_ImprovingEndtoEndSingleChannel} & {---} & 23.4 & {---} && {---} & {---} & {---} && {---} & {---} & {---} \\\\\n    \\midrule\n    \\rn{19} &Oracle ASR result based on ground truth data & 2.1 & 4.7 & {$+\\infty$} && 2.1 & 4.7 & {$+\\infty$} && 2.0 & 4.6 & {$+\\infty$} \\\\\n    \\bottomrule\n\\end{tabular}\n\\vspace{-\\baselineskip}\n\\end{table*}\n\n\\noindent\nThe same models are evaluated for counting accuracy in \\cref{tab:counting}.\nCounting for the 2+3 talker TasNet is performed by an energy-based threshold on its third output.\nAll thresholds are chosen to maximize the accuracy on \\enquote{cv} and \\enquote{dev} data.\n\nThe \\enquote{DPRNN 2+3 talker} model performs well in discriminating between two- and three-talker mixtures, but the threshold does not generalize well to the single-talker scenario due to higher energy levels in the estimated signals.\nA counting accuracy of over \\SI{90}{\\%} is possible with an adjusted threshold, but this degrades the accuracy on two- and three-speaker mixtures.\n\nThe OR-PIT detects the number of sources correctly in most cases.\nThe threshold model performs slightly better for the numbers of talkers it was trained on, but the stop-flag model generalizes better to larger numbers of talkers.\n\nCompared to the model selection scheme in \\cite{Nachmani2020_VoiceSeparationUnknown}, our system performs better in source counting but worse in separation, where their model achieves \\SI{20.1}{dB} SI-SDR improvement and our system \\SI{17}{dB} on two talkers.\nSimilar modifications as \\cite{Nachmani2020_VoiceSeparationUnknown} could be applied to the OR-PIT to possibly achieve a similar separation performance.\n\nAlthough not directly comparable, the stopping criteria introduced in this work seem to perform comparable to the Alexnet classifier with \\SI{95.7}{\\%} accuracy for less than three talkers \\cite{Takahashi2019_RecursiveSpeechSeparation} while being simpler.\n\n\\subsection{Single-talker speech recognition}\n\nWe use a configuration similar to \\cite{Seki2018_PurelyEndtoEndSystem} without the speaker dependent layers for the speech recognizer.\nThis results in two CNN layers followed by two \\gls{BLSTM}P layers with $1024$ units for the encoder, one LSTM layer with 300 units for the decoder and a feature dimension of $80$.\nWe use a location-aware attention mechanism and ADADELTA \\cite{Zeiler2012_ADADELTAadaptivelearning} as optimizer.\nDecoding is performed with a word-level RNN language model.\nOur ASR system achieves a WER of \\SI{6.4}{\\percent} on the WSJ eval92 set.\n\n\\subsection{Two-talker speech recognition}\n\nWe evaluate the speech recognition performance of the TasNet-based recognizer in rows 1 to 9 of \\cref{tab:or-pit-recog}.\nWe observe similar tendencies as were found in \\cite{Heymann2017_BeamnetEndtoendTraining} and \\cite{vonNeumann2020_EndtoEndTrainingTime}.\nFine-tuning the ASR part gives a greater improvement than fine-tuning the TasNet but fine-tuning both jointly gives the best performance on two speakers (compare rows 1 to 3) of \\SI{58}{\\%} relative improvement compared to not fine-tuning.\nThe final \\SI{7.5}{\\%} WER (3) forms a new state-of-the-art result on WSJ0-2mix and is a huge improvement over previous techniques \\cite{vonNeumann2020_EndtoEndTrainingTime}.\nSimilar to the separation evaluation, the two-speaker TasNet performs better than the two- and three-speaker model (rows 7 to 9).\n\n    \n\n\n\\subsection{Multi-talker speech recognition with counting}\n\nAfter having shown that the joint system performs well with a separator for a fixed number of talkers, we evaluate joint training for the OR-PIT in rows 10 to 15 of \\cref{tab:or-pit-recog}.\nWe use the stop-flag OR-PIT due to better counting accuracy on \\emph{max} data.\n\n\nThe OR-PIT shows an odd behavior that is not present in the TasNet.\nIn regions where only a single talker is active, one output should be silent.\nThis works for the TasNet, but the OR-PIT outputs the speech signal scaled down heavily.\nThe ASR system picks it up and creates insertion errors.\nAn energy-based voice activity detection (VAD) improves the WER substantially to \\SI{11.5}{\\%} (11) for two talkers without fine-tuning (10).\n\nFine-tuning the ASR system with the single-iteration scheme (12) can improve the performance, but the mismatch created by fine-tuning the FE this way degrades the performance compared to fine-tuning ASR (13).\nUsing the multi-iteration fine-tuning scheme (14 and 15) can prevent this, but the overall performance is not comparable to the two-speaker TasNet model, although the OR-PIT achieves a better separation performance, according to Tab.~\\ref{tab:separation}.\nThis might be caused by the more complex training procedure.\n\nThe performance on larger numbers of speakers is listed in the last two columns in \\cref{tab:or-pit-recog}.\nThe OR-PIT again performs better than the TasNet on more than two speakers.\nIt generalizes to the unseen larger number of four speakers.\nContradicting the results for two speakers, fine-tuning just the ASR part performs best for more than two speakers (rows 5, 8, 12 and 14).\nThese results suggest that it is beneficial to only fine-tune the ASR back-end in some scenarios, especially when the separation task is more challenging, while fine-tuning the front-end contributes to over-fitting to a specific scenario.\n\n\\section{Conclusion}\n\nWe build the first joint end-to-end system that performs source number counting and multi-talker speech recognition.\nOur specialized model to two talkers provides a new state-of-the-art \\gls{WER} for the WJS0-2mix database of \\SI{7.5}{\\%}.\nThe OR-PIT-based system that can count the numbers of speakers performs better than the TasNet-based model on three speakers and generalizes well to an unknown number of speakers, i.e., four speakers.\nEvaluation of the source counting abilities show very promising performance.\n\n\\section{Acknowledgements}\nComputational resources were provided by the Paderborn Center for Parallel Computing.\n\n\\bibliographystyle{IEEEtran}\n\\pagebreak \n\\balance\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:intro}\n\nThe two determinant identities we generalise, due respectively to Gessel and Viennot and to Aitken, are as follows.\nFor \\(n\\) a nonnegative integer, we write \\(\\intervalwz{n}=\\set{0,1,\\ldots,n}\\).\n\n\\begin{restatable}[Binomial determinant duality theorem {\\cite[Proposition 7]{GESSEL1985300}}]{theorem}{binomialidentity}\n\\label{thm:binomial_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\). Then\n\\[\n    \\det\\mleft(%\n        \\binom{b}{a}%\n    \\mright)_{a \\in A, b \\in B}\n    =\n    \\det\\mleft(%\n        \\binom{a'}{b'}%\n    \\mright)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\n\\begin{restatable}[Symmetric function duality theorem {\\cite[Section 2]{aitken_1931}}]\n{theorem}{symfunctionidentity}\n\\label{thm:sym_function_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nThen\n\\[\n    \\det\\Bigl(%\n        h_{b - a} %\n    \\Bigr)_{a \\in A, b \\in B}\n    =\n    \\det\\Bigl(%\n        e_{a' - b'} %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\nHere \\(h_i\\) and \\(e_i\\) denote the complete homogeneous and elementary symmetric functions of degree \\(i\\) respectively, where by convention \\(e_0 = h_0 = 1\\) and \\(e_d = h_d = 0\\) for \\(d < 0\\).\nFor the purposes of indexing matrices, we consider finite subsets of \\(\\mathbb{Z}\\) to be ordered smallest element to largest.\n\nWe state our main theorem here in a slightly weakened form to avoid a technical condition whose necessity becomes clear only in the proof; the full statement with a weaker requirement on the parameters is given in \\autoref{thm:main_theorem}.\n\n\\begin{restatable}{theorem}{maintheorem}\\label{thm:intro_det_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nLet \\(\\alpha\\) and \\(\\beta\\) be partitions with \\(n\\) parts (with parts equal to \\(0\\) permitted) such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i \\in [n]\\).\nSuppose each part of \\(\\alpha\\) and \\(\\beta\\) is at most \\(1\\) less than the preceding part.\nThen the determinants\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}}) %\n    \\Bigr)_{a \\in A, b \\in B}\n\\]\nand\n\\[\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n\\]\nare equal.\n\\end{restatable}\nWe illustrate this statement in \\autoref{eg:intro_thm_example} at the end of this section.\nNote that symmetric functions of positive degree over an empty set of variables are considered to equal~\\(0\\).\n\nOur proof, which comprises \\autoref{section:proof}, is by counting paths on lattices and constructing a bijection between sets of such paths.\nThis argument, which includes what is now known as the Lindstr\\\"{o}m--Gessel--Viennot lemma, is similar to that used by Gessel and Viennot to prove \\autoref{thm:binomial_identity}.\nHowever, we consider lattices of more general shape and with weighted edges.\nWeighted edges allow us to replace binomial coefficients with symmetric polynomials;\nthe more general shape of our lattices allows us to vary the number of variables in each symmetric polynomial. \n\nWe explain how \\autoref{thm:binomial_identity} and \\autoref{thm:sym_function_identity} can be deduced from \\autoref{thm:intro_det_identity} in \\autoref{section:corollaries}.\nGessel and Viennot's binomial duality theorem corresponds to a staircase-shaped lattice, while Aitken's symmetric function duality theorem corresponds to a rectangular lattice.\nThis paper thus provides a unifying framework for these two results, and since our theorem also permits lattices of shapes intermediate between staircases and rectangles, it generalises them significantly.\n\nWe also obtain lifts of the binomial duality theorem to \\(q\\)-binomial coefficients and to symmetric polynomials, which we state here and prove in \\autoref{section:corollaries}.\n\n\\begin{restatable}\n{corollary}{qbinomialidentity}\n\\label{cor:q-binomial_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nThen\n\\[\n    \\det\\mleft(%\n        \\qbinom{b}{a}%\n    \\mright)_{a \\in A, b \\in B}\n    =\n    \\det\\mleft(%\n        q^{\\binom{a'-b'}{2}} \\qbinom{a'}{b'}%\n    \\mright)_{a' \\in A^\\mathsf{c},b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\n\\begin{restatable}\n{corollary}{sympolybinomidentity}\n\\label{cor:sym_poly_binomial_identity}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nThen\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{1}, x_{2}, \\ldots, x_{a+1}) %\n    \\Bigr)_{a \\in A, b \\in B}\n    =\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{1}, x_{2}, \\ldots, x_{a'}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\n\\end{restatable}\n\nAitken's proof of \\autoref{thm:sym_function_identity} is of an entirely different flavour to the methods in this paper: it is an application of an elementary result in linear algebra due to Jacobi, which we refer to as Jacobi's complementary minor formula.\nAs we show in \\autoref{section:JT_connection}, this formula is insufficient to prove our main theorem.\n\n\n\\begin{example}\n\\label{eg:intro_thm_example}\nSuppose \\(n=4\\), \\(A = \\set{0,1,2}\\), \\(B = \\set{1,3,4}\\), \\(\\alpha = (1,1,0,0)\\) and \\(\\beta = (4,3,3,2)\\).\nThen the two determinants that \\autoref{thm:intro_det_identity} states are equal are\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{cc>{\\scriptstyle}c>{\\scriptstyle}c>{\\scriptstyle}c}\n      &   & b=1 & b=3 & b=4 \\\\\n      &   & \\beta_{b}=4 & \\beta_{b}=3 & \\beta_{b}=2\\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}c|ccc|}\n    a=0 & \\alpha_{a+1}+1=2 & h_1(x_2, x_3, x_4) & h_3(x_2, x_3) & h_4(x_2) \\\\\n    a=1 & \\alpha_{a+1}+1=2 & 1 & h_2(x_2, x_3) & h_3(x_2) \\\\\n    a=2 & \\alpha_{a+1}+1=1 & 0 & h_1(x_1, x_2, x_3) & h_2(x_1, x_2) \\\\\n    \\end{block}\n\\end{blockarray}\n\\]\nand\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{ccc>{\\scriptstyle}c>{\\scriptstyle}cc}\n      &   && b'=0 & b'=2 &\\\\\n      &   && \\beta_{b'+1}=4 & \\beta_{b'+1}=3& \\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}cc|cc|c}\n    a'=3 & \\alpha_{a'}+1=1 && e_3(x_1, x_2, x_3, x_4) & e_1(x_1, x_2, x_3)& \\\\\n    a'=4 & \\alpha_{a'}+1=1 && e_4(x_1, x_2, x_3, x_4) & e_2(x_1, x_2, x_3)&.\\\\\n    \\end{block}\n\\end{blockarray}\n\\]\n\\end{example}\n\n\n\\section{Proof of the main theorem}\n\\label{section:proof}\n\nThe steps in our proof are:\n\\begin{enumerate}[label={\\arabic*)}]\n    \\item\n    define two lattices and paths upon them;\n    \\item\n    show that the weighted counts of tuples of paths on the lattices equal the determinants in the main theorem;\n    \\item\n    construct a weight-preserving bijection from tuples of paths on one lattice to the other.\n\\end{enumerate}\n\nWe adopt the notation of \\autoref{thm:intro_det_identity} throughout: let \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal size with complements in \\(\\intervalwz{n}\\) denoted \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\), and let \\(\\alpha\\) and \\(\\beta\\) be partitions with \\(n\\) parts (with parts equal to zero permitted) such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i \\in [n]\\).\nAdditionally, let \\(l = \\@ifstar{\\oldabs}{\\oldabs*}{A} = \\@ifstar{\\oldabs}{\\oldabs*}{B}\\), and let \\(r = \\@ifstar{\\oldabs}{\\oldabs*}{A^\\mathsf{c}} = \\@ifstar{\\oldabs}{\\oldabs*}{B^\\mathsf{c}} = n+1-l\\).\n\n\\subsection{Definition of lattices}\n\\label{section:lattices}\n\nWe picture Young diagrams as lying in a plane with the \\(x\\)-direction being downward and the \\(y\\)-direction being rightward, and the \\(1 \\times 1\\) square whose bottom-right corner is the point \\((i,j)\\) is referred to as the \\emph{box} \\(\\Ybox{i}{j}\\).\n(Though it is common to refer to a box simply by its coordinates, we denote boxes in this manner since we have need to distinguish between boxes and points.)\nThe skew Young diagram of \\(\\beta\/\\alpha\\) is then \\(\\Y{\\beta\/\\alpha} = \\setbuild{ \\Ybox{i}{j} }{1 \\leq i \\leq n,\\ \\alpha_i +1 \\leq j \\leq \\beta_i}\\).\n\nWe use \\(\\Y{\\beta\/\\alpha}\\) to construct two lattices, \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\).\nIn both lattices, the set of nodes is the set of points in the plane which occur as some corner of some box in \\(\\Y{\\beta\/\\alpha}\\).\nThe edges in each lattice are described below.\nWe give some edges a \\emph{weight} which will be a formal variable.\nThe weight of a path is then given by the product of the weights of the steps, and the weight of a tuple of paths is the product of the weights of each path.\n\nIn \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\), we have horizontal edges and vertical edges:\n\\begin{itemize}\n    \\item the horizontal edges are the horizontal sides of the boxes in \\(\\Y{\\beta\/\\alpha}\\), and are directed rightward;\n    \\item the vertical edges are the right-hand sides of the boxes in \\(\\Y{\\beta\/\\alpha}\\), are directed downward, and have weight \\(x_j\\) where \\(j\\) is the column of the corresponding box.\n\\end{itemize}\nIn  \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), we have horizontal edges and diagonal edges:\n\\begin{itemize}\n    \\item\n    the horizontal edges are the horizontal sides of the boxes in \\(\\Y{\\beta\/\\alpha}\\), and are directed leftward;\n    \\item\n    the diagonal edges are the top-right to bottom-left diagonals of the boxes in \\(\\Y{\\beta\/\\alpha}\\), are directed left-and-downward, and have weight \\(x_j\\) where \\(j\\) is the column of the corresponding box.\n\\end{itemize}\nAn example of each of these lattices is depicted in \\autoref{fig:lattices}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\begin{subfigure}{0.49\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{L-lattice.pdf}\n        \\caption{\\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\)}\n        \\label{subfig:L-lattice}\n    \\end{subfigure}\n    \\begin{subfigure}{0.49\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{R-lattice.pdf}\n        \\caption{\\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\)}\n        \\label{subfig:R-lattice}\n    \\end{subfigure}\n    \\caption{The relevant lattices when \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\) and \\(\\beta = (6,6,5,4,4,3)\\).\n    }\n    \\label{fig:lattices}\n\\end{figure}\n\nWe now define \\emph{sources} and \\emph{sinks} for paths on these lattices.\n\nIn \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\), take as \\emph{sources} the left-most node in each horizontal line indexed by \\(A\\),\nand take as \\emph{sinks} the right-most node in each horizontal line indexed by \\(B\\).\nExplicitly, the sources are the points \\(\\setbuild{(a, \\alpha_{a+1})}{a \\in A}\\) (where we interpret \\(\\alpha_{n+1} = \\alpha_n\\) if \\(n \\in A\\)) and the sinks are the points \\(\\setbuild{(b, \\beta_{b})}{b \\in B}\\) (where we interpret \\(\\beta_0 = \\beta_1\\) if \\(0 \\in B\\)).\n\nIn \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), take as \\emph{sources} the right-most node in each horizontal line indexed by \\(B^\\mathsf{c}\\),\nand take as \\emph{sinks} the left-most node in each horizontal line indexed by \\(A^\\mathsf{c}\\).\nExplicitly, the sources are the points \\(\\setbuild{(b', \\beta_{b'})}{b' \\in B^\\mathsf{c}}\\) (where we interpret \\(\\beta_0 = \\beta_1\\) if \\(0 \\in B^\\mathsf{c}\\)) and the sinks are the points \\(\\setbuild{(a', \\alpha_{a'+1})}{a' \\in A^\\mathsf{c}}\\) (where we interpret \\(\\alpha_{n+1} = \\alpha_n\\) if \\(n \\in A^\\mathsf{c}\\)).\n\nWe are interested in tuples of paths in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) that join the sources to the sinks in a matching; we call such tuples in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) \\emph{blue connectors} and such tuples in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) \\emph{red connectors}.\nWe furthermore describe sources, sinks and paths in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) as \\emph{blue} and in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) as \\emph{red}.\nA blue connector and a red connector are depicted in \\autoref{fig:paths}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\begin{subfigure}{0.495\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{lpath}\n        \\caption{A blue connector.}\n        \\label{subfig:l-path}\n    \\end{subfigure}\n    \\begin{subfigure}{0.495\\linewidth}\n        \\centering\n        \\includegraphics[scale=0.78]{rpath.pdf}\n        \\caption{A red connector.}\n        \\label{subfig:r-path}\n    \\end{subfigure}\n    \\caption{Examples of a blue connector and a red connector, when \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\), \\(\\beta = (6,6,5,4,4,3)\\),\n    \\(A = \\set{0,1,3,4}\\) and \\(B = \\set{1,3,5,6}\\).\n    Sources are indicated by circles, sinks by crosses.\n    }\n    \\label{fig:paths}\n\\end{figure}\n\n\\subsection{Enumeration of connectors}\n\nWe count the \\emph{non-intersecting} blue connectors and red connectors.\nThey key result we use for this is the Lindstr\\\"{o}m--Gessel--Viennot Lemma, stated below.\nThe lemma was first articulated in the context of Markov chains in \\cite{karlin1959}, and later in the context of matroid theory in \\cite{lindstroem1973}.\nIt was subsequently used to deduce various combinatorial identities in \\cite{GESSEL1985300}.\nFor an illuminating illustration of the argument behind the lemma, see \\cite{CountingOnDeterminants}.\n\n\\begin{theorem}[Lindstr\\\"om--Gessel--Viennot Lemma]\n\\label{thm:lgv_lemma}\nLet \\(G\\) be a directed acyclic graph with \\(m\\) designated sources sinks, where \\(m\\) is a nonnegative integer.\nLet \\(M\\) be the \\(m \\times m\\) matrix whose \\((i, j)\\)th entry is the number of paths, counted with weight, from the \\(i\\)th source to the \\(j\\)th sink.\nSuppose \\(G\\) is nonpermutable.\nThen the number of non-intersecting \\(m\\)-tuples of paths from sources to sinks, counted with weight, is equal to the determinant of \\(M\\).\n\\end{theorem}\n\nHere, \\emph{nonpermutable} means that a non-intersecting \\(m\\)-tuple of paths must connect the \\(i\\)th source to the \\(i\\)th sink.\nThe lattices \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) are clearly acyclic and nonpermutable, so \\autoref{thm:lgv_lemma} applies.\nWe thus want to count paths from the \\(i\\)th source to the \\(j\\)th sink in each lattice.\n\n\\begin{proposition}\n\\label{prop:l-path_count}\nThe weighted count of non-intersecting blue connectors in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) is\n\\[\n    \\det\\mleft(%\n        h_{b - a}(x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}}) %\n    \\mright)_{a \\in A,b \\in B}.\n\\]\n\\end{proposition}\n\n\\newcommand{(b, \\beta_{b})}{(b, \\beta_{b})}\n\\newcommand{(a, \\alpha_{a+1})}{(a, \\alpha_{a+1})}\n\n\\begin{proof}\nBy \\autoref{thm:lgv_lemma}, it suffices to show that the weighted count of paths from \\((a, \\alpha_{a+1})\\) to \\((b, \\beta_{b})\\) is \\(h_{b - a}(x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}})\\), for all \\(a \\in A\\) and \\(b \\in B\\).\n\nSuppose first that \\(a \\geq b\\).\nIf the inequality is strict, then \\((a, \\alpha_{a+1})\\) is below \\((b, \\beta_{b})\\), so there are no paths between them, and \\(h_{b - a} = 0\\) as required.\nIf equality holds, then \\((a, \\alpha_{a+1})\\) and \\((b, \\beta_{b})\\) are in the same row so there is a unique horizontal path between them, and \\(h_{b-a} = 1\\) as required.\n\nSuppose next \\(a < b\\) and \\(\\alpha_{a+1} \\geq \\beta_{b}\\).\nThen \\((a, \\alpha_{a+1})\\) is further right that \\((b, \\beta_{b})\\) (or in the same column, but without vertical edges joining them), so there are no paths between them.\nMeanwhile the corresponding polynomial is a symmetric function over an empty range of variables and hence is zero, as required.\n\nNow suppose \\(a < b\\) and \\(\\alpha_{a+1} < \\beta_{b}\\).\nA path from \\((a, \\alpha_{a+1})\\) to \\((b, \\beta_{b})\\) must make exactly \\(b - a\\) vertical steps.\nObserve that the vertical steps in such a path must be made down (the right-hand sides of) boxes in the rectangular region whose vertices are the boxes \\(\\Ybox{a+1}{\\alpha_{a+1}+1}\\), \\(\\Ybox{a+1}{\\beta_{b}}\\), \\(\\Ybox{b}{\\beta_{b}}\\) and \\(\\Ybox{b}{\\alpha_{a+1}+1}\\), as exemplified in \\autoref{fig:rectangular_region}.\nSince \\(\\alpha\\) and \\(\\beta\\) are partitions, we have \\(\\alpha_{i} \\leq \\alpha_{a+1}\\) and \\(\\beta_{i} \\geq \\beta_{b}\\) for all \\(a+1 \\leq i \\leq b\\), and so all these boxes are indeed contained in the Young diagram \\(\\Y{\\beta\/\\alpha}\\).\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{rectangular_region.pdf}\n    \\caption{The collection of boxes which must contain the vertical steps of any path from \\((a, \\alpha_{a+1})\\) to \\((b,\\beta_b)\\) when \\(a=1\\), \\(b=5\\), \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\) and \\(\\beta = (6,6,5,4,4,3)\\).\n    }\n    \\label{fig:rectangular_region}\n\\end{figure}\n\n\nThus the choice of \\(b-a\\) columns in which vertical steps take place can be made freely with repetition from \\(\\alpha_{a+1}+1, \\alpha_{a+1}+2, \\ldots, \\beta_{b}\\) (and each such choice uniquely determines a path).\nThen since the vertical edges in column \\(j\\) each have weight \\(x_j\\), the weighted count of possible paths is the required polynomial.\n\\end{proof}\n\n\\newcommand{\\parallelogramcondition}{%\nSuppose that for all \\(a' \\in A^\\mathsf{c}\\) and \\(b' \\in B^\\mathsf{c}\\), either:\n\\begin{itemize}\n    \\item\n    \\(a' - b' \\leq 0\\); or\n    \\item\n    \\(a' - b' > \\beta_{b'+1} - \\alpha_{a'}\\); or\n    \\item\n    for all \\(i\\) such that \\(b'+1 \\leq i \\leq a'\\), we have \\(\\alpha_{i} - \\alpha_{a'} \\leq a'-i\\) and \\(\\beta_{b'+1} - \\beta_i \\leq i-b'-1\\).\n\\end{itemize}\n}\n\n\\begin{proposition}\n\\label{prop:r-path_count}\n\\parallelogramcondition\nThen the weighted count of non-intersecting red connectors in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) is\n\\[\n    \\det\\mleft(%\n        e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}}) %\n    \\mright)_{a' \\in A^\\mathsf{c},b' \\in B^\\mathsf{c}}.\n\\]\n\\end{proposition}\n\n\\renewcommand{(b, \\beta_{b})}{(b', \\beta_{b'})}\n\\renewcommand{(a, \\alpha_{a+1})}{(a', \\alpha_{a'+1})}\n\n\\begin{proof}\nBy \\autoref{thm:lgv_lemma}, it suffices to show that the weighted count of paths from \\((b, \\beta_{b})\\) to \\((a, \\alpha_{a+1})\\) is \\(e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}})\\), for all \\(a' \\in A^\\mathsf{c}\\) and all \\(b' \\in B^\\mathsf{c}\\).\n\nSuppose first that \\(a' \\leq b'\\).\nIf the inequality is strict, then \\((b, \\beta_{b})\\) is below \\((a, \\alpha_{a+1})\\), so there are no paths between them, and \\(e_{a' - b'} = 0\\) as required.\nIf equality holds, then \\((b, \\beta_{b})\\) and \\((a, \\alpha_{a+1})\\) are in the same row so there is a unique horizontal path between them, and \\(e_{a'-b'} = 1\\) as required.\n\nSuppose next that\n\\(a' - b' > \\beta_{b'+1} - \\alpha_{a'}\\).\nThen any path starting at \\((b, \\beta_{b})\\) reaches the left-hand side of the lattice and terminates before it reaches the \\(a'\\)th row, so there are no paths between \\((b, \\beta_{b})\\) and \\((a, \\alpha_{a+1})\\). Meanwhile the corresponding polynomial is an elementary symmetric function in fewer variables than its degree and hence is zero, as required.\n\nNow suppose\n\\(a' - b' \\leq \\beta_{b'} - \\alpha_{a'}\\).\nA path from \\((b, \\beta_{b})\\) to \\((a, \\alpha_{a+1})\\) must make exactly \\(a'- b'\\) diagonal steps.\nObserve that the diagonal steps in such a path must be made across boxes in the parallelogram-shaped region whose vertices are the boxes \\(\\Ybox{b'+1}{\\beta_{b'+1}}\\), \\(\\Ybox{b'+1}{\\alpha_{a'}+a'-b'}\\), \\(\\Ybox{a'}{\\alpha_{a'}+1}\\) and \\(\\Ybox{a'}{\\beta_{b'+1}+b'-a'+1}\\), as exemplified in \\autoref{fig:parallelogram_region}.\nThe condition for this collection of boxes to be contained in the Young diagram is that for all \\(i \\in \\set{b'+1, b'+1, \\ldots, a'}\\) we have \\(\\alpha_i \\leq \\alpha_{a'} + (a'-i)\\) and \\(\\beta_i \\geq \\beta_{b'+1} - (i-b'-1)\\), which is exactly the assumed condition on \\(\\alpha\\) and \\(\\beta\\).\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{parallelogram_region.pdf}\n    \\caption{The collection of boxes which must contain the diagonal steps of any path from \\((b', \\beta_{b'})\\) to \\((a',\\alpha_{a'+1})\\) when \\(a'=5\\), \\(b'=2\\), \\(n=6\\), \\(\\alpha = (2,1,1,0,0,0)\\) and \\(\\beta = (6,6,5,4,4,3)\\).\n    }\n    \\label{fig:parallelogram_region}\n\\end{figure}\n\n\nThus the choice of \\(a'-b'\\) columns in which diagonal steps take place can be made freely without repetition from \\(\\alpha_{a'}+1\\), \\(\\alpha_{a'}+2\\), \\ldots, \\(\\beta_{b'+1}\\) (and each such choice uniquely determines a path).\nThen since the diagonal edges in column \\(j\\) each have weight \\(x_j\\), the weighted count of possible paths is the required polynomial.\n\\end{proof}\n\n\n\\subsection{Bijection between connectors}\n\\label{section:path_bijection}\n\nWe now define a bijection between the non-intersecting blue connectors in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and the non-intersecting red connectors in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), for any skew-partition \\(\\beta\/\\alpha\\).\n(In fact, the arguments in this section hold for \\(\\alpha\\) and \\(\\beta\\) any compositions such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i\\).)\n\nIt is convenient to overlay the lattices \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) and \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), so we can compare paths on one with paths on the other.\n\nThe bijection is via the following construction.\nWe define the construction of a red connector from a blue connector; the inverse construction is analogous.\n\n\\begin{definition}\nGiven a non-intersecting blue connector, define the \\emph{complementary} red connector to be the collection of paths constructed as follows:\nbeginning at each red source, take a horizontal step from each node unless the blue connector takes a vertical step from that node,\nin which case take a diagonal step.\n\\end{definition}\n\n\\begin{example}\nThe red connector in \\autoref{subfig:r-path} is complementary to the blue connector in \\autoref{subfig:l-path}, as illustrated in \\autoref{fig:complementary_paths}.\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{landrpath.pdf}\n    \\caption{The blue connector and red connector from \\autoref{fig:paths} overlayed, illustrating that the red connector is complementary to the blue connector.\n   \n    }\n    \\label{fig:complementary_paths}\n\\end{figure}\n\\end{example}\n\nIt is not obvious that the complementary red connector is indeed a red connector (that is, that each path reaches a distinct red sink).\nWe will show that it is, and that it is non-intersecting.\nTo do so, we use the following lemma.\n\n\\begin{lemma}\n\\label{lemma:complementary_lemma}\nA non-intersecting blue connector and its complementary red connector intersect only at nodes from which a vertical blue step is taken.\n\\end{lemma}\n\n\\begin{proof}\nSuppose, towards a contradiction, there exists a node which lies in both a non-intersecting blue connector and its complementary red connector and from which there is not a vertical blue step.\nConsider a right-most such node \\((i,j)\\).\n\nFirst observe \\((i,j)\\) is not a blue sink: in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\), the right-most nodes are not the endpoints of any edges, so no path in \\(\\Lambda_{\\text{R}}(\\beta\/\\alpha)\\) starting at a red source can contain a blue sink.\n\nTherefore there must be a horizontal blue step from \\((i,j)\\), to \\((i,j+1)\\).\nIn particular, \\((i,j)\\) is not right-most in its row, so it is not a red source.\nThus there is a red step to \\((i,j)\\), either horizontally from \\((i,j+1)\\) or diagonally from \\((i-1,j+1)\\).\nWe explain, and illustrate beneath each explanation, how both of these possibilities lead to a contradiction.\n\nIf the red step is horizontal, then by the construction of the complementary red connector there is no vertical blue step from \\((i,j+1)\\).\nThis contradicts our choice of \\((i,j)\\) as the right-most intersection from which there is not a vertical blue step.\n\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick,color=myblue,postaction={half mid arrow}] (0.5-\\nudge,0) -- (0.5-\\nudge,1);\n\\draw[thick,color=myred,postaction={half mid arrow}] (0.5+\\nudge,1) -- (0.5+\\nudge,0);\n\n\\node at (0.5,2.4) {\\(\\implies\\)};\n\n\\draw[thick,color=myblue,postaction={half mid arrow}] (0-\\nudge,4) -- (0-\\nudge,5);\n\\draw[thick,color=myred,postaction={half mid arrow}] (0+\\nudge,5) -- (0+\\nudge,4);\n\\draw[thick,color=paleblue,postaction={mid arrow}] (0,5) -- (1,5);\n\\node[cross out, line width=0.9pt, draw=darkgray, minimum size=0.32cm, at={(0.45,5)}] {};\n\n\\foreach \\x\/\\y in {0.5\/0, 0.5\/1, 1\/4, 1\/5, 0\/4, 0\/5} {\n    \\drawnode{(\\x,\\y)}\n}\n\\node[at={(0.5,0)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\node[at={(0,4)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\n\\end{tikzpicture}\n\\end{center}\n\nIf the red step is diagonal, then by the construction of the complementary red connector, there is a vertical blue step from \\((i-1, j+1)\\) to \\((i,j+1)\\). This contradicts that the blue connector is non-intersecting.\n\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick,color=myblue,postaction={mid arrow}] (0,0) -- (0,1);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,1) -- (0,0);\n\n\\node at (-0.5,2.4) {\\(\\implies\\)};\n\n\\draw[thick,color=myblue,postaction={mid arrow}] (0,4) -- (0,5);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,5) -- (0,4);\n\\draw[thick,color=myblue,postaction={mid arrow}] (-1,5) -- (0,5);\n\n\\foreach \\x\/\\y in {-1\/0, -1\/1, 0\/0, 0\/1, -1\/4, -1\/5, 0\/4, 0\/5} {\n    \\drawnode{(\\x,\\y)}\n}\n\\node[at={(0,0)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\node[at={(0,4)},label={[shift={(0.8,0)}]\\(\\scriptstyle(i,j)\\)}] {};\n\n\\end{tikzpicture}\n\\end{center}\n\nIn either case we have a contradiction, so no such intersection exists.\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:complements_exist}\nThe complementary red connector to a non-intersecting blue connector is a non-intersecting red connector.\n\\end{proposition}\n\n\\begin{proof}\nFirst observe that in \\(\\Lambda_{\\text{L}}(\\beta\/\\alpha)\\) the only edges out of the left-most nodes are horizontal, and thus the first step in every blue path is horizontal.\nTherefore, by \\autoref{lemma:complementary_lemma}, the complementary red connector does not contain any blue sources.\nSince the blue sources and the red sinks partition the left-most nodes in the lattices, we deduce that the paths of the complementary red connector reach red sinks (not necessarily distinct).\n\nWe next show that the complementary red connector is non-intersecting.\nThis implies that the red sinks the red paths reach are distinct, and hence that it is indeed a red connector.\n\nSuppose, towards a contradiction, that the complementary red connector has an intersection.\nConsider a right-most intersection \\((i,j)\\).\nNote that \\((i,j)\\) cannot be a red source (or a blue sink): if it were, then it would be the right-most node in its row, and so there would be no edges from \\((i,j+1)\\) or \\((i-1,j+1)\\) for a red path to arrive from (and so it could not be an intersection of two red paths).\n\nSince we assumed \\((i,j)\\) to be a right-most intersection, the incoming red paths must come from distinct vertices: there is both a horizontal and a diagonal red step to \\((i,j)\\).\nThen, by the construction of the complementary red connector and as illustrated below, there must be a vertical blue step to \\((i,j+1)\\) and there cannot be a vertical blue step out of \\((i,j+1)\\).\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick,color=myred,postaction={mid arrow}] (0,1) -- (0,0);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,1) -- (0,0);\n\n\\node at (0,2.2) {\\(\\implies\\)};\n\n\\draw[thick,color=myred,postaction={mid arrow}] (0,5) -- (0,4);\n\\draw[thick,color=myred,postaction={mid arrow}] (-1,5) -- (0,4);\n\\draw[thick,color=myblue,postaction={mid arrow}] (-1,5) -- (0,5);\n\\draw[thick,color=paleblue,postaction={mid arrow}] (0,5) -- (1,5);\n\\node[cross out, line width=0.9pt, draw=darkgray, minimum size=0.32cm, at={(0.45,5)}] {};\n\n\\foreach \\x\/\\y in {0\/0, 0\/1, -1\/0, -1\/1, 1\/0, 1\/1, 0\/4, 0\/5, -1\/4, -1\/5, 1\/4, 1\/5} {\\drawnode{(\\x,\\y)}\n}\n\n\\node[at={(0,0)},label={[shift={(0.5,-0.5)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\node[at={(0,4)},label={[shift={(0.5,-0.5)}]\\(\\scriptstyle(i,j)\\)}] {};\n\\end{tikzpicture}\n\\end{center}\nThen \\((i,j+1)\\) lies in both the blue connector and its complementary red connector, but there is no vertical blue step out of it, contradicting \\autoref{lemma:complementary_lemma}.\n\\end{proof}\n\n\\begin{proposition}\nThe map from the set of non-intersecting blue connectors to the set of non-intersecting red connectors defined by taking the complementary red connector is a weight-preserving bijection.\n\\end{proposition}\n\n\\begin{proof}\nWrite \\(\\Sigma C\\) for the sum of the elements of a set \\(C\\).\nObserve that \\(\\Sigma A^\\mathsf{c} - \\Sigma B^\\mathsf{c} = \\Sigma B - \\Sigma A\\),\nand hence that the number of vertical steps made by a blue connector equals the number of diagonal steps made by a red connector.\nThus every vertical blue step in a non-intersecting blue connector must give rise to a diagonal red step in its complementary red connector.\nThat is, the nodes from which a non-intersecting blue connector takes vertical steps are precisely the nodes from which its complementary red connector takes diagonal steps (and these are precisely the intersections of the connectors).\n\nIt is then clear that a non-intersecting blue connector has the same weight as its complementary red connector, and that taking the analogous construction of a complementary blue connector provides an inverse.\n\\end{proof}\n\n\n\\subsection{Conclusion}\n\nCombining the enumerations given by Propositions \\ref{prop:l-path_count} and \\ref{prop:r-path_count} with the bijection described in \\autoref{section:path_bijection}, we obtain our main theorem, stated in full below.\n\n\\begin{theorem}\\label{thm:main_theorem}\nLet \\(n\\) be a nonnegative integer, let \\(A\\) and \\(B\\) be subsets of \\(\\intervalwz{n}\\) of equal siz\n, and let \\(A^\\mathsf{c}\\) and \\(B^\\mathsf{c}\\) be their complements in \\(\\intervalwz{n}\\).\nLet \\(\\alpha\\) and \\(\\beta\\) be partitions with \\(n\\) parts (with parts equal to \\(0\\) permitted) such that \\(\\alpha_i \\leq \\beta_i\\) for all \\(i \\in \\intervalwz{n}\\).\n\\parallelogramcondition\nThen the determinants\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{\\alpha_{a+1}+1}, x_{\\alpha_{a+1}+2}, \\ldots, x_{\\beta_{b}}) %\n    \\Bigr)_{a \\in A, b \\in B}\n\\]\nand\n\\[\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{\\alpha_{a'}+1}, x_{\\alpha_{a'}+2}, \\ldots, x_{\\beta_{b'+1}}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n\\]\nare equal.\n\\end{theorem}\n\n\\begin{remark}\nThe hypothesis in \\autoref{thm:main_theorem} is precisely the hypothesis in \\autoref{prop:r-path_count} which ensures that each entry of the second matrix counts the number of red paths correctly (by requiring that all minimal parallelograms between appropriate sinks and sources lie inside the Young diagram).\nThis hypothesis is necessary and sufficient for each individual entry to give a correct count.\nHowever, this hypothesis is not necessary for the determinant to correctly count the total number of non-intersecting red connectors.\nFor example, suppose there exists \\(m \\in \\intervalwz{n}\\) such that \\(\\@ifstar{\\oldabs}{\\oldabs*}{A \\cap [0,m]} = \\@ifstar{\\oldabs}{\\oldabs*}{B \\cap [0,m]}\\).\nThen no non-intersecting connector crosses the \\((m+1)\\)th row, and so the count of non-intersecting connectors is the product of the the counts of non-intersecting connectors on each half of the lattice.\nMeanwhile the matrices whose entries count paths are block triangular, and the entries of the off-diagonal block are irrelevant to the determinant, and so it is not necessary for the hypothesis to hold for pairs with indices on both sides of \\(m+1\\).\n\\end{remark}\n\nPartitions whose parts are at most \\(1\\) less than the preceding part clearly satisfy the hypothesis of \\autoref{thm:main_theorem}, and so we recover \\autoref{thm:intro_det_identity} given in the introduction.\n\n\\section{Specialisations of the main theorem}\\label{section:corollaries}\n\nIn this section we indicate how to recover Gessel and Viennot's binomial duality theorem (\\autoref{thm:binomial_identity}) and Aitken's symmetric function duality theorem (\\autoref{thm:sym_function_identity}) from our main thereom.\nWe also deduce lifts of Gessel and Viennot's theorem to \\(q\\)-binomial coefficients (\\autoref{cor:q-binomial_identity}) and to symmetric polynomials (\\autoref{cor:sym_poly_binomial_identity}).\n\nTo deduce \\autoref{thm:binomial_identity}, \\autoref{cor:q-binomial_identity} and \\autoref{cor:sym_poly_binomial_identity}, we use staircase-shaped lattices.\n\n\\sympolybinomidentity*\n\n\\begin{proof}\nIn \\autoref{thm:main_theorem}, take \\(\\beta = (n^{n})\\) and take \\(\\alpha\\) to be the staircase given by \\(\\alpha_i = n-i\\) for \\(i \\in [n]\\).\nThen the left-hand matrix is \\((h_{b-a}(x_{n-a}, \\ldots, x_{n}))_{a \\in A,b \\in B}\\) and the right-hand matrix is \\((e_{a'-b'}(x_{n-a'+1}, \\ldots, x_{n}))_{a' \\in A^\\mathsf{c},b \\in B^\\mathsf{c}}\\).\nRelabelling the variables (via \\(x_i \\mapsto x_{n+1-i}\\)) gives the result.\n\\end{proof}\n\nThe \\(q\\)-binomial coefficients (also known as Gaussian coefficients) are polynomials in \\(q\\), and are a generalisation of the usual binomial coefficients in the sense that setting \\(q=1\\) yields the corresponding binomial coefficients.\nFor a definition, see (for example) \\cite[Section 3]{genbinoms}.\n\n\\qbinomialidentity*\n\n\\begin{proof}\nRecall that the \\(q\\)-binomial coefficients are related to the symmetric polynomials in the following way \\cite[Section 3, equations 17 and 18]{genbinoms}:\n\\begin{align*}\n    h_{k}(1, q, \\ldots, q^{n-1}) &= \\qbinom{n+k-1}{k}, \\\\\n    e_{k}(1, q, \\ldots, q^{n-1}) &= q^{\\binom{k}{2}} \\qbinom{n}{k}.\n\\end{align*}\nThus, set \\(x_i = q^{i-1}\\) in \\autoref{cor:sym_poly_binomial_identity} and the matrix entries become, respectively,\n\\[\nh_{b-a}(1,q, \\ldots, q^{a})\n    = \\qbinom{b}{b-a} = \\qbinom{b}{a}\n\\]\nand\n\\[\ne_{a'-b'}(1,q, \\ldots, q^{a'-1})\n    = q^{\\binom{a'-b'}{2}} \\qbinom{a'}{a'-b'} = q^{\\binom{a'-b'}{2}}\\qbinom{a'}{b'},\n\\]\nas required.\n\\end{proof}\n\nSetting \\(q=1\\) in \\autoref{cor:q-binomial_identity} recovers Gessel and Viennot's binomial duality theorem (\\autoref{thm:binomial_identity}).\n\nTo recover Aitken's symmetric function duality theorem (\\autoref{thm:sym_function_identity}), we use rectangular lattices.\nIn \\autoref{thm:main_theorem}, take \\(\\alpha = 0\\) and \\(\\beta = (m^{n})\\), for a positive integer \\(m\\), to obtain\n\\[\n    \\det\\Bigl(%\n        h_{b - a} (x_{1}, x_{2}, \\ldots, x_{m}) %\n    \\Bigr)_{a \\in A, b \\in B}\n    =\n    \\det\\Bigl(%\n        e_{a' - b'} (x_{1}, x_{2}, \\ldots, x_{m}) %\n    \\Bigr)_{a' \\in A^\\mathsf{c}, b' \\in B^\\mathsf{c}}\n    .\n\\]\nSince this holds for arbitrary positive integers \\(m\\),\n\\autoref{thm:sym_function_identity} follows.\n\n\n\n\\section{Insufficiency of Jacobi's complementary minor formula}\n\\label{section:JT_connection}\n\n\\autoref{thm:sym_function_identity} was proved in \\cite{aitken_1931} using Jacobi's complementary minor formula and a form of Newton's identity.\nThese two results are stated below.\nWe here outline Aitken's proof, and show that this method is not sufficient to deduce our main theorem.\n\nFor our purposes it is convenient to index matrix rows and columns from \\(0\\).\nGiven a \\((d+1) \\times (d+1)\\) matrix \\(M\\) and subsets \\(A,B \\subseteq \\intervalwz{d}\\), let \\(M_{A,B}\\) denote the matrix obtained by retaining only those rows indexed by elements of \\(A\\) and those columns indexed by elements of \\(B\\).\nWrite \\(\\Sigma A\\) for the sum of the elements of \\(A\\).\n\n\\begin{proposition}[Jacobi's complementary minor formula { \\cite[Lemma A.1(e), p.~96]{Caracciolo_2013}}]\n\\label{prop:Jacobi_comp_minor}\nLet \\(M\\) be an invertible \\((d+1) \\times (d+1)\\) matrix and let \\(A,B \\subseteq \\intervalwz{d}\\) be subsets of equal size.\nThen\n\\[\n    \\det ( M_{A,B} ) = (-1)^{\\Sigma A + \\Sigma B} \\det (M) \\det \\mleft( {\\mleft((M^{-1})^{\\top}\\mright)}_{A^\\mathsf{c}, B^\\mathsf{c}} \\mright).\n\\]\n\\end{proposition}\n\n\\begin{proposition}[Newton's identity {\\cite[Equation (\\ensuremath{2.6'}), p.~21]{macdonald1998symmetric}}]\n\\label{prop:Newton_identity}\nLet \\(d > 0\\).\nThen\n\\[\n    \\sum_{i=0}^d {(-1)}^i e_i h_{d-i} = 0.\n\\]\n\\end{proposition}\n\nIn \\cite{aitken_1931}, Aitken obtains his identity (\\autoref{thm:sym_function_identity}) by applying \\autoref{prop:Jacobi_comp_minor} to the matrix \\({( h_{j-i} )}_{0\\leq i,j \\leq n}\\).\nIts inverse is \\({( {(-1)}^{i+j} e_{j-i})}_{0\\leq i,j \\leq n}\\), as can be verified using \\autoref{prop:Newton_identity}. \n\nIf we attempt to use this method to prove \\autoref{thm:main_theorem}, we would be required to show, given partitions \\(\\alpha\\) and \\(\\beta\\) satisfying the hypotheses, that the matrices\n\\begin{align*}\nH(\\beta \/ \\alpha) &= {\\Bigl(\n    h_{j-i}(x_{\\alpha_{i+1}+1}, \\ldots, x_{\\beta_j})\n\\Bigr)}_{0\\leq i,j \\leq n},\\\\\n\\intertext{and}\nE(\\beta \/ \\alpha) &= {\\mleft(\n    {(-1)}^{i+j} e_{j-i}(x_{\\alpha_j+1}, \\ldots, x_{\\beta_{i+1}})\n\\mright)}_{0\\leq i,j \\leq n}\n\\end{align*}\nare inverse.\nHowever, \\autoref{thm:main_theorem} can provide a determinant identity when this is not the case.\n\nFor example, let \\(n=3\\) and let \\(\\alpha = (2, 0, 0)\\) and \\(\\beta = (3,3,1)\\).\nWe have\n\\begin{align*}\n    {\\mleft(E(\\beta \/ \\alpha)H(\\beta \/ \\alpha)\\mright)}_{0,2}\n    &= h_2(x_3) - e_1(x_3)h_1(x_1, x_2, x_3) + e_2(x_1, x_2, x_3) \\\\\n    &= x_1 x_2 \\\\\n    &\\neq 0,\n\\end{align*}\nso the matrices \\(H(\\beta \/ \\alpha)\\) and \\(E(\\beta \/ \\alpha)\\) are not inverse.\nNevertheless, choosing \\(A = \\set{0,1,2}\\) and \\(B = \\set{1,2,3}\\) meets the hypotheses of \\autoref{thm:main_theorem}, and so we find that the determinants\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{cc>{\\scriptstyle}c>{\\scriptstyle}c>{\\scriptstyle}c}\n      &   & b=1 & b=2 & b=3 \\\\\n      &   & \\beta_{b}=3 & \\beta_{b}=3 & \\beta_{b}=1\\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}c|ccc|}\n    a=0 & \\alpha_{a+1}+1=3 & h_1(x_3) & h_2(x_3) & 0 \\\\\n    a=1 & \\alpha_{a+1}+1=1 & 1 & h_1(x_1, x_2, x_3) & h_2(x_1) \\\\\n    a=2 & \\alpha_{a+1}+1=1 & 0 & 1 & h_1(x_1) \\\\\n    \\end{block}\n\\end{blockarray}\n\\]\nand\n\\[\n\\hspace*{-6.5em}\n\\begin{blockarray}{ccc>{\\scriptstyle}c}\n      &   && b'=0 &\\\\\n      &   && \\beta_{b'+1}=3 & \\\\[4pt]\n    \\begin{block}{>{\\scriptstyle}c>{\\scriptstyle}cc|c|}\n    a'=3 & \\alpha_{a'}+1=1 && e_3(x_1, x_2, x_3)\\\\\n    \\end{block}\n\\end{blockarray}\n\\]\nare equal.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{} if # should be set in lowercase\n\n\n\n\n\n\n\\dropcap{T}he polymerization of molecular nitrogen under pressure has been very\nactively researched over the past two decades and has stimulated many\nexperimental \\cite{Goncharov2000,Eremets2001,\n  Gregoryanz2001,Gregoryanz2002,Eremets2004,Eremets2004a,Eremets2007,\n  Gregoryanz2007,Lipp2007} and theoretical\n\\cite{McMahan1985,Martin1986,Lewis1992,Mailhiot1992,\n  Mitas1994,Alemany2003,Mattson2004,Zahariev2005,Oganov2006,Uddin2006,Zahariev2006,\n  Caracas2007,Wang2007a,Zahariev2007,Yao2008,Pickard2009-nitrogen,Ma2009-nitrogen,Wang2012-diamondoid-N}\ninvestigations.  Density functional theory (DFT) studies suggested\nthat dissociation of nitrogen could occur at high pressures which\nwere, nevertheless, attainable in diamond anvil cell (DAC)\nexperiments.  In a ground-breaking paper, Mailhiot \\textit{et al.}\\\n\\cite{Mailhiot1992} predicted polymerization of nitrogen molecules\nunder pressure, leading to the formation of the ``cubic gauche''\nframework structure.  After considerable efforts\n\\cite{Goncharov2000,Eremets2001}, cubic gauche nitrogen was finally\nsynthesized by Eremets \\textit{et al.}\\ \\cite{Eremets2004}, a decade\nafter its prediction.  The polymeric phase is quench recoverable to\nambient conditions, where it exists as a rather unstable\nhigh-energy-density material (HEDM) \\cite{Eremets2004}. Under ambient\nconditions an energy of roughly 1.5 eV\/atom ($\\approx$ 10330 J\/g) is\nreleased on converting cubic gauche nitrogen to molecular N$_2$, which\nis more than twice the energy density of TNT (about 4686 J\/g).\nPolymeric nitrogen \\cite{Eremets2004}, or high--N content salts\n\\cite{Haiges2004}, are therefore expected to be excellent candidates\nas high-energy-density materials.\n\nThe N$\\equiv$N triple bond is one of the strongest known and breaking\nit requires surmounting a substantial energetic barrier.  However, at\nhigh temperatures the triple bond breaks at pressures above about 110\nGPa \\cite{Eremets2004}, a much lower pressure than predicted for the\nsingle bond in H$_2$ \\cite{Pickard2007-hydrogen} and the double bond\nin O$_2$, which is predicted to survive up to 2 TPa\n\\cite{Sun2012-O2,Zhu2012}. Once the N$\\equiv$N triple bond is broken,\na wide variety of structures can be adopted, similar to phosphorus and\narsenic, and many studies aimed at elucidating the phase diagram of\nnitrogen have been published.\n  \n\nThe set of candidate structures includes simple cubic, the\nisoelectronic ``black phosphorus'' and $\\alpha$-arsenic or cis-trans\nand $Cmcm$ chain structures, etc. \\cite{McMahan1985,Mailhiot1992,\n  Martin1986,Lewis1992,Alemany2003,Mattson2004,Mattson2004,Zahariev2005,Oganov2006,Yao2008}\nNevertheless, only the ``black phosphorus'' structure seemed feasible,\nas it was predicted to become more stable than cubic gauche above\n$\\sim$210 GPa \\cite{Mailhiot1992,Mattson2004,Wang2007a}.\nThe advent of structure searching using DFT methods marked a\nsignificant advance; under conditions in which standard chemical\nintuition may no longer be reliable, new stable polymeric nitrogen\nphases beyond cubic gauche have been proposed, including the layered\n$Pba2$ and framework $P2_12_12_1$ structures\n\\cite{Ma2009-nitrogen,Pickard2009-nitrogen}. ``Black phosphorus'' was\nsubsequently ruled out on energetic grounds.\nThese phases have in common a large DFT band gap that hardly closes\nwith pressure.\nNitrogen compounds could form metallic phases at high pressures and\/or\ntemperatures, which could have important consequences for\nunderstanding planetary processes.\n\n\nLarge compressions normally lead to an increase in the atomic\ncoordination numbers and metallization, as predicted in, for example,\nhydrogen \\cite{Pickard2007-hydrogen, McMahon2011}, carbon\n\\cite{Sun2009-carbon,Martinez2012-carbon},\nand oxygen \\cite{Sun2012-O2},\nas well as observed in many other systems. \\cite{Grochala2007}\nThe coordination number of the atoms in solid nitrogen increases from\none to three on transformation from the molecular to cubic gauche\nforms and three-fold coordination persists in higher pressure\nstructures.\n\nAll high-pressure nitrogen structures proposed so far are insulating.\nAlthough an increase in coordination number with pressure is\nphysically reasonable, it is by no means universal.\nFor example, aluminum transforms into more open structures at TPa\npressures as the valence electrons move away from the ions in the\nformation of ``electride structures'' \\cite{Pickard2010-Al} although,\nas must occur in a pressure-induced equilibrium phase transition, the\ndensity increases at the transition, and \nwell-packed structures are strongly disfavored in oxygen up to at\nleast 25 TPa \\cite{Sun2012-O2}.\nThe coordination number of nitrogen has not been determined at\npressures above 400 GPa,\nand a metallic phase of nitrogen has not yet been reported.\n\nMaterials under TPa pressures are important in planetary science, for\nexample, the pressure at the center of Jupiter is estimated to be\nabout 7 TPa \\cite{Jeanloz2007}.\nDynamical shock wave \\cite{Jeanloz2007,Knudson2008,Eggert2010} and\nramped compression experiments \\cite{Hawreliak2007,Bradley2009} are\nincreasingly being used to investigate materials at TPa pressures.\nEven more extreme conditions are attainable in laser ignition\nexperiments \\cite{Kritcher2011}.\nExperimental determinations of structures are not, however, currently\npossible at TPa pressures and therefore theoretical predictions are\nparticularly important.\n\\cite{Sun2009-carbon,McMahon2011,Pickard2010-Al,Martinez2012-carbon,Sun2012-O2}\nIn this work we focus on the structural phases of nitrogen at \nmulti-TPa pressures.\n\n\n\n\n\n\\vspace*{5mm}\n\\noindent{\\bf\\large Results}\n\n\\noindent{\\bf Crystal structures and charge transfer.}\nWe have used \\textit{ab initio} random structure searching (AIRSS)\n\\cite{Pickard2006-silane,Pickard2011-review} and DFT methods to find\ncandidate structures of nitrogen up to multi-TPa pressures.\nThese searches have enabled us to identify novel candidate nitrogen\nstructures that are more stable than previously known ones.\nFour of them are thermodynamically stable within certain pressure\nranges, namely, $P4\/nbm$ ($Z$=14), $Fdd2$ ($Z$=48), $R\\bar{3}m$\n($Z$=1) and $I4_1\/amd$ ($Z$=4).\nMost interestingly, the tetragonal $P4\/nbm$ structure is characterized\nby the presence of N$_2$ pairs and N$_5$ tetrahedra, while the\northorhombic $Fdd2$ structure is modulated, as depicted in Fig.\\\n\\ref{fig:lattice}(a) and (b).\n\nAt 2.5 TPa, the N-N bond length within the N$_5$ tetrahedra of\n$P4\/nbm$ is 1.13 \\AA, and the shortest distance between the corners of\nthe tetrahedra and N$_2$ dimers is 1.26 \\AA.  The N-N bond length\nwithin the dimer of about 1.17 \\AA\\ is significantly shorter than the\nN-N separation between adjacent dimers (about 1.29 \\AA), which implies\nthat the dimers are well separated.\n\nThe unique structure of the $P4\/nbm$ phase led us to investigate the\ncharges on the different atomic sites.\nWe used the wave-function-based Mulliken population and charge-density\nbased Bader \\cite{Bader} methods to analyze the charge transfer.\nAs summarized in Table\\ \\ref{table:charge}, the two methods lead to\nthe same conclusions; the N$_5$ tetrahedra are negatively charged and\nserve as anions while the N$_2$ dimers are positively charged and act\nas cations, \nand thus the $P4\/nbm$ structure resembles an all-nitrogen salt.\nThe resultant charges (about $\\pm$0.22$e$ from Mulliken analysis and\nabout $\\pm$0.37$e$ from Bader analysis) are substantial, although they\ndepend somewhat on the definition of the atomic charges.\n\nThe complicated $Fdd2$ structure has a 48-atoms conventional cell, and\nthe atoms are arranged in undulating layers, as depicted in Fig.\\\n\\ref{fig:lattice}(b).\nThe charge distribution of $Fdd2$ also exhibits large distortions.  As\nshown in Fig.\\ \\ref{fig:lattice}(b) and Table\\ \\ref{table:charge},\nalthough all the atoms are at 16b sites, each orbit has a different\ncharacter: there is \na group of strong electron acceptors (Bader charge $Q_B$=-0.36$e$,\nred), a group of strong electron donors ($Q_B$=0.38$e$, blue), and a\ngroup of nearly neutral atoms ($Q_B$=-0.02$e$, green).\nDue to the undulation, with the inequivalent $z$ positions averaging\nat $0.1254$,\nthe $Fdd2$ structure is very similar to a charge density wave (CDW)\nmodulation.  The appearance of strong charge transfer effects in an\nelement at such high pressures is most unexpected, as the resulting\nCoulomb energy is substantial.\n\n\nWe also found a layered $Cmca$ structure similar to ``black\nphosphorus'', which becomes thermodynamically stable at about 2 TPa.\nAs depicted in the supplementary information, the $Cmca$ structure\nconsists of three-fold-coordinated\nnitrogen atoms and has zig-zag layers, with a shortest N-N distance of\nabout 1.15 \\AA\\ at 2.5 TPa, which is much shorter than the N-N\nseparation between the layers of about 1.56 \\AA.\nLayered structures also appear in the high-pressure phases of other\nsmall molecules of first row atoms, e.g., CO \\cite{Sun2011-CO} and\nCO$_2$ \\cite{Sun2009-CO2}.\nThe nitrogen atoms in the $R\\bar{3}m$ structure are\nsix-fold-coordinated and form distorted octahedra.\nPure four-fold coordinated structures often appear in our searches at\n3 TPa, but they are less favorable than $Cmca$ and $P4\/nbm$.\nThe crystal structures and charge transfers of the newly predicted\nstable phases are summarized in Table\\ \\ref{table:charge}.\n\n\n\n\n\n\n\n\n\\noindent{\\bf Energetics.} As can be seen in the enthalpy-pressure\nrelations of Fig.\\ \\ref{fig:eos}(a), solid nitrogen undergoes a series\nof structural phase transitions beyond the previously-known framework\n$P2_12_12_1$ polymeric phase\n\\cite{Pickard2009-nitrogen,Ma2009-nitrogen}\nand recently discovered diamondoid structure ($I\\bar{4}3m$)\n\\cite{Wang2012-diamondoid-N}: \n$I\\bar{4}3m \\xrightarrow{\\rm 2.1\\ TPa} Cmca \\xrightarrow{\\rm 2.5\\ TPa}\nP4\/nbm \\xrightarrow{\\rm 7.1\\ TPa} Fdd2 \\xrightarrow{\\rm 11.5\\ TPa}\nR\\bar{3}m \\xrightarrow{\\rm 30\\ TPa} I4_1\/amd$. \nThis partially ionic $P4\/nbm$ phase is stable over a wide pressure\nrange, but compression to about 7.1 TPa leads to the modulated $Fdd2$\nstructure.\nThe $Fdd2$ structure is the most favorable phase from about 7.1--11.5\nTPa, whereupon it transforms into a six-fold-coordinated hexagonal\n$R\\bar{3}m$ phase.  As shown in the inset in Fig.\\ \\ref{fig:eos},\nnitrogen forms a $P4_1\/amd$ structure similar to Cs-IV\n\\cite{Takemura1982-Cs-IV} at about 30 TPa.\n\nThe volume-pressure relations of the most stable phases are shown in\nFig.\\ \\ref{fig:eos}(b). The volume decreases by about 2.5\\% at the\ntransition from $I\\bar{4}3m$ to $Cmca$ at 2.1 TPa, 0.7\\% at the\ntransition from $Cmca$ to $P4\/nbm$ at 2.5 TPa, 0.6\\% at the transition\nfrom $P4\/nbm$ to $Fdd2$ at 7.1 TPa, and 0.5\\% at the $Fdd2$ to\n$R\\bar{3}m$ transition at 11.5 TPa.\n\n\n\n\n\n\n\n\n\\noindent{\\bf Electronic structures.}\nThe electronic band structures and projected densities of states\n(PDOS) of $Cmca$ at 2.0 TPa, $P4\/nbm$ at 3.0 TPa, and $Fdd2$ at 8.0\nTPa, are shown in Fig.\\ \\ref{fig:band}, decomposed into $s$ and $p$\ncomponents.\nThe $P2_12_12_1$ and $I\\bar{4}3m$ structures are semiconducting at 2.5\nTPa, but the band structures of $Cmca$, $P4\/nbm$, and $Fdd2$ shown in\nFig.\\ \\ref{fig:band} and $R\\bar{3}m$ are metallic, see the\nsupplementary material. The insulating state of nitrogen persists to\nabout 2.0 TPa, which is considerably higher than in other light\nelements such as hydrogen, carbon and oxygen. \\cite{Sun2012-O2}\n\nThe band structure of $Cmca$ at 2.0 TPa features a partially filled\nDirac cone between the $Y$ and $\\Gamma$ points, similar to the one\nthat appears in\nelectron doped graphene. \\cite{Giovannetti2008} \nThe projected density of states in these metallic structures arises\nfrom the $s$ and $p$ orbitals which extend over essentially the same\nenergy interval, with both $s$ and $p$ orbitals contributing to the\nelectronic density of states at the Fermi level and conduction bands,\nwhich indicates strong $sp$ hybridization in the $Cmca$, $P4\/nbm$ and\n$Fdd2$ structures.\nThe densities of states at the Fermi levels of $P4\/nbm$ and $Fdd2$ are\nhigher than in $Cmca$, implying that $Cmca$ is less metallic.\n\n\n\\vspace*{5mm}\n\\noindent{\\bf\\large Discussion}\n\n\n\nThe dynamical stability of the newly predicted stable structures was\ninvestigated by calculating their phonon dispersion relations.  As\nshown in Fig.~\\ref{fig:phonon} and in the supplementary material, all\nof the structures are predicted to be mechanically and dynamically\nstable at the specified pressures.\nThe phonon dispersion relation of $P4\/nbm$ shows steep acoustic\nbranches together with fairly flat optical modes. This supports the\nview that $P4\/nbm$ is formed of atomic units: the N--N vibrons as\nwell as the breathing mode of the tetrahedra are essentially\ndispersionless, and well screened by the electronic cloud.\n\nOn the other hand, the phonon mode of $Fdd2$ corresponding to the\namplitude variation of the undulating layers drops to a very low\nfrequency of about 190 cm$^{-1}$.  For comparison, the next\nlowest-lying mode (the relative sliding mode) has a much higher\nfrequency of about 1300 cm$^{-1}$, because the shear mode brings atoms\nof the same charge closer together.\n\n\n\n\n\nThe formation of N$_2$ pairs and N$_5$ tetrahedra in $P4\/nbm$ was\ndeduced from the calculated bond lengths and confirmed by the charge\ndensity plot in Fig.\\ \\ref{fig:charge_density}(b), where the atoms\nbetween the tetrahedron center and corners, and between the two atoms\nwithin the dimers, form strong covalent bonds.\nThe electron localization function (ELF) shown in the supplementary\nmaterial provides additional evidence for the formation of tetrahedra\nand dimers in $P4\/nbm$.\n\n\n\n\nThe formation of charged units from an elementary compound has also\nbeen observed in $\\gamma$--Boron \\cite{Oganov2009}.  The pressures of\ninterest in the present study are much larger than those at which\n$\\gamma$--Boron has been observed and, besides, $P4\/nbm$--nitrogen is\na very different system.  The charge transfer in boron satisfies its\ntendency to form electron deficient icosahedra.  Based on the\nstructural richness of phosphorus and arsenic one might assume that\nnitrogen could form similarly rich bonding patterns once the triple\nbond is broken. At 2 TPa, however, packing efficiency is crucial and\none would expect structures with high coordination numbers to be\nformed. \nFor example, the coordination number of $Cmca$ is 3, compared with 6\nin the $R\\bar{3}m$ phase.  It is therefore reasonable to expect that\nthe average coordination of nitrogen in the $P4\/nbm$ and $Fdd2$\nstructures would be between 3 to 6.\nIn $P4\/nbm$, if one employs a longer bond criterion, e.g., 1.30 \\AA,\nthe coordination number for the corners of the tetrahedra, their\ncenters and the nitrogen atoms in the chains are 3, 4, and 6,\nrespectively, as shown in Fig.\\ S5 of the supplementary information.\nIn this metallic phase the variation in coordination (or the\ninhomogeneity of the density of ions) leads to charge transfer from\nthe highly-coordinated atoms (``N$_2$ pairs'') to the lower\ncoordinated ones (``tetrahedra'' corners).\nThis charge transfer allows the $P4\/nbm$ structure to form a unique\nmetallic all-nitrogen salt.\nEach atom in the 1D chains perpendicular to the tetrahedra layers has\nfour long bonds to the corners of the tetrahedra, thus only one\nelectron is left.\nThe 1D chain with uniform N--N distances is unstable and undergoes a\nPeierls distortion, similar to that in lithium \\cite{Neaton1999}, and\nN$_2$ pairs are formed.\n\n\nAlthough $Fdd2$ also shows clear variation in coordination and charge\ntransfer, the atomic arrangement is completely different from that in\n$P4\/nbm$.  This phase has undulating layers, formed from distorted\nparallelepipeds, resembling a CDW.  CDWs have been observed previously\nin chalcogenides under pressure \\cite{Degtyareva2005a}.\nIn light of the charge transfer, the buckling of layers could be\nviewed not only as a symmetry breaking, but also as allowing the\npositive and negative ions to approach one another and reduce the\nenergy.\nIn addition, the charge transfer might create atomic sites similar to\nthose in carbon at comparable pressures, although the $Fdd2$\nbandstructure shows very different behaviour\n\\cite{Martinez2012-carbon}. While carbon expels charge from the $2s$\nlevels at high compression, eventually forming an electride, the\nbottom of the $Fdd2$ valence band shows a large $2s$ population.\nCharge depletion from the $2s$ levels will only arise with the\ntransition to the superconducting $R\\bar{3}m$ phase, at almost 12 TPa.\nElectride structures are found for example in carbon\n\\cite{Martinez2012-carbon} and aluminum \\cite{Pickard2010-Al} at very\nhigh pressures, but they do not to appear in nitrogen up to at least\n100 TPa.\nWhile intuition suggests that close-packed structures should be\nfavored under extreme pressures, this is not the case for nitrogen\neven at 100 TPa where, for example, the fcc structure is almost 2 eV\nper atom higher in enthalpy than $I4_1\/amd$.\n\n\n\n\\vspace*{5mm}\n\\noindent{\\bf\\large Methods}\n\n\\vspace*{2mm}\n\\noindent{\\bf Structure search.} \nThe {\\it ab initio} random structure searching (AIRSS) method\n\\cite{Pickard2006-silane,Pickard2011-review} was used to identify\nlow-enthalpy structures of nitrogen at multi-TPa pressures.\nExtensive searches were performed at selected combinations\nof 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,\n18, 19, 20, 21, 22, 24 nitrogen atoms per cell \nat 1, 2, 3, 4, 5, 7, 10, 15, 20, 50 and 100 TPa.\nMore than 47,000 structures in total were optimized using DFT.\nAdditional searches were performed using initial structures consisting\nof random packings of N$_2$ dimers and N$_5$ tetrahedra, in order to\nstudy the partially-ionic structures in more detail.  No better\nstructure than $P4\/nbm$ was found.\n\n\\vspace*{2mm}\n\\noindent{\\bf {\\it Ab initio} calculations.} We used the\n\\textsc{castep} plane-wave DFT code \\cite{CASTEP} and an ultrasoft\npseudopotential generated with a small core radius to avoid\noverlapping of the core radii at the high-pressures studied. The\nenthalpy-pressure relations of the structures were recalculated using\nvery hard Projector Augmented Wave (PAW) pseudopotentials and the VASP\ncode \\cite{vasp} with a plane-wave basis set cutoff energy of 1000 eV.\nPhonon and electron-phonon coupling calculations were performed using\nDFT perturbation theory \\cite{DFPT} with the \\textsc{ABINIT}\n\\cite{ABINIT} and Quantum ESPRESSO codes \\cite{Giannozzi2009}. Very\nhigh cutoff energies (1632 eV for \\textsc{ABINIT} and 952 eV for\nQuantum ESPRESSO), together with large numbers of k-points, were used\nto obtain convergence of the enthalpy differences between phases to\nbetter than $5 \\times 10^{-3}$ eV per atom.\nCalculations were performed with the different codes to cross-check\nthe transition pressures.\n\n\n\n\\begin{acknowledgments}\n\nJ.S.\\ gratefully acknowledges financial support from the Alexander von\nHumboldt (AvH) foundation and Marie Curie actions.\nM.M.C.\\ C.J.P.\\ and R.J.N.\\ were supported by the EPSRC.\nThe calculations were carried out at {\\sc Bovilab@RUB} (Bochum),\nLIDOng (Dortmund), on the supercomputers at NRC (Ottawa) and UCL\n(London).\n\\end{acknowledgments}\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}