|
{: sector encoded in the Standard Model (SM) of particle physics, it is reasonable to consider whether DM resides within a ``dark sector\ approximation. In this case, for the energies and momentum transfers of interest we can ignore the atomic binding energies, as long as we only consider the outer shell electrons. For the case of xenon atoms, used as target material in the current and planned large scale DM detectors, this \ncorresponds to electrons in the $n = 4$ and 5 levels, for a total effective charge of $Z_{eff} = 26$. In an approximation where the electrons are treated as free and \ninitially at rest, we find the differential cross section for DM electron scattering (for some relevant formalism, see for example Ref.~\\cite{Essig:2015cda}) \n\\begin{equation}\n\\frac{\\text{d} (\\sigma_e \\,v)}{\\text{d} E_R} =\\frac{8 \\pi \\,m_e\\, \\alpha\\, \\alpha_D \\epsilon^2 \\, Z_{eff}}{v\\, (2 m_e E_R + m_D^2)^2}\\Theta(2\\mu_{\\chi e}^2v^2\/m_e-E_R)\\,,\n\\label{diff-sigmav}\n\\end{equation}\nwhere the electron recoil energy is given by $E_R = |\\vec{q}|^2\/(2 m_e)$, with the magnitude of the three-momentum transfer denoted by $|\\vec{q}|$.\nThe step function provides the kinematic limit.\n\n\nWe can get the total cross section by integrating \\eq{diff-sigmav}. In order to regulate the infrared behavior of the cross section, we will introduce a threshold energy \n$E_{th}$, below which events are not registered by the experiment. We then find,\n\\begin{equation}\n(\\sigma_e \\, v) = \\frac{16 \\pi\\, \\alpha\\, \\alpha_D \\epsilon^2 Z_{eff} (\\mu_{\\chi e}^2 v^2 - m_e\\, E_{th}\/2)}{v\\, (2 m_e \\,E_{th} + m_D^2)(4 \\mu_{\\chi e}^2 v^2 + m_D^2)}\\,,\n\\label{sigmav}\n\\end{equation}\nwhere $\\mu_{e \\chi}$ is the reduced mass of the electron-DM system, $1\/\\mu_{\\chi e} \\equiv 1\/m_e + 1\/m_\\chi$. In the above, the maximum recoil energy is given by $E^{max}_R = 2 (\\mu_{\\chi e}^2\/m_e) v^2$. \n\nUsing \\eq{sigmav}, we can write down the expected rate per detector mass and year,\n\\begin{equation}\n\\frac{\\text{d} R}{\\text{d} t \\,\\text{d} M} = n_T\\, n_\\chi (\\sigma_e v)\\,,\n\\label{dRdtDM}\n\\end{equation}\nwhere $n_T = 6.02 \\times 10^{23}\\, \\text{g}^{-1}\/A$ is the number of target atoms per gram, with $A$ the target atomic mass, and $n_\\chi=r_v \\rho_\\chi\/m_\\chi$ is the number density of DM particles; the DM energy density is $\\rho_\\chi \\approx 0.3$~GeV cm$^{-3}$ \\cite{Tanabashi:2018oca} and the enhancement of the number density $r_v$ from \\eq{eq:rv}.\n\n\nIn the above, due to the nearly uniform boost of all DM to $v\\sim 0.1$ at the detector, we may approximate the DM velocity distribution by a delta function\n\\begin{equation}\nf(v) \\approx \\delta [v - v(R_\\oplus)]\\,, \n\\end{equation}\nnear the surface of the Earth. \n\n For light dark photons with $m_D\\lesssim 10$ keV the cross section is independent of the dark photon mass whereas for large dark photon masses the signal rates depends on $m_D^{-4}$.\nThese results need to be compared to constraints on the mass of a dark photon and its kinetic mixing taken from \\cite{Essig:2013lka}. We will restrict ourselves to the region between $100~\\text{eV}<m_D<1 $ MeV where the decay of the dark photon into SM fermions is not kinematically allowed.\nIn the region between $m_D>1$ eV up to 0.1 MeV strong constraints on the kinetic mixing come from stellar cooling of the Sun, of stars in the horizontal branch (HB), and for red giants (RG).\nWe note that there is a slight hint of new physics in HB cooling measurements which could be explained by a dark photon for parameters shown in fig.~\\ref{fig:results} \\cite{Giannotti:2015kwo}.\n\nBetween $m_D\\sim $ 0.1 and 1 MeV constraints on the kinetic mixing from the diffuse photon background (DPB) apply. However, in our model these constraints can be evaded by assuming a light dark fermion that would allow prompt {\\it invisible} decays of the dark photon. This may seem to lead to conflict with the number of relativistic degrees of freedom allowed during Big Bang Nucleosynthesis (BBN). However, for values of $\\epsilon \\lesssim 10^{-11}$ of interest in our work, the dark sector and the SM sector would not be in equilibrium and the dark sector could be much ``cooler\ which would lead to significant anisotropy of the signal. This hypothesis could be tested in experiments that have directional sensitivity \\cite{Sekiya:2003wf,Sciolla:2008vp,Daw:2011wq,Miuchi:2012rma,Santos:2013hpa,Battat:2013gma,Cappella:2013rua,DAmbrosio:2014arr,Couturier:2016isu,Hochberg:2016ntt,Kadribasic:2017obi,Rajendran:2017ynw,Budnik:2017sbu,Griffin:2018bjn,Coskuner:2019odd} as the anisotropy is different from both solar neutrinos and the conventional isotropic DM blizzard.\n\nThe long range interaction component of this model provides another unique, although difficult to test, prediction.\nConfirming a direct detection signal of DM would require multiple independent detections of the signal.\nDue to the velocity gain as DM falls into the Earth, this model predicts that the detection rate will be altitude dependent.\nThat is, we expect a very slightly higher rate at detectors in underground mines such as LZ at SURF which is 1.5 km below the surface than those at the surface such as XENON1T at Gran Sasso.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe have presented a unique model of dark matter (DM) wherein the Earth provides an attractive force on it due to an ultralight mediator.\nWhile this does not significantly modify the evolution of DM in the Galaxy, this potential does provide a large effect on the velocity distribution near the Earth, in particular by considerably adding to the velocity of 100\\% of the DM and yielding a nearly radial flux.\nThus, instead of $v\\sim10^{-3}$, all of the DM could have much higher velocities which considerably changes the phenomenology of low target mass recoil experiments such as electron recoils.\nIn addition, the resultant velocity distribution is highly peaked.\nWe have included a dark photon sector in our model to provide a testable interaction between DM and electrons.\nThis model is consistent with known astrophysical, cosmological, and laboratory experiments and possibly explains a tension in stellar cooling data.\n\nOur scenario is testable at low-threshold large-volume DM direct detection experiments such as XENON1T.\nIn light of the fact that XENON1T has recently seen a tantalizing excess of events at low recoils, we investigated the compatibility of this model with those data.\nWe found a good fit to the data for model parameters that are consistent with other bounds.\nIn addition, this model makes several distinguishing predictions.\nAlthough some would be extremely difficult to test without some rather extreme experiments -- such as a XENON1T like experiment in space or on the moon -- others are much more down to Earth. In particular, the scenario entails a nearly radial flux of high velocity DM at Earth surface, giving rise to ``dark matter rain,\,:{:}} |
|
{:\npart of the bosonized Hamiltonian has the form of coupled sine-Gordon terms\n\\begin{equation}\nH_{int}^{B}=\\sum_{\\alpha}g_{\\alpha}\\cos(a_{i}^{(\\alpha)}\\widetilde{\\psi}_{i})+\\sum_{\\beta}g_{\\beta}\\cos(A_{i}^{(\\beta)}\\widetilde{\\vartheta}_{i}),\\label{eq:5-1}\n\\end{equation}\nwhere $\\alpha=1-3$, $7-9$ and $\\beta=4-6$, and the coefficients,\n\\begin{align*}\na^{(1)} =\\left(\\begin{array}{ccc}\n\\frac{2\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\phi}}}, & 0, & 0\\end{array}\\right),\\\\\na^{(2)} =\\left(\\begin{array}{ccc}\n\\frac{\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\phi}}}, & \\frac{\\sqrt{6}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\phi}}}, & 0\\end{array}\\right),\\\\\na^{(3)} =\\left(\\begin{array}{ccc}\n\\frac{\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\phi}}}, & -\\frac{\\sqrt{6}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\phi}}}, & 0\\end{array}\\right),\\\\\na^{(7)} =\\left(\\begin{array}{ccc}\n0, & \\frac{4\\sqrt{\\pi}}{\\sqrt{6}\\sqrt{K_{\\bot}^{\\phi}}} & \\frac{4\\sqrt{\\pi}}{\\sqrt{3}\\sqrt{K_{0}^{\\phi}}}\\end{array}\\right),\\\\\na^{(8)} =\\left(\\begin{array}{ccc}\n\\frac{\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\phi}}}, & \\frac{2\\sqrt{\\pi}}{\\sqrt{6}\\sqrt{K_{\\bot}^{\\phi}}}, & -\\frac{4\\sqrt{\\pi}}{\\sqrt{3}\\sqrt{K_{0}^{\\phi}}}\\end{array}\\right),\\\\\n\\end{align*}\n\\begin{align*}\na^{(9)} =\\left(\\begin{array}{ccc}\n\\frac{\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\phi}}}, & -\\frac{2\\sqrt{\\pi}}{\\sqrt{6}\\sqrt{K_{\\bot}^{\\phi}}}, & \\frac{4\\sqrt{\\pi}}{\\sqrt{3}\\sqrt{K_{0}^{\\phi}}}\\end{array}\\right),\\\\\nA^{(4)} =\\left(\\begin{array}{ccc}\n\\frac{2\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\theta}}}, & 0, & 0\\end{array}\\right),\\\\\nA^{(5)} =\\left(\\begin{array}{ccc}\n\\frac{\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\theta}}}, & \\frac{\\sqrt{6}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\theta}}}, & 0\\end{array}\\right),\\\\\nA^{(6)} =\\left(\\begin{array}{ccc}\n\\frac{\\sqrt{2}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\theta}}}, & -\\frac{\\sqrt{6}\\sqrt{\\pi}}{\\sqrt{K_{\\bot}^{\\theta}}}, & 0\\end{array}\\right),\n\\end{align*}\nwhere the effective couplings $g_{\\alpha}(\\alpha=1-9)$ are linear\ncombinations of the couplings $g_{i}^{(j)}$ (see Appendix \\ref{app:A}). \nThe validity of the perturbative RG analysis we shall perform below requires the coupling constants $g_\\alpha$ to be small, and we assume this to be the case for the rest of the paper. However this limitation does not extend to the stiffnesses $K^{\\phi,\\theta},$ which may depart significantly from the noninteracting value $K^{\\phi,\\theta}=1,$ remaining within the purview of perturbative RG.\nIndeed, given our motivation of understanding electronic phase competition in the quantum limit in low-carrier density (and consequently strongly correlated) semimetals such as bismuth, in the rest of the paper we will largely focus on regimes where the stiffnesses appreciably depart from unity.\nNote that we allow the\npossibility of the coupling constants in the sine-Gordon model to\nbreak the $C_{3}$ permutation symmetry in the following analysis.\nThe same can also be done in the quadratic part and the two are equivalent. During the RG\nprocedure, the vectors $\\widehat{a}$ and $\\widehat{A}$, in general,\nrotate and stretch. The scaling dimensions for the interaction\nterms in Eq. \\ref{eq:5-1} depend on the values of the Luttinger parameters\n$K_{0}^{\\phi,\\theta}$ and $K_{\\perp}^{\\phi,\\theta}$, and in our\nanalysis, we only retain the most relevant interaction terms (with the smallest\nscaling dimensions). This further reduces the number of parameters\nwe need to consider in our model. \n\n\n\\section{\\label{sec:renormalization-group-analysis}renormalization-group\nanalysis}\n\nThe renormalization group follows the standard Wilsonian procedure\nof elimination of fast degrees of freedom, restoration of the cutoff,\nrescaling of the couplings and the renormalization of the fields.\nThis gives rise to off-diagonal\ncorrections in the stiffness matrices, which then take the form $Z_{\\mu\\nu}$. To keep the Gaussian\nfixed point unchanged, we rotate the $Z_{\\mu\\nu}^{\\theta,\\phi}$\nmatrices, to diagonalize them, and then rescale the fields $\\widetilde{\\phi}_{i}$ or $\\widetilde{\\theta}_{i}$ (using the eigenvalues of these matrices) such that\nthe matrices become proportional to identity. Note that the above\nrotation does not change the scaling dimensions of the sine-Gordon interaction\nterms. Now, in the new basis obtained after the rotation and the subsequent rescaling\nof the fields, we once again compute the one-loop corrections\nand the resulting changes in the diagonal and off-diagonal elements of the stiffness matrices,\nand repeat the aforementioned steps throughout the RG process. An\nequivalent procedure has been followed in Ref. \\onlinecite{PhysRevB.78.075124},\nwhere, instead of keeping the Gaussian fixed point unchanged, the\nfields are kept unchanged and the renormalization process leads to\nrotations and stretching of eigenvalues of the $Z_{\\mu\\nu}^{\\theta,\\phi}$\nmatrices. We simplify our analysis by considering the anisotropic\nlimits $K_{\\bot}^{\\phi}\\gg K_{0}^{\\phi}$ or $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$,\nwhich allows us to drop certain terms (which have higher scaling dimensions) in the interacting Hamiltonian\nin Eq. \\ref{eq:5-1} in each of these limits. However, the formulation may be readily extended to the most general case.\nWe note that the anisotropic limits $K_{\\bot}^{\\phi}\\gg K_{0}^{\\phi}$ or $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$ necessarily mean we are far from the noninteracting limit where $K^{\\phi,\\theta}\\approx 1.$ Our remaining analysis thus corresponds to a strong coupling limit of the model. \nBelow we discuss the results obtained by incorporating one-loop corrections\nto the matrices $Z_{\\mu\\nu}^{\\phi}$ and $Z_{\\mu\\nu}^{\\theta}$ in\nthe two aforementioned anisotropic parameter regimes. At any given stage of the\nRG, the matrix $Z_{\\mu\\nu}^{\\phi}$, with the one-loop corrections\nincorporated, is given by\n\n\\begin{widetext}\n\n\\begin{equation}\nZ^{\\phi}=\\left(\\begin{array}{ccc}\n\\frac{1}{2}+\\sum_{\\alpha}\\frac{g_{\\alpha}^{2}dy}{16\\pi}((a_{1}^{(\\alpha)})^{2}+(a_{-1}^{(\\alpha)})^{2})(a_{1}^{(\\alpha)})^{2} & \\sum_{\\alpha}\\frac{g_{\\alpha}^{2}dy}{16\\pi}((a_{1}^{(\\alpha)})^{2}+(a_{-1}^{(\\alpha)})^{2})(a_{1}^{(\\alpha)})(a_{-1}^{(\\alpha)}) & 0\\\\\n\\sum_{\\alpha}\\frac{g_{\\alpha}^{2}dy}{16\\pi}((a_{1}^{(\\alpha)})^{2}+(a_{-1}^{(\\alpha)})^{2})(a_{1}^{(\\alpha)})(a_{-1}^{(\\alpha)}) & \\frac{1}{2}+\\sum_{\\alpha}\\frac{g_{\\alpha}^{2}dy}{16\\pi}((a_{1}^{(\\alpha)})^{2}+(a_{-1}^{(\\alpha)})^{2})(a_{-1}^{(\\alpha)})^{2} & 0\\\\\n0 & 0 & \\frac{1}{2}\n\\end{array}\\right).\\label{eq:5-3}\n\\end{equation}\n\n\\end{widetext}\n\nNote that the above matrix is block-diagonal - a consequence of the\nnature of the interaction terms and\/or approximations employed in\nthe parameter regimes considered in our analysis. While the corrections\naccumulated are infinitesimal, the rotations involved in restoring\nthe matrices with off-diagonal contributions are finite rotations\nwhich cannot be accounted for in the RG flow equations. In our approach,\nwe are always in the rotating frame, where these large rotations are\nabsent, and only small incremental changes to the components along\nthe field directions need to be tracked. These amount to slow changes\nin the orientation and length, in the rotating frame, upon scaling.\nIn the limit where $K_{0}^{\\phi}\\ll K_{\\perp}^{\\phi}$, we find that\nwe only need to retain the couplings $g_{\\alpha}(\\alpha=1-3)$, based\non their lower scaling dimensions. In this case, we calculate one-loop\ncorrections to the $Z^{\\phi}$ matrices due to the terms $g_{1}$,\n$g_{2}$ and $g_{3}$ in the interaction Hamiltonian, and likewise,\nto the $Z^{\\theta}$ matrices due to the terms $g_{4}$, $g_{5}$\nand $g_{6}$. The corresponding matrix turns out to be block-diagonal\ndue to the symmetry of the interaction terms in this regime. On the\nother hand, in the limit where $K_{\\perp}^{\\phi}\\ll K_{0}^{\\phi}$,\nonly the couplings $g_{\\alpha}(\\alpha=7-9)$ need to be retained for\nour analysis. Here we obtain one-loop corrections to the $Z^{\\phi}$ matrices\narising from the couplings $g_{7}$, $g_{8}$ and $g_{9}$, and, once\nagain, to the $Z^{\\theta}$ matrices due to the terms $g_{4}$, $g_{5}$\nand $g_{6}$. In this case, the matrix $Z^{\\phi}$ generally comprises nonzero corrections to every matrix element.\nHowever, in the limit $K_{\\perp}^{\\phi}\\ll K_{0}^{\\phi}$,\nwe can drop certain terms and it reduces to a block-diagonal form\nsimilar to Eq. \\ref{eq:5-3} above with $\\alpha=7-9$. \n\nIn our analysis, we track the scaling equations for the interaction couplings, as well as the coefficients of the fields in the sine-Gordon terms. The eigenvalues of the matrix $Z_{\\mu\\nu}$ in Eq. \\ref{eq:5-3}\nabove are denoted by $z_{1}$, $z_{-1}$ and $z_{0}$. We diagonalize the\nmatrix and then rescale the fields using these eigenvalues. At any given\nstage of the RG flow, the coefficients of the fields in the cosine terms evolve in the manner $a_{i}^{(\\alpha)}\\rightarrow\\frac{(Ra^{(\\alpha)})_{i}}{\\sqrt{z_{i}}}$,\nwhere $R$ is the rotation which diagonalizes the matrix $Z_{\\mu\\nu}$.\nWe continue to denote the interaction terms as $g_{\\alpha}\\cos[\\widehat{a}_{i}^{(\\alpha)}\\widetilde{\\psi}_{i}]$\nor $g_{\\alpha}\\cos[\\widehat{A}_{i}^{(\\alpha)}\\widetilde{\\vartheta}_{i}]$, and write down the RG equations for the coefficients ${a}_{i}^{(\\alpha)}$, $A_{i}^{(\\alpha)}$ and the couplings $g_{\\alpha}$.\nAs an example, proceeding in incremental steps, the RG equations for the coefficients $a_{1}^{(1)}$\nand $a_{-1}^{(1)}$ (corresponding to the coupling $g_{1}$) due to the rescaling process described above, are given by \n\\begin{align}\n\\frac{da_{1}^{(1)}}{dy} & =-a_{1}^{(1)}\\Lambda_{1}\\nonumber \\\\\n\\frac{da_{-1}^{(1)}}{dy} & =-a_{-1}^{(1)}\\Lambda_{-1}\\label{eq:14-2}\n\\end{align}\n where $z_{1}=1\/2+\\Lambda_{1}dy$ and $z_{-1}=1\/2+\\Lambda_{-1}dy$, with $\\Lambda_{1}$ and $\\Lambda_{-1}$ depending upon all\n the coupling constants and the coefficients of all the fields in the sine-Gordon terms (see Appendix \\ref{app:A}, for the explicit expressions of $\\Lambda_{1}$ and $\\Lambda_{-1}$). The leading corrections are quadratic in the coupling constants. This is not surprising since the RG equations of Eq. \\ref{eq:14-2} essentially describe the renormalization of the stiffness constants $K^{\\phi,\\theta}$, which do not have $O(g)$ tree-level corrections.\nThe RG flow equations for the rest of the components $a_{i}^{(\\alpha)}$\nalso behave in the same way. \n\nThe tree-level contributions to the RG flows\nfor the sine-Gordon couplings $g_{\\alpha}$ are obtained in terms of the scaling dimensions of the respective sine-Gordon terms, and the\none-loop contributions are obtained using the Operator Product Expansion (OPE).\nThe RG equations for the couplings $g_{\\alpha},\\alpha=1-3$ are\n\\begin{align}\n\\frac{dg_{1}}{dy} & =g_{1}(2-\\frac{1}{4\\pi}((a_{1}^{(1)})^{2}+(a_{-1}^{(1)})^{2}),\\nonumber \\\\\n & +\\frac{1}{8\\pi}(a_{1}^{(2)}a_{1}^{(3)}+a_{-1}^{(2)}a_{-1}^{(3)})g_{2}g_{3},\\nonumber \\\\\n\\frac{dg_{2}}{dy} & =g_{2}(2-\\frac{1}{4\\pi}((a_{1}^{(2)})^{2}+(a_{-1}^{(2)})^{2}),\\nonumber \\\\\n & -\\frac{1}{8\\pi}(a_{1}^{(1)}a_{1}^{(3)}+a_{-1}^{(1)}a_{-1}^{(3)})g_{1}g_{3},\\nonumber \\\\\n\\frac{dg_{3}}{dy} & =g_{3}(2-\\frac{1}{4\\pi}((a_{1}^{(3)})^{2}+(a_{-1}^{(3)})^{2}),\\nonumber \\\\\n & -\\frac{1}{8\\pi}(a_{1}^{(1)}a_{1}^{(2)}+a_{-1}^{(1)}a_{-1}^{(2)})g_{1}g_{2}\\label{eq:13-1}\n\\end{align}\nThe RG equations for the rest\nof the couplings $g_{\\alpha}(\\alpha=4-9)$ are also easily obtained and have a similar form as Eq. \\ref{eq:13-1}.\n\n\\subsection*{One-loop corrections to the RG equations}\n\nThe $O(g^2)$ one-loop (or OPE) contributions to the renormalization of the coupling constants $g_{\\alpha}$ in Eq. \\ref{eq:13-1} above are perturbatively smaller than the leading tree-level term. In contrast, the OPE contribution is the leading one in the RG equation for the coefficients of the fields in the cosine terms, given in Eq. \\ref{eq:14-2}, which determine the scaling dimensions of the interaction terms. In general, one-loop corrections can have a significant effect on the RG flows when the tree-level term is small \\textendash{} the usual motivation for considering higher order corrections in the perturbative RG. However, we found that if the initial values of the sine-Gordon couplings are small, and the initial stiffnesses are appreciably different from unity (reflecting the strongly correlated nature of our problem), the RG equations with or without the one-loop corrections generically give very similar solutions (see Fig. \\ref{fig:tlol}). If the initial scaling dimensions of the interaction terms are close to two (i.e. the tree-level contribution is small), or the bare values of the couplings are not sufficiently small (so that the one-loop and tree-level terms are comparable), then the one-loop terms need to be taken into account. This requires a separate, more detailed study and is not attempted in the present work.\n\\iffalse\nNote that, while in general, the one-loop corrections would have the nontrivial effect of coupling different interaction channels with each other, and are thus expected to give\nqualitatively different results, in our case, such a coupling is already present at the tree-level order.\n\\fi\n\nWe solve the coupled\ndifferential equations 7 and 8 numerically and obtain the fixed-point values\nfor the various couplings $g_{\\alpha}$ and the coefficients $a_{i}^{(\\alpha)}.$\nWe consider weak repulsive interactions in every channel, and study\nthe nature of the RG flows as a function of the initial conditions\non the interactions and the value of the Luttinger liquid parameter\n$K_{\\bot}^{\\phi,\\theta}.$ In general, we find that the couplings\n$g_{\\alpha}$ either diverge or flow to zero in the course of the\nRG flow. From Eq. \\ref{eq:14-2} above, it is clear that the coefficients\n$a_{1}^{(\\alpha)}$ and $a_{-1}^{(\\alpha)}$ obey different RG equations,\nand show qualitatively different behavior as a function of the RG\nflow parameter. In other words, the coefficients of the different\nfields rescale differently in the course of the RG flow, following\nthe rotation of the stiffness matrix. \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=1.0\\columnwidth]{fig2}\n\\par\\end{centering}\n\\caption{\\label{fig:The-figure-shows}The figure shows a schematic illustration\nof our renormalization group procedure. The stiffness matrix, which\nis initially diagonal, develops off-diagonal corrections in the course\nof the RG flow and takes the general form $Z_{\\mu\\nu}$. This matrix\nis diagonalized, which leads to a rotation $R$ of the coefficients\n$a^{(\\alpha)}$ of the sine-Gordon interaction terms. The diagonal elements\nare then absorbed in the respective sine-Gordon fields, which brings\nthe stiffness matrix back to unity, and leads to a rescaling of the\nrotated coefficients $a^{(\\alpha)}$. }\n\\end{figure}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{fig3}\n\\par\\end{centering}\n\\caption{\\label{fig:tlol}The figure compares \nthe generic scaling behavior of the coupling $g_1$ with and without considering the effect of\nthe one-loop corrections in the scaling equations for the coupling constants. The parameters have been chosen such that the initial value of the tree-level term exceeds the one-loop contribution. The blue and red circles correspond to the cases with and without the one-loop contributions, respectively. The initial values of the couplings considered are $g_{1}=0.3$, $g_{2}=0.1$, $g_{3}=0.05$, and\nthe value of the Luttinger parameter $K_{\\perp}^{\\phi}=0.1.$ Clearly, the two sets of equations, with or without the one-loop contributions, give very similar results in this regime.}\n\\end{figure}\n\n\\begin{figure}\n\\begin{centering}\n(a)\\includegraphics[width=0.9\\columnwidth]{fig4a}\n\\par\\end{centering}\n\\begin{centering}\n(b)\\includegraphics[width=0.9\\columnwidth]{fig4b}\n\\par\\end{centering}\n\\caption{\\label{fig:rgflows} The figure shows the RG flows for the couplings $g_{1}$,\n$g_{2}$ and $g_{3}$ for the Luttinger parameter $K_{\\perp}^{\\phi}=0.1$ and initial conditions $g_{1}^{0}=0.3$, $g_{2}^{0}=0.1$, $g_{3}^{0}=0.05$. While $g_{1}$ grows monotonously (see (a))\nunder these conditions, $g_{2}$ and $g_{3}$ show a decline (see (b)). In general, any one or more of the couplings $g_{\\alpha}$ may diverge, depending on the initial conditions chosen. The RG flows of the couplings $g_{i}$,$i=4-9$ behave in a manner qualitatively similar to that of $g_{1}$, $g_{2}$ and $g_{3}$.}\n\\end{figure}\n\n\\section{\\label{sec:order-parameters-and}phase diagram and critical behavior}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=1.0\\columnwidth]{fig5}\n\\par\\end{centering}\n\\caption{\\label{fig:scalingcollapse}The figure shows a scaling collapse plot\nof the RG flow parameter $y\\sim\\mathrm{ln}[\\xi]$ (where $\\xi$ is\nthe correlation length) as a function of $\\frac{1}{\\sqrt{K_{\\bot}^{\\phi}-K_{c}}}$.\n$K_{c}$ denotes the critical value of the Luttinger liquid parameter\n$K_{\\bot}^{\\phi}$, where the system undergoes a phase transition.\nThe plot shows results for five different sets of initial conditions\non the interactions, with one or more of the couplings $g_{\\alpha}$\ntaking non-zero values initially, and indicates that the phase transitions\noccuring in this system are continuous in nature and belong to the\nBKT universality class. }\n\\end{figure}\n\nThe order parameters considered in our analysis are fermionic bilinear\noperators characterized by chirality and band indices. There are two\nclasses of order parameters in our system. These are defined in the\nparticle-hole channel (density wave), \\cite{PhysRevB.94.205129}\n\\begin{equation}\n{\\rm Re}[O_{ph}^{i0}]\\propto\\sum_{mm^{\\prime}}\\lambda_{mm^{\\prime}}^{i}\\psi_{Rm}^{\\dagger}\\psi_{Lm^{\\prime}}+{\\rm h.c,}\\label{eq:6-1-1-1}\n\\end{equation}\nand in the\nparticle-particle channel (superconductivity), \n\\begin{equation}\n{\\rm Re}[O_{pp}^{i0}]\\propto\\sum_{mm^{\\prime}}\\lambda_{mm^{\\prime}}^{i}\\psi_{Rm}^{\\dagger}\\psi_{Lm^{\\prime}}^{\\dagger}+{\\rm h.c},\\label{eq:7}\n\\end{equation}\nwhere $\\lambda^{i} (i=1...8)$ correspond to the Gell-Mann matrices\n(see Appendix \\ref{app:B} for details), $\\lambda^{0}$ denotes the 3x3 unit\nmatrix, and $\\psi_{pm}$($\\psi_{pm}^{\\dagger}$) is the electron annihilation\n(creation) operator with chirality $p$ and band $m$.\nWe follow the convention used by Ref.\n\\onlinecite{PhysRevB.94.205129} ; \nhowever, in both the Eqs. \\ref{eq:6-1-1-1} and \\ref{eq:7}, no spin\nindices are present, due to the spinless nature of the fermions, indicated by the second index being $0$ for the order parameters.\nNote that we consider ordered states arising from scattering or pairing\nin opposite chiralities in this analysis, and we have checked that\nequal-chirality interband pairing terms show a qualitatively similar\nbehavior. The order parameters in Eqs. \\ref{eq:6-1-1-1} and \\ref{eq:7}\nabove are expressed in terms of the bosonic fields. A total of\neighteen order parameters are obtained in the particle-hole and particle-particle\nchannels in the spinless case (see Appendix \\ref{app:B} for expressions of the order parameters in terms of the bosonic fields). \n\nWe now discuss the physical meaning of the electronic phases corresponding to the above order parameters. In the anisotropic strong coupling regime that we study (where the initial $K_{\\perp}^{\\phi}$ value is often far from unity and the initial $g_i$ are generically unequal), the phases that are obtained are typically associated with the breaking of valley permutation or bond permutation symmetries. However, we also find phases with the $C_{3}$ symmetry restored, not slaved to the initial conditions where this is explicitly broken (see Appendix B). Interband pairing \nin the particle-hole channel corresponds to a bond-ordered (BO) phase, while in the particle-particle channel it\ngives rise to superconductivity at a finite wavevector (FFLO) equal to the separation between two small Fermi \npockets in momentum space, $Q$. The intraband order parameters correspond to linear combinations of the fermionic bilinears on\nthe three different pockets. One of them is a symmetric linear combination ($s-$wave, denoted by SW) while the other two are nematic,\ncorresponding to angular momentum $l=2$ ($d-$wave order). If we ascribe the angular positions of the three patches in momentum space as $\\delta=0$, $\\delta=2\\pi\/3$ and $\\delta=4\\pi\/3$,\nthe phases of the order parameters on the three valleys go either as $\\cos(2\\delta)$ or $\\sin(2\\delta)$, both of the $d-$wave type. \nIt is also possible to have chiral orders, with phases going as $\\exp(\\pm i \\delta)$, as a linear combination of nematic orders. These linear combinations are not\nunique, and depending on the initial conditions, the actual order parameter may be some combination of these. \nIntraband pairing in the particle-hole channel has an ordering wavevector $2k_{F}$, much less than $Q$, and is generally incommensurate. Depending on the initial conditions,\nthe CDW (charge density wave) order could involve a linear combination of the CDW orders on the three different patches. If $C_{3}$ symmetry is not broken, then the orders may have $s-$wave (uniform CDW, denoted by UCDW), or a \ndoubly degenerate $d-$wave symmetry ($d-$density wave). As was the case for superconductivity, the $d-$density wave order can be either nematic (denoted by NCDW) or chiral type (denoted by cCDW). The order parameters corresponding to different types of order are listed in Table \\ref{tab:phases}. \n\nTo study the dominant electronic orders, we introduce, in the disordered phase, test vertices corresponding to various order parameter fluctuations\nand determine how they grow or shrink upon scaling. The evolution of any particular order parameter is governed by a certain combination of couplings, and the one \nwith the smallest scaling dimension, such that the divergence is strongest upon scaling, is the dominant order. Those order parameters that initially have a large scaling dimension do not grow under scaling and correspond to short-range order. We also take into account the corrections to the scaling dimensions to leading order, $O(g)$ in the couplings, as these terms sometimes lead to shifts in the scaling dimensions of order parameters that have identical RG equations at the tree-level order, resulting in the lifting of degeneracies, with one of them becoming long-range ordered and the other short-range ordered (see Appendix \\ref{app:B}). \n\nIn order to determine the winning\norder parameters, we consider the behavior of the couplings $g_{\\alpha}$\nand the corresponding coefficients of the fields $a_{i}^{(\\alpha)}$ near\nthe fixed point of the RG for a given set of initial conditions and find\nthat both quantities play a crucial role in deciding the nature of\nthe dominant electronic orders. In some cases, we find that none of\nthe order parameters we studied grows under RG, implying the absence\nof any quasi-long range ordered state despite the presence of interactions. Such situations also come up in the context of floating phases in coupled sine-Gordon models. The advantage of our method is that it not only gives us the dominant\norder parameters, but also yields the scaling dimension at the fixed point which is essentially the exponent of power law correlations of the order parameter fields\nin the quasi-long range ordered state. Later, we will show that the transitions, where they occur, belong to the Berezinskii-Kosterlitz-Thouless (BKT) universality\nclass, and that the correlation functions diverge upon \napproaching the critical point, in accordance with the BKT law. \n\n\\begin{table*}\n\\begin{centering}\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n\\multicolumn{2}{|c|}{Type of order} & Order parameter & Name of order\\tabularnewline\n\\hline \n\\hline \n\\multirow{4}{*}{Interband} & \\multirow{2}{*}{p-p } & $O_{pp}^{10}$,$O_{pp}^{40}$,$O_{pp}^{60}$ & FFLO(wavevector $Q$)\\tabularnewline\n\\cline{3-4} \\cline{4-4} \n & & $O_{pp}^{20}$,$O_{pp}^{50}$,$O_{pp}^{70}$ & FFLO(wavevector $Q$)\\tabularnewline\n\\cline{2-4} \\cline{3-4} \\cline{4-4} \n & \\multirow{2}{*}{p-h } & $O_{ph}^{10}$,$O_{ph}^{40}$,$O_{ph}^{60}$ & Bond order (BO)(wavevector $Q$)\\tabularnewline\n\\cline{3-4} \\cline{4-4} \n & & $O_{ph}^{20}$,$O_{ph}^{50}$,$O_{ph}^{70}$ & Bond order (BO)(wavevector $Q$)\\tabularnewline\n\\hline \n\\multirow{4}{*}{Intraband} & \\multirow{2}{*}{p-p } & \\multirow{2}{*}{$O_{pp}^{00}$,$O_{pp}^{30}$,$O_{pp}^{80}$} & \\multirow{2}{*}{$s-$wave (SW), Nematic $d-$wave, Chiral $d-$wave}\\tabularnewline\n & & & \\tabularnewline\n\\cline{2-4} \\cline{3-4} \\cline{4-4} \n & \\multirow{2}{*}{p-h } & \\multirow{2}{*}{$O_{ph}^{00}$,$O_{ph}^{30}$,$O_{ph}^{80}$} & \\multirow{2}{*}{Uniform(U) CDW , nematic (N) $d-$CDW , chiral (c) $d-$CDW }\\tabularnewline\n & & & \\tabularnewline\n\\hline \n\\end{tabular}\n\\par\\end{centering}\n\\caption{\\label{tab:phases} Table showing electronic phases corresponding to each of the order parameters considered in our analysis. Here particle-particle (p-p) refers to superconductivity, while particle-hole (p-h) refers to density wave orders. Interband pairing between different\npairs of bands in the particle-hole channel leads to bond order (denoted by BO) while\nthe corresponding pairing in the particle-particle channel leads to\na finite-momentum pairing (denoted by FFLO) state with the wavevector $Q$, equal to the separation between two small Fermi pockets in momentum space. Intraband pairing can correspond\nto a situation with different phases on different Fermi pockets and\nlead to uniform charge density wave (denoted by UCDW) or nematic $d-$density wave\norder (denoted by NCDW) in the particle-hole channel, and $s-$wave or nematic $d-$wave\nsuperconductivity in the particle-particle channel. In the case where\nthese different order parameters are degenerate, a combination of\nthem which is chiral in nature gives rise to the lowest\nenergy configuration. In such a situation, a\nchiral $d-$density wave (denoted by cCDW) or chiral $d-$wave superconductivity can\nbe realized. Despite choosing initial conditions that generically break $C_3$ permutation symmetry, one nevertheless finds that in some parameter regimes (see text, Fig. \\ref{fig:pd}), phases with the $C_{3}$ symmetry restored, such as the chiral orders, are dominant.}\n\n\\end{table*}\n\n\nWe classify the nature of the dominant orders in different parameter regimes depending upon the relative sizes of $K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}$, considering the two broad classes of parameters, $K_{0}^{\\phi}\\gg K_{\\perp}^{\\phi}$ and $K_{\\perp}^{\\phi}\\gg K_{0}^{\\phi}$. Clearly, this implies some $K_{\\perp}^{\\phi}$ values must necessarily take values far from the noninteracting point $K_{\\perp}^{\\phi}=1,$ i.e., we are in a strong-correlation regime that is nevertheless accessible by perturbative RG. Within each of these classes, we further examine situations with either $K_{0}^{\\phi}\\gg1$ or $K_{0}^{\\phi}\\ll1$.\nThe case with $K_{0}^{\\phi}\\sim1$, involving a competition between different types of orders, depending upon the initial conditions, requires a more detailed study,\nand has not been addressed here. In the regime where $K_{0}^{\\phi}\\gg K_{\\perp}^{\\phi}$\nand $K_{0}^{\\phi}\\ll1$, the dominant instabilities are found in the\nintraband particle-particle channel. Similarly, in the regime where\n$K_{\\perp}^{\\phi}\\gg K_{0}^{\\phi}$ and $K_{0}^{\\phi}\\gg1$, the dominant\ninstabilities are found in the intraband particle-hole channel. Note that in these two parameter regimes, $K_{\\perp}^{\\phi}$ is automatically constrained to be numerically very small or very large. We\nnow consider the remaining two cases, which allow us to tune $K_{\\bot}^{\\phi}$\nover a wide range of values, giving rise to both intraband and interband orders. \n\nWe find that for $K_{0}^{\\phi}\\gg K_{\\perp}^{\\phi}$ and $K_{0}^{\\phi}\\gg1$,\nthe particle-hole orders are more relevant than the particle-particle orders, due to smaller scaling\ndimensions of the corresponding order parameters, and for $K_{\\perp}^{\\phi}\\gg K_{0}^{\\phi}$\nand $K_{0}^{\\phi}\\ll1$, the particle-particle orders are likewise found to\nbe more important. Within the regimes considered by us,\nthe phase diagram is affected primarily by two factors: the magnitude\nof the Luttinger liquid parameter $K_{\\bot}^{\\phi}$ and the set of\ninitial conditions considered for the interactions $g_{\\alpha}$. The nature of the phase transitions is studied using a numerical scaling analysis. The scaling of the correlation length $\\xi$ at the critical point\nis determined by identifying the characteristic scale $y$ where the\ncouplings $g_{\\alpha}(y)$ cross a designated value $\\apprge1$. We\nobtain continuous transitions as a function of $K_{\\bot}^{\\phi}$,\nbelonging to the Berezinskii-Kosterlitz-Thouless (BKT) universality\nclass, which is confirmed by demonstrating the universal BKT scaling\ncollapse for the behavior of the correlation length close to the critical\npoint (see Fig. \\ref{fig:scalingcollapse}). Note that the critical\nvalue $K_{c}$ of the Luttinger parameter $K_{\\bot}^{\\phi}$ is different\nfor different initial conditions on the couplings $g_{\\alpha}$, as\nshown in Fig. \\ref{fig:scalingcollapse}, each of which give rise to the same\ncritical behavior. \n\n\n\\begin{figure*}\n\\begin{centering}\n\\includegraphics[width=1.6\\columnwidth]{fig6}\n\\par\\end{centering}\n\\caption{\\label{fig:pd} The figure shows the phase diagram for a system of\nthree coupled spinless Luttinger liquids as a function of $K_{\\bot}^{\\phi}$, considering the parameter regimes\n(a) $K_{\\perp}^{\\phi}\\gg K_{0}^{\\phi}$ and $K_{0}^{\\phi}\\ll1$, where\nonly particle-particle (p-p) orders are considered due to the smaller\nscaling dimensions of the corresponding order parameters, (b) $K_{\\perp}^{\\phi}\\gg K_{0}^{\\phi}$ and $K_{0}^{\\phi}\\gg1$, where\nthe dominant instabilities belong to the intraband\nparticle-hole channel, (c) $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$\nand $K_{0}^{\\phi}\\ll1$ where the dominant instabilities occur in the intraband particle-particle channel, and\n (d) $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$\nand $K_{0}^{\\phi}\\gg1$, where only particle-hole (p-h) orders are\nconsidered in our analysis, due to smaller scaling dimensions of the corresponding terms. In cases (a) \nand (d), we can tune $K_{\\bot}^{\\phi}$ over a large range of values,\nand for $K_{\\bot}^{\\phi}\\sim1$, various interband and intraband\norders compete with one another, the winner being determined\nby the initial conditions on the interactions. Note that our results are not reliable for $K_{\\bot}^{\\phi}=1$ in regime (a), where the one-loop corrections must be taken into account. The orders indicated in the figure have \nbeen denoted in the paper as SW for $s-$wave, FFLO for finite-momentum pairing, UCDW as a CDW order with s-wave symmetry, NCDW as nematic $d-$density wave, cCDW as chiral $d-$density wave and BO as bond order. The shaded (gray) portions of the phase diagram demarcate the parameter regimes which can be understood from our analysis.The boundaries of different types of phases are flexible in nature, and can change depending on the initial conditions chosen for the couplings. }\n\\end{figure*}\n\n\nBelow we discuss the salient features of the phase diagram for the aforementioned two parameter regimes, $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\gg1$,\nor $K_{0}^{\\phi}\\ll K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\ll1$, each corresponding to a range of values of $K_{\\bot}^{\\phi}$. Since\n$K_{\\bot}^{\\theta}$ is inversely related to $K_{\\bot}^{\\phi}$ in\nour model, it does not constitute an independent parameter in the\nphase diagram. \n\n\\paragraph*{$K_{\\bot}^{\\phi}\\ll1$:}\n\nIn this regime, for $K_{0}^{\\phi}\\ll K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\ll1$,\nthe intraband particle-particle orders (SW, Nematic, Chiral) are found to be more relevant, whereas\nfor $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\gg1$, \nno electronic orders are present when we consider extremely small values of $K_{\\bot}^{\\phi}$,\nand for larger values of $K_{\\bot}^{\\phi}$, a particular pair of\ninterband particle-hole orders (BO) dominates, depending upon the initial\nconditions being considered for the interactions. \n\n\\paragraph*{$K_{\\bot}^{\\phi}\\sim1$: }\n\nFor $K_{\\bot}^{\\phi}\\sim1$, various intraband and interband particle-particle (FFLO, SW, Nematic, Chiral) orders compete with each other in the regime $K_{0}^{\\phi}\\ll K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\ll1$, and likewise, various particle-hole (UCDW, NCDW, cCDW,BO) orders compete with each other in the regime $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\gg1$, and it is in this part of the phase diagram that the winning phases\nare dependent most sensitively on the initial conditions chosen for\nthe interactions. However, at $K_{\\bot}^{\\phi}=1$ for $K_{0}^{\\phi}\\ll K_{\\bot}^{\\phi}$, or very close to this point, the one-loop corrections should be taken into account, and our analysis in this regime requires further work. \n\n\\paragraph*{$K_{\\bot}^{\\phi}\\gg1$: }\n\nIn this case, for $K_{0}^{\\phi}\\ll K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\ll1$,\na particular pair of interband particle-particle orders (FFLO) is found to\ndominate, depending on the initial conditions\nchosen for the interactions, and no order is found to be present when we consider extremely\nlarge values of $K_{\\bot}^{\\phi}$, whereas for $K_{0}^{\\phi}\\gg K_{\\bot}^{\\phi}$ and $K_{0}^{\\phi}\\gg1$,\nthe intraband particle-hole orders (UCDW, NCDW, cCDW) are found to be more relevant. \n\nThe types of electronic orders occurring in different parameter\nregimes, considered in our analysis, are schematically shown in Fig. \\ref{fig:pd}. \n\n\\section{\\label{sec:discussion-and-conclusions}discussion and conclusions}\n\nIn summary, we have studied competing electronic phases and phase\ntransitions in a system of three coupled spinless Luttinger liquids\nusing a renormalization group analysis of the bosonized interactions\nthat takes into account off-diagonal contributions arising from one-loop corrections to the stiffness matrices. This is done by introducing a series of rotations and rescalings of the fields (or equivalently, the\ncoefficients of different fields in the sine-Gordon interaction terms) in the course of the RG flow.\nThese rotations and rescalings are found to depend on all the\ncouplings as well as coefficients of all the fields present in the system. They couple the different interaction channels even at the tree-level order. To determine the most dominant electronic orders, we introduce,\nin the disordered phase, test vertices corresponding to\nvarious order parameter fluctuations and study their evolution under the renormalization group. We find that the overall nature\nof the winning orders in different parameter regimes is governed by\nthe RG flows of the couplings, as well as those of the coefficients\nof the fields in the sine-Gordon terms. Notably, for a range of values of the Luttinger liquid parameter $K_{\\bot}^{\\phi}$, which depart appreciably from the noninteracting limit $K_{\\bot}^{\\phi}=1,$ interband\norders involving any one\npair of bands are found to be dominant, the specific pair being determined by the initial conditions\nfor the couplings. This is an example of valley symmetry breaking. At $K_{\\bot}^{\\phi}=1$ for $K_{\\perp}^{\\phi}\\gg K_{0}^{\\phi}$, one-loop corrections to the RG equations must be taken into account, and this aspect of our analysis requires further work. In the regions where intraband orders are the most relevant, they can be chiral in nature. Such orders restore the original $C_{3}$ symmetry of the system, broken explicitly through the initial conditions for the couplings. In the regimes where $K_{\\bot}^{\\phi}\\sim1$,\nthe nature of the dominant orders is found to be sensitively determined by the initial conditions on the interaction couplings, with multiple orders competing closely. For simplicity of analysis, we have considered\nthe strong correlation regimes of $K_{0}^{\\phi}\\gg1$ or $K_{0}^{\\phi}\\ll1$, and the more involved\ncase of $K_{0}^{\\phi}\\sim1$ has not been discussed, where the particle-particle and particle-hole channels compete\nwith each other and the results are likely to be sensitive to the\ninitial conditions considered. This will be taken up in a future work.\nWe also determine the nature of the phase transitions as a function of\nthe Luttinger parameter $K_{\\perp}^{\\phi}$ as well as the initial\nconditions on the interactions $g_{\\alpha}$ using a numerical scaling\nanalysis. The system hosts continuous transitions belonging to the BKT universality class, where the critical value of $K_{\\bot}^{\\phi}$ differs with the initial values of the couplings.\n\nFrom an experimental point of view, our analysis is expected to be relevant for studying electronic interaction effects in semimetals with three small Fermi pockets under conditions of high magnetic fields such that the bands are effectively in the quantum limit, and may be regarded as one-dimensional. Examples include bismuth, the graphite intercalation compounds and possibly the heavy fermion semimetal UTe$_{2}$ at high magnetic fields. For bismuth, when the magnetic field is aligned along the highest symmetry axis (the trigonal axis), a field of 9 T allows one to attain the\nquantum limit putting carriers in their lowest Landau level.\\cite{Yang2010} In this situation,\nCoulomb interaction effects play an important role in determining the electronic phase. The presence of anomalous features in the magnetization \\cite{Li2008} and the Nernst response \\cite{Behnia1729} of bismuth at high fields, beyond the quantum limit, points towards the importance of examining possible electronic instabilities due to interaction effects in this regime. \nFurthermore, there has been experimental evidence for valley symmetry breaking at high magnetic fields in bismuth, \\cite{Kuchler2014} and the importance of electron correlations for the same has been recognized.\nFrom recent magnetoresistance studies, one or two valleys have been observed to become completely empty above a threshold magnetic field. \\cite{Zhu2018}\nMoreover, in semi-metallic bismuth the flow of Dirac fermions along the trigonal axis \nis extremely sensitive to the orientation of in-plane magnetic field. \nIn the vicinity of the quantum limit, the orientation of magnetic field significantly affects the distribution of carriers in each valley, \nand the valley polarization is induced by the magnetic field. As the temperature is decreased or the magnetic field increased, the symmetry between the three valleys is spontaneously lost. We expect our technique to be useful for theoretically describing such a situation in bismuth, incorporating the features known from experiment, and predicting possible electronic instabilities. \n\nIn graphite intercalates, the Fermi level often naturally lies in the vicinity of the M-points in the Brillouin zone, which gives rise to another system with three small Fermi pockets. Superconductivity has been predicted and observed experimentally in multiple graphite intercalation compounds, such as CaC$_{6}$,YbC$_{6}$ and KC$_{8}$, \\cite{Weller2005} but the possibility of realizing superconductivity or a density wave order under a high magnetic field in such materials has not received much attention in the literature. The case of pure graphite is different; there is evidence for a high field-induced CDW transition \\cite{Yoshioka1981} resulting from the enhancement of interactions due to the confinement effect of the magnetic field. However valley-symmetry breaking in graphite occurs between the K and K$^{\\prime}$ points, which is not the subject of this paper.\nCorresponding field-induced phase transitions in graphite intercalates may, however, be accessible using our analysis. \n\nThe recently discovered heavy fermion triplet superconductor UTe$_{2}$,\\cite{Ran684, Aoki2019, Metz2019,Ishizuka2019,Jiao2019, Sundar2019} with a transition temperature $T_{sc}$=1.6 K, exhibits two independent high-field superconducting phases,\\cite{Ran2019} one of which has an upper critical field exceeding 65 T, and lies within a field-polarized phase. Such re-entrant superconducting phases are observed for selective ranges of orientation of the field. \\cite{Ran2019, Knebel2019} High-resolution ARPES data for UTe$_{2}$ indicates that it has three small Fermi pockets.\\cite{Miao2020} A quasi-1D bandstructure has been indicated both by bandstructure calculations and the ARPES studies. Our analysis is expected to be applicable at the highest fields, with electrons fully spin-polarized and in the quantum limit. A field of 65 T corresponds to a magnetic length of about 3.2 nm, which would require a carrier density of about 7x10$^{18}$ cm$^{-3}$ to be in the quantum limit, typical for semimetallic systems\\cite{Akiba2015}. \n\nIn the present work, we have not studied the case where $K_{0}^{\\phi}\\sim K_{\\bot}^{\\phi}$,\nwith the rotations being in general O(3) matrices. The rotation matrices\nin that case are non-abelian and it would interesting to see if this\ngives qualitatively new insights into the problem. In this regime,\nwe also have the possibility of an additional Ising-type symmetry\nbreaking due to the symmetry between the $\\widetilde{\\theta}$ and\n$\\widetilde{\\phi}$ fields when $K_{\\bot}^{\\phi}=K_{0}^{\\phi}=1$,\nwhich has not been considered in this paper. We hope to study the implications of our approach for\nthe spinful three-band case, and compare our results with Ref. \\onlinecite{PhysRevB.94.205129},\nwhere the rotations of the matrices $Z_{\\mu\\nu}$ were not taken into\naccount in the RG analysis. We would also like to consider the case\nof special fillings where intraband Umklapp scattering terms are possible.\nAt first sight, these terms have higher scaling dimensions than the\ninteractions considered by us, and so, at the tree level, they are\nnot relevant. However, more work needs to be done to see the effect\nthey have on the conclusions of this paper. \n\\begin{acknowledgments}\nVT acknowledges DST for a Swarnajayanti grant (No. DST\/SJF\/PSA-0212012-13). \n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev4-1}\nmetaredpajama_set_nameRedPajamaArXiv |
|
text\\section{Physical meaning of correlation functions}\n\n\\label{A}\n\nCorrelations for two particles are often seen as a measure of predictability of local results when knowing the other result.\nYet, this simple statement has to be used carefully.\nA non-vanishing $n$-partite correlation function indicates that we can make an educated guess of the $n$th result from the product of the other $n-1$ results.\nThe converse statement does not hold and we provide an example of a state with vanishing correlation functions where the inference is still possible.\n\nLet us denote by $r_j = \\pm 1$ the result of the $j$th observer.\nWe assume that $n-1$ parties cannot infer from the product of their outcomes, $r_1 \\dots r_{n-1}$, the result of the last observer, $r_n$, i.e., the following conditional probabilities hold:\n\\begin{equation}\nP(r_n | r_1 \\dots r_{n-1}) = \\frac{1}{2}.\n\\label{COND_PROB}\n\\end{equation}\nWe show that this implies that the corresponding correlation function, $T_{j_1 \\dots j_n}$, vanishes.\nThe correlation function is defined as expectation value of the product of all local outcomes\n\\begin{equation}\nT_{j_1 \\dots j_n} = \\langle r_1 \\dots r_n \\rangle = P(r_1 \\dots r_n = 1) - P(r_1 \\dots r_n = -1).\n\\end{equation}\nUsing Bayes' rule\n\\begin{eqnarray}\nP(r_1 \\dots r_n = \\pm 1) &=& \\sum_{r = \\pm 1} P(r_n = \\pm r | r_1 \\dots r_{n-1} = r) \\nonumber \\\\\n&\\times& P(r_1 \\dots r_{n-1} = r). \n\\end{eqnarray}\nAccording to assumption (\\ref{COND_PROB}) we have $P(r_n = \\pm r | r_1 \\dots r_{n-1} = r) = \\frac{1}{2}$, giving $P(r_1 \\dots r_n = \\pm 1) = \\frac{1}{2}$ and $T_{j_1 \\dots j_n} = 0$.\n\nAs an example of a state with vanishing correlation functions yet allowing to make an educated guess of the result, let us consider the two-qubit mixed state\n\\begin{equation}\n\\frac{1}{2} \\proj{00} + \\frac{1}{4} \\proj{01} + \\frac{1}{4} \\proj{10},\n\\end{equation}\nwhere $\\ket{0}$ and $\\ket{1}$ are the eigenstates of the Pauli operator $\\sigma_z$ with eigenvalues $+1$ and $-1$, respectively.\nAll correlation functions $T_{kl}$, with $k,l=x,y,z$, of this state vanish. Yet, whenever Alice (Bob) observes outcome $-1$ in the $\\sigma_z$ measurement, she (he) is sure the distant outcome is $+1$, i.e., $P(r_2 = +1 | r_1 = -1) = 1$.\nSimilar examples exist for multiple qubits, but we note that the states $\\rho^{nc}_{\\phi}$ of the main text are an equal mixture of a state and its anti-state. In this case, the vanishing $n$-party correlations lead to the impossibility of inferring the $n$-th result.\n\n\\section{Criterion for genuine multipartite entanglement}\n\n\\label{B}\n\nTo evaluate entanglement we use the following criterion (see main text) where, $T^{{exp}} = T$, i.e., assuming the ideal experiment producing the required state described by the correlation tensor $T$:\n\\begin{equation}\n\\max_{T^{{bi-prod}}} (T,T^{{bi-prod}}) < (T,T).\n\\end{equation}\nThe maximization is performed over all bi-product states keeping in mind also all possible bipartitions.\nThe inner product between two correlation tensors of three qubit states is defined as\n\\begin{equation}\n(V,W) \\equiv \\sum_{\\mu,\\nu,\\eta = 0}^3 V_{\\mu \\nu \\eta} W_{\\mu \\nu \\eta}.\n\\end{equation}\n\n\\subsection{Tripartite entanglement}\n\nTo keep the statement as general as possible, we prove that all states $\\rho^{nc}_{\\phi} = \\frac{1}{2} \\proj{\\phi} + \\frac{1}{2} \\proj{\\overline \\phi}$ with\n\\begin{eqnarray}\n\\ket{\\phi} \\!& = &\\! \\sin \\beta \\cos \\alpha \\ket{001} + \\sin \\beta \\sin \\alpha \\ket{010} + \\cos \\beta \\ket{100}, \\label{PSI-PSIBAR} \\\\\n\\ket{\\overline \\phi} \\!& = &\\! \\sin \\beta \\cos \\alpha \\ket{110} + \\sin \\beta \\sin \\alpha \\ket{101} + \\cos \\beta \\ket{011}, \\nonumber\n\\end{eqnarray}\nare genuinely tripartite entangled as soon as $\\ket{\\phi}$ is genuinely tripartite entangled. \\\\\nFirst, note that $\\ket{\\phi}$ is a bi-product state if at least one amplitude vanishes, i.e., if either\n\\begin{enumerate}\n\\item $\\beta = 0$ (full product state),\n\\item $\\beta = \\frac{\\pi}{2}$ and $\\alpha = 0$ (full product state),\n\\item $\\beta = \\frac{\\pi}{2}$ and $\\alpha = \\frac{\\pi}{2}$ (full product state),\n\\item $\\beta = \\frac{\\pi}{2}$ and $\\alpha \\in (0,\\frac{\\pi}{2})$ (bi-product $A|BC$),\n\\item $\\alpha = 0$ and $\\beta \\in (0,\\frac{\\pi}{2})$ (bi-product $B|AC$),\n\\item $\\alpha = \\frac{\\pi}{2}$ and $\\beta \\in (0,\\frac{\\pi}{2})$ (bi-product $C|AB$).\n\\end{enumerate}\nThe correlation tensor of the state $\\rho^{nc}_{\\phi}$ contains only bipartite correlations:\n\\begin{eqnarray}\nT_{xx0} & = & T_{yy0} = \\sin(2 \\beta) \\sin(\\alpha), \\nonumber \\\\\nT_{x0x} & = & T_{y0y} = \\sin(2 \\beta) \\cos(\\alpha), \\nonumber \\\\\nT_{0xx} & = & T_{0yy} = \\sin^2(\\beta) \\sin(2 \\alpha), \\nonumber \\\\\nT_{zz0} & = & \\cos(2 \\alpha) \\sin^2(\\beta) - \\cos^2(\\beta), \\nonumber \\\\\nT_{z0z} & = & - \\cos(2 \\alpha) \\sin^2(\\beta) - \\cos^2(\\beta), \\nonumber \\\\\nT_{0zz} & = & \\cos(2 \\beta),\n\\label{T_MIX}\n\\end{eqnarray}\nand $T_{000}=1$. Using these expressions, the right-hand side of the entanglement criterion is\n\\begin{equation}\nR = (T,T) = 4.\n\\label{eq:R4}\n\\end{equation}\nTo find the maximum of the left-hand side, we shall follow a few estimations.\nConsider first the bi-product state in a fixed bipartition, say $AB|C$, i.e., of the form $\\ket{\\chi}_{AB} \\otimes \\ket{c}$,\nwhere $\\ket{\\chi}_{AB} = \\cos(\\theta) \\ket{00} + \\sin(\\theta) \\ket{11}$, when written in the Schmidt basis.\nLet us denote the correlation tensor of $\\ket{\\chi}_{AB}$ with $P$ and its local Bloch vectors by $\\vec a$ and $\\vec b$.\nWe therefore have:\n\\begin{eqnarray}\nL &=& 1 + T_{xx0} (P_{xx} + P_{yy}) + T_{zz0} P_{zz} + T_{x0x}(a_x c_x + a_y c_y) \\nonumber \\\\\n&+& T_{z0z} a_z c_z + T_{0xx}(b_x c_x + b_y c_y) + T_{0zz} b_z c_z. \n\\label{L}\n\\end{eqnarray}\nBy optimizing over the states of $\\ket{c}$ we get the following upper bounds:\n\\begin{equation}\nT_{x0x}(a_x c_x + a_y c_y) + T_{z0z} a_z c_z \\le \\sqrt{T_{x0x}^2 (a_x^2 + a_y^2) + T_{z0z}^2 a_z^2},\n\\end{equation}\nand\n\\begin{equation}\nT_{0xx}(b_x c_x + b_y c_y) + T_{0zz} b_z c_z \\le \\sqrt{T_{0xx}^2 (b_x^2 + b_y^2) + T_{0zz}^2 b_z^2}.\n\\label{OVER_C}\n\\end{equation}\nThe Schmidt decomposition implies for local Bloch vectors:\n\\begin{equation}\na_x^2 + a_y^2 + a_z^2 = b_x^2 + b_y^2 + b_z^2 = \\cos^2(2 \\theta),\n\\end{equation}\nand therefore\n\\begin{equation}\n\\vec a = \\cos(2 \\theta) \\vec n, \\quad \\vec b = \\cos(2 \\theta) \\vec m,\n\\end{equation}\nwhere $\\vec n$ and $\\vec m$ are normalized vectors with directions along the local Bloch vectors.\nThis gives the bound\n\\begin{eqnarray}\n&& \\sqrt{T_{x0x}^2 (a_x^2 + a_y^2) + T_{z0z}^2 a_z^2} + \\sqrt{T_{0xx}^2 (b_x^2 + b_y^2) + T_{0zz}^2 b_z^2} \\nonumber \\\\\n&& = \\cos(2 \\theta) ( \\sqrt{T_{x0x}^2 (n_x^2 + n_y^2) + T_{z0z}^2 n_z^2} \\\\\n&& +\\sqrt{T_{0xx}^2 (m_x^2 + m_y^2) + T_{0zz}^2 m_z^2} ) \\nonumber \\\\\n&& \\le \\cos(2 \\theta) ( \\max(|T_{x0x}|,|T_{z0z}|) + \\max(|T_{0xx}|,|T_{0zz}|) ),\\nonumber\n\\end{eqnarray}\nwhere the maxima follow from convexity of squared components of a normalized vector.\n\nNow let us focus on the terms depending on the correlations of $\\ket{\\chi}_{AB}$.\nIn order to maximize \\eqref{L}, the Schmidt basis of $\\ket{\\chi}_{AB}$ has to be either $x$, $y$, or $z$ as otherwise off-diagonal elements of $P$ emerge leading to smaller values entering \\eqref{L}.\nFor the diagonal correlation tensor we have $|P_{xx}| = \\sin(2 \\theta)$, $|P_{yy}| = \\sin(2 \\theta)$, and $P_{zz} = 1$, and with indices permuted.\nTherefore, there are three cases to be considered in order to optimize $T_{xx0} (P_{xx} + P_{yy}) + T_{zz0} P_{zz}$:\n\\begin{itemize}\n\\item[(i)] $|P_{xx}| = 1$ and $|P_{yy}| = |P_{zz}| = \\sin(2 \\theta)$ with their signs matching those of $T_{xx0}$ and $T_{zz0}$ respectively,\n\\item[(ii)] $|P_{zz}| = 1$ and $P_{xx} = P_{yy} = \\sin(2 \\theta)$,\n\\item[(iii)] $|P_{zz}| = 1$ and $P_{xx} = - P_{yy} = \\sin(2 \\theta)$.\n\\end{itemize}\nEach of these cases leads to an upper bound on $L$.\nFor example, for the first case we find\n\\begin{widetext}\n\\begin{eqnarray}\nL_{\\mathrm{(i)}} &=& 1 + |T_{xx0}| + \\sin(2 \\theta) (|T_{xx0}| + |T_{zz0}|)\n+ \\cos(2 \\theta) (\\max(|T_{x0x}|,|T_{z0z}|) + \\max(|T_{0xx}|,|T_{0zz}|)) \\nonumber\\\\\n &\\le& 1 + |T_{xx0}|\n+ \\sqrt{(|T_{xx0}| + |T_{zz0}|)^2 + (\\max(|T_{x0x}|,|T_{z0z}|) + \\max(|T_{0xx}|,|T_{0zz}|))^2},\n\\label{eq:Li}\n\\end{eqnarray}\n\\end{widetext}\nwhere in the last step we optimized over $\\theta$. The same procedure applied to the other two cases gives:\n\\begin{widetext}\n\\begin{eqnarray}\nL_{\\mathrm{(ii)}} & \\le & 1 + |T_{zz0}| + \\sqrt{4 T_{xx0}^2 + (\\max(|T_{x0x}|,|T_{z0z}|) + \\max(|T_{0xx}|,|T_{0zz}|))^2}, \\label{eq:Lii}\\\\\nL_{\\mathrm{(iii)}} & \\le & 1 + |T_{zz0}| + \\max(|T_{x0x}|,|T_{z0z}|) + \\max(|T_{0xx}|,|T_{0zz}|). \\label{eq:Liii}\n\\end{eqnarray}\n\\end{widetext}\nIf instead of the bipartition $AB|C$ another one was chosen, the bounds obtained are given by those above with the indices correspondingly permuted.\nSince there are three possible bipartitions, altogether we have nine bounds out of which we should finally choose the maximum as the actual upper bound on the left-hand side.\n\n\\subsubsection*{Numerical derivation of bounds}\n\nA first approach is to numerically evaluate Eqs.~(\\ref{eq:Li})-(\\ref{eq:Liii}).\nFig.~\\ref{FIG_CRIT} shows that only for states $\\ket{\\phi}$ that are bi-product the left-hand side reaches $L=4$.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.46\\textwidth]{tarcza4b.pdf}\n\\caption{Contour plot showing the maximal value of the left-hand side of our entanglement criterion for the states $\\rho^{nc}_{\\phi}$ defined above (\\ref{PSI-PSIBAR}).\nWhenever the value is below $4$, i.e., the right-hand side value as given in \\eqref{eq:R4}, the criterion detects genuine tripartite entanglement.\nThis shows that all the states $\\rho^{nc}_{\\phi}$ are genuinely tripartite entangled except for those arising from bi-product states $\\ket{\\phi}$, i.e., for $\\alpha, \\beta = 0$ or $\\pi\/2$.\nNumerical optimizations over all bi-separable states yield the same plot.}\n\\label{FIG_CRIT}\n\\end{figure}\n\n\n\nFor the $W$ state we thus obtain $\\max L = 10\/3$ which is achieved by the bi-product state $(\\cos\\theta \\ket{++} - \\sin\\theta \\ket{--}) \\otimes \\ket{+}$,\nwhere $\\ket{\\pm} = \\frac{1}{\\sqrt{2}}(\\ket{0} \\pm \\ket{1})$ and $\\tan(2 \\theta) = 3\/4$ in order to optimize case (i) which is the best for the $W$ state.\nThis bound is used in the main text.\n\n\n\n\\subsubsection*{Analytic argument}\n\nThe last step of the proof, showing that only bi-separable states can achieve the bound of $4$ in our criterion, involved numerical optimization (Fig.~\\ref{FIG_CRIT}).\nOne may complain that due to finite numerical precision there might be genuinely tripartite entangled states for values of $\\alpha$ or $\\beta$ close to $0$ and $\\pi\/2$ that already achieve the bound of $4$.\nHere, we give a simple analytical argument showing that $\\rho^{nc}_{{\\phi}}$ is genuinely tripartite entangled if and only if $\\ket{\\phi}$ is so.\n\n\n\nWe first follow the idea of Ref.~\\cite{PhysRevLett.101.070502} and note that a mixed state $\\rho^{nc}_{\\phi}$ can only be bi-separable if there are bi-product pure states in its support.\nThe support of $\\rho^{nc}_{\\phi}$ is spanned by $\\ket{\\phi}$ and $| \\overline{\\phi} \\rangle$, i.e., $\\rho^{nc}_{\\phi}$ does not have any overlap with the orthogonal subspace $\\openone - \\proj{\\phi} - | \\overline{\\phi} \\rangle \\langle \\overline{\\phi} |$. \nAccordingly any decomposition of $\\rho^{nc}_{\\phi}$ into pure states can only use pure states of the form\n\\begin{equation}\n\\ket{\\Phi} = a \\ket{\\phi} + b | \\overline{\\phi} \\rangle.\n\\end{equation}\nWe now give a simple argument that $\\ket{\\Phi}$ is bi-product, and hence $\\rho^{nc}_{\\phi}$ is bi-separable, if and only if $|\\phi\\rangle$ is bi-product.\nIn all other infinitely many cases, the no-correlation state is genuinely tripartite entangled.\nAssume that $|\\Phi\\rangle$ is bi-product in the partition $AB|C$. \nAccordingly, all its correlation tensor components factor across this partition.\nIn particular, \n\\begin{eqnarray}\n&&T_{0xx} = W_{0x} V_x, \\,\\,\\,\\,\\,\\,\\, T_{0yy} = W_{0y} V_y, \\,\\,\\,\\,\\,\\,\\, \\\\\n&&T_{0xy} = W_{0x} V_y, \\,\\,\\,\\,\\,\\,\\, T_{0yx} = W_{0y} V_x \\nonumber\n\\end{eqnarray}\nwhere $W$ is the correlation tensor of the state of $AB$ and $V$ is the correlation tensor corresponding to the state of $C$.\nOne directly verifies that for such a bi-product state we have \n\\begin{equation}\nT_{0xx} T_{0yy} = T_{0xy} T_{0yx}.\n\\label{T_BIPROD}\n\\end{equation}\nEvaluating condition (\\ref{T_BIPROD}) for the states $\\ket{\\Phi}$ gives the following condition on the amplitudes of $|\\phi\\rangle$:\n\\begin{equation}\n\\sin^2(2 \\alpha) \\sin^4(\\beta) = 0,\n\\end{equation}\nand indicates that at least one amplitude must be zero.\nSimilar reasoning applies to other partitions and we conclude that $\\ket{\\Phi}$ is bi-product if and only if $\\ket{\\phi}$ is bi-product.\n\n\n\\subsubsection*{Alternative entanglement criterion}\n\nAlternativly we can apply a witness of genuine tripartite entanglement based on angular momentum operators\n~\\cite{JOptSocAmB.24.275},\n\\begin{equation}\n\\mathcal{W}_3 = J_x^2 + J_y^2,\n\\end{equation}\nwhere e.g. $J_x = \\frac{1}{2}(\\sigma_x \\otimes \\openone \\otimes \\openone + \\openone \\otimes \\sigma_x \\otimes \\openone + \\openone \\otimes \\openone \\otimes \\sigma_x)$.\nMaximization of this quantity over bi-separable states gives~\\cite{JOptSocAmB.24.275}:\n\\begin{equation}\n\\max_{\\rho^{\\mathrm{bi-sep}}} \\langle \\mathcal{W}_3 \\rangle = 2+\\sqrt{5}\/2 \\approx 3.12.\n\\label{J_BISEP}\n\\end{equation} \nThis criterion detects entanglement of the states $\\ket{\\phi}$ and $\\ket{\\overline \\phi}$, and, consequently, \nsince it uses two-party correlations only, also of the state $\\rho_{\\phi}^{nc}$. However, entanglement\nis detected only for a range of roughly $\\alpha \\in [0.59,1.3]$ and $\\beta \\in [0.33,1.2]$.\n\n\n\n\\subsection{Five-partite entanglement}\n\nIn order to obtain the five-partite bound given in the main text, i.e., $\\max_{T^{{bi-prod}}} (T,T^{{bi-prod}}) = 12.8$,\nwe have numerically optimized over all bi-product states keeping $T$ as the correlation tensor of an equal mixture of Dicke states $| D_5^{(2)} \\rangle$ and $|D_5^{(3)} \\rangle$, where\n\\begin{equation}\n| D_n^{(e)} \\rangle = \\frac{1}{\\sqrt{{n \\choose e}}} \\sum_i | \\mathcal{P}_i(1,\\dots,1,0\\dots,0) \\rangle,\n\\end{equation}\nwith $\\mathcal{P}_i$ denoting all distinct permutations of $e$ ones and $n-e$ zeros.\n\nBelow, we generalize the analytical argument given above to prove genuine multipartite entanglement of arbitrary mixtures of Dicke and anti-Dicke states.\nThe anti-Dicke state has exchanged roles of zeros and ones as compared with the Dicke state, i.e., it has $n-e$ ones (excitations).\nOne easily verifies that the Dicke state of $n$ qubits with $e$ excitations has the following bipartite correlations:\n\\begin{eqnarray}\nT_{0 \\dots 0 xx} & = & T_{0 \\dots 0 yy} = \\frac{2 {n-2 \\choose e-1}}{{n \\choose e}} = \\frac{2 e (n-e)}{n (n-1)}, \\nonumber \\\\\nT_{0 \\dots 0 xy} & = & T_{0 \\dots 0 yx} = 0.\\label{D-DBAR}\n\\end{eqnarray}\nThe correlations of an anti-Dicke state, with $n-e$ excitations, are the same due to the symmetry $e \\leftrightarrow n-e$ of these correlations.\nAssume that $n$ is odd so that (i) the Dicke and anti-Dicke states are orthogonal and (ii) the parity of the number of excitations, i.e., whether there is an even or odd number of them, is opposite in the Dicke and anti-Dicke states.\nFor arbitrary superposition $\\alpha | D_n^{(e)} \\rangle + \\beta | D_n^{(n-e)} \\rangle$ the correlations read:\n\\begin{eqnarray}\nT_{0 \\dots 0 jk} &=& |\\alpha|^2 T_{0 \\dots 0 jk}^D + |\\beta|^2 T_{0 \\dots 0 jk}^{\\overline{D}} \\nonumber \\\\\n &+& \\alpha^* \\beta \\langle D_n^{(e)} | \\openone \\otimes \\dots \\openone \\otimes \\sigma_j \\otimes \\sigma_k | D_n^{(n-e)} \\rangle \\\\\n&+& \\alpha \\beta^* \\langle D_n^{(n-e)} | \\openone \\otimes \\dots \\openone \\otimes \\sigma_j \\otimes \\sigma_k | D_n^{(e)} \\rangle. \\nonumber\n\\end{eqnarray}\nSince applying $\\sigma_j \\otimes \\sigma_k$ with $j,k=x,y$ to the Dicke states does not change the parity of their excitations, the last two terms vanish, and for the first two terms we have $T_{0 \\dots 0 jk}^D = T_{0 \\dots 0 jk}^{\\overline{D}}$.\nTherefore, an arbitrary superposition of Dicke and anti-Dicke states has the same correlations as in (\\ref{D-DBAR}) and therefore none of such superposed states is bi-product.\nSince the Dicke states are invariant under exchange of parties (and so are their superpositions), the same holds for other partitions.\nFinally, the lack of bi-product states in a subspace spanned by Dicke and anti-Dicke states implies that their mixtures are also genuinely multipartite entangled.\n\n\\section{Genuine tripartite correlations}\n\n\\label{C}\n\n\nWhile the conventional full correlation function vanishes for $\\rho^{nc}_{\\phi}$, this is not necessarily so for other types of correlation functions introduced recently.\nFor a comparison we analyze the correlation content of the states of our family also according to the three measures given in Ref. \\cite{PhysRevLett.107.190501},\nnamely: (a) genuine tripartite correlations $T^{(3)}(\\rho^{nc}_{\\phi})$,\n(b) genuine tripartite classical correlations $J^{(3)}(\\rho^{nc}_{\\phi})$,\nand (c) genuine tripartite quantum correlations $D^{(3)}(\\rho^{nc}_{\\phi})$.\nThe results are presented and discussed in Fig. \\ref{ecor}.\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.49\\textwidth]{GENUINE_CORRELATIONS_ABC_bitmaps.pdf\n\\caption{\\label{ecor}\nCorrelation content~\\cite{PhysRevLett.107.190501} of the states $\\rho^{nc}_{\\phi} = \\frac{1}{2} \\proj{\\phi} + \\frac{1}{2} \\proj{\\overline \\phi}$ with the pure states given in Eq.~(\\ref{PSI-PSIBAR}).\n(a) \\emph{Total genuine tripartite correlations}.\nThe genuine tripartite correlations vanish only for mixtures of bi-product states. The highest value ($1.2516$) is obtained for the state $(|W\\rangle \\langle W |+ |\\overline{W}\\rangle \\langle \\overline{W}|)\/2$.\n(b) {\\em Genuine tripartite classical correlations}. The genuine classical correlations also vanish only for mixtures of bi-product states. The highest value (1.0) is observed for fully separable states. The local maximum (0.8127) is achieved by the state $(|W\\rangle \\langle W |+ |\\overline{W}\\rangle \\langle \\overline{W}|)\/2$.\n(c) {\\em Genuine tripartite quantum correlations}. The genuine quantum correlations vanish for mixtures of bi-product states and for fully separable states. The highest values (0.6631) correspond to the mixture of the state\n$\\sqrt{1\/6} |001\\rangle + \\sqrt{1\/6}|010\\rangle + \\sqrt{2\/3}|100\\rangle$ with its antistate (and permutations). The state $(|W\\rangle \\langle W |+ |\\overline{W}\\rangle \\langle \\overline{W}|)\/2$ achieves the local maximum (0.4389).\n}\n\\end{figure}\n\n\n\n\\section{Experimental three and five qubit states}\n\n\\label{D}\n\nThe experimentally prepared states $\\ket{W}^{exp}$, $\\ket{\\overline{W}}^{exp}$, $\\rho_W^{nc,exp}$, and $\\rho^{nc,exp}_{D_{5}^{(2)}}$ were characterized by means of quantum state tomography. Their corresponding density matrices can be seen in Fig.~\\ref{3QUBITS} and Fig.~\\ref{5QUBITS}.\nThe fidelities of the observed three qubit states with respect to their target states are $0.939\\pm0.011$ for $\\ket{W}^{exp}$, $0.919\\pm0.010$ for $\\ket{\\overline{W}}^{exp}$, and $0.961\\pm0.003$ for $\\rho^{nc,exp}_W$. \nNote that the value of the fidelity for the state $\\rho^{nc,exp}_W$ was obtained from a maximum likelihood (ML) reconstruction together with non-parametric bootstrapping. This value thus might be slightly incorrect due to the bias of the maximum likelihood data evaluation~\\cite{arxiv}.\n\nFig.~\\ref{5QUBITS} shows the real part of the tomographically determined no-correlation state from which all further five qubit results are deduced.\nThe five-qubit fidelity of $\\rho^{nc,exp}_{D_{5}^{(2)}}$ is determined via a ML reconstruction from five-fold coincidences to be $0.911\\pm0.004$.\n\nTo obtain a correlation function value, e.g., $T_{zzz}=\\operatorname{Tr}(\\rho~ \\sigma_z \\otimes \\sigma_z \\otimes \\sigma_z)$, we analyze the three photons in the respective set of bases (here all $\\hat{z}$). Fig.~\\ref{FIG_CORRS} shows the relative frequencies for observing all the possible results for such a polarization analysis. Clearly one recognizes the complementary structure of the the detection frequencies for the states $\\ket{W}^{exp}$ and $\\ket{\\overline{W}}^{exp}$ which results in approximately the same magnitude of the correlations, yet with different sign. Mixing the two states, one thus obtains a vanishingly small correlation. Fig.~[2] of the main text then shows the full set of correlations.\n\nFor the analysis of the five qubit no correlation state, we see from an eigen decomposition that this state indeed comprises of a mixture of two states ($|\\Theta^{(2)} \\rangle^{exp}$ and $|\\Theta^{(3)} \\rangle^{exp}$), which are in very good agreement with $| D_{5}^{(2)} \\rangle$ and $| D_{5}^{(3)} \\rangle$.\nFig.~\\ref{FIG_5P} (a) and (b) show all symmetrized correlations for the five-qubit states $| \\Theta^{(2)} \\rangle$ and $| \\Theta^{(3)} \\rangle$ and $\\rho_{D_5^{(2)}}^{{nc,exp}}$ with good agreement with the ideal states. \nAlso the respective fidelity of the eigenvectors of the experimentally determined state are quite high ($F_{| D_{5}^{(2)} \\rangle}(| \\Theta^{(2)} \\rangle)=0.978\\pm0.012$ and $F_{| D_{5}^{(3)} \\rangle}(| \\Theta^{(3)} \\rangle^{exp})=0.979\\pm0.012$). Equally mixing the states $|\\Theta^{(2)} \\rangle^{exp}$ and $|\\Theta^{(3)}\\rangle^{exp}$ indeed would result in a state with vanishingly small correlations as seen in Fig.~\\ref{FIG_5P} (c). However, due to asymmetry in the coupling of signal and idler states from the down conversion source~\\cite{SignalIdler} the correlations are still present, albeit smaller by a factor of 10 compared with $| D_{5}\n^{(2)} \\rangle$ and $| D_{5}^{(3)} \\rangle$.\nIn the main text we show that the very same state is genuinely five-party entangled.\n\n\n\n\\begin{figure*}[!ht]\n\\includegraphics[width=0.8\\textwidth]{3qubit_states.png}\n\\caption{\\label{3QUBITS}\nExperimental three qubit states as obtained from the state $|D_4^{(2)}\\rangle^{exp}$. (a) The state $\\ket{W}^{exp}$ is obtained by projection of the fourth qubit of $|D_4^{(2)}\\rangle^{exp}$ on $V$. (b)\nThe state $\\ket{\\overline{W}}^{exp}$ is prepared by projecting the fourth qubit of $|D_4^{(2)}\\rangle^{exp}$ on $H$. (c) When the fourth qubit of $|D_4^{(2)}\\rangle^{exp}$ is traced out, a mixture of\n$\\ket{W}^{exp}$ and $\\ket{\\overline{W}}^{exp}$ is obtained, i.e., the state $\\rho_W^{nc,exp}$.\nThe corresponding fidelities with respect to their target states are $0.939\\pm0.011$ for $\\ket{W}^{exp}$, $0.919\\pm0.010$ for $\\ket{\\overline{W}}^{exp}$, and $0.961\\pm0.003$ for $\\rho^{nc,exp}_W$.\n}\n\\end{figure*}\n\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.48\\textwidth]{5qubit_state_Wiesiu.png}\n\\caption{\\label{5QUBITS}\nExperimental state $\\rho^{nc,exp}_{D_{5}^{(2)}}$ determined from five-fold coincidences together with permutational invariant tomography~\\cite{PhysRevLett.113.040503}.\nThe fidelity with respect to the target state is $0.911\\pm0.004$.\n}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.46\\textwidth]{probs_zzz_basis_redblue_3-eps-converted-to.pdf}\n\\caption{(color online). Detection frequencies when observing the states $\\ket{W}^{exp}$ (red) and $\\ket{\\overline{W}}^{exp}$ (blue) and $\\rho_W^{nc,exp}$ (red and blue) in the $\\sigma_z^{\\otimes 3}$ basis. From these data $T_{zzz}$ values can be calculated showing how the correlations of $\\ket{W}^{exp}$ and $\\ket{\\overline{W}}^{exp}$ average to approximately 0. For comparison, the theoretically expected values are shown in gray. \nThe correlation value $T_{zzz}$ of the state $\\rho_W^{nc,exp}$ was determined as the weighted sum of the correlation values $T_{zzz}$ of the states $\\ket{W}^{exp}$ and $\\ket{\\overline{W}}^{exp}$. The state $\\ket{W}^{exp}$ was observed with a slightly lower probability ($0.485$) than the state $\\ket{\\overline{W}}^{exp}$ ($0.515$) leading to a value of $T_{zzz} = 0.022$ for the state $\\rho_W^{nc,exp}$.\nIn contrast, in Fig.~2 of the main text the states $\\ket{W}^{exp}$ and $\\ket{\\overline{W}}^{exp}$ were obtained from the state $| D_{4}^{(2)} \\rangle^{exp}$ by projection of the fourth qubit onto horizontal\/vertical polarization, i.e., from measuring $\\sigma_z$ on the fourth qubit. There, $\\rho_W^{nc,exp}$ was obtained by tracing out the fourth qubit and hence measurements of $\\sigma_x, \\sigma_y, \\sigma_z$ on the fourth qubit of $| D_{4}^{(2)} \\rangle^{exp}$ contribute, leading to approximately three times better statistics for the state $\\rho_W^{nc,exp}$. \n}\n\\label{FIG_CORRS}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.49\\textwidth]{fivepartite_corrs_errorbars_3-eps-converted-to.pdf}\n\\caption{\\label{FIG_5P}\nExperimental five-partite symmetric correlations for the two most prominent states (a) $|\\Theta^{(2)} \\rangle^{exp}$ and (b) $|\\Theta^{(3)} \\rangle^{exp}$ in the eigen decomposition of the experimental density matrix $\\rho^{nc,exp}_{D_{5}^{(2)}}$ shown in Fig. \\ref{5QUBITS}. The correlations of these states are compared with the ones of the states (a) $| D_{5}^{(2)} \\rangle$ and (b) $| D_{5}^{(3)} \\rangle$, respectively, shown in gray. The agreement between the actual and expected correlations is evident and also the fidelities of $|\\Theta^{(2)} \\rangle^{exp}$ and $|\\Theta^{(3)} \\rangle^{exp}$ with the respective target states are high: $F_{| D_{5}^{(2)} \\rangle}(| \\Theta^{(2)} \\rangle^{exp})=0.978\\pm0.012$ and $F_{| D_{5}^{(3)} \\rangle}(| \\Theta^{(3)} \\rangle^{exp})=0.979\\pm0.012$. (c) When both states are evenly mixed, the resultant state has practically vanishing correlations. (d) Since the collection efficiencies for signal and idler photons generated via spontaneous parametric down-conversion differ \nslightly \\cite{SignalIdler}, the \nstates $|\\Theta^{(2)} \\rangle^{exp}$ and $|\\Theta^{(3)} \n\\rangle^{exp}$ are observed with relative weights of $0.54$ and $0.46$ leading to largely suppressed but not entirely vanishing full correlations. Hence, the experimentally prepared state $\\rho_{D_5^{(2)}}^{{nc,exp}}$ is a very good approximation to a no-correlation state. Please note that the correlations shown in (c) and (d) are magnified by a factor of $10$ compared with the scale of (a) and (b).\nThe errors given in subfigures (a)-(c) were obtained by non-parametric bootstrapping~\\cite{bootstrapping} whereas for (d) Gaussian error propagation was used.\n}\n\\end{figure}\n\n\n\n\\section{Statistical analysis}\n\\label{E}\n\n\\subsection{Error analysis}\n\nIn order to carry out $n$-qubit quantum state tomography, we measured in the eigenbases of all $3^n$ combinations of local Pauli settings $s_i$ with $s_1 = x...xx$,\n$s_2 = x...xy$, ..., $s_{3^n} = z...zz$.\nIn each setting $s_i$ we performed projection measurements on all the $2^n$ eigenvectors of the corresponding operators.\nThe single measurement results are enumerated by $r_j$ representing the binary numbers from $0$ to $2^n-1$ in increasing order, i.e., $r_1 = 0...00$, $r_2 = 0...01$, ..., $r_{2^n} = 1...11$.\nThe observed counts for the outcome $r_j$ when measuring $s_i$ are labeled as $c_{r_j}^{s_i}$ and the total number of counts $N_{s_i}$ for setting $s_i$ is given by $N_{s_i} = \\sum\\limits_{j=1}^{2^n} c_{r_j}^{s_i}$.\nFrom these data the density matrix can be obtained as\n\\begin{equation}\n\\rho = \\sum\\limits_{i=1}^{3^n}\\sum\\limits_{j=1}^{2^n} \\frac{c_{r_j}^{s_i}}{N_{s_i}} M_{r_j}^{s_i}\n\\end{equation}\nwhere the elements of the generating set of operators $M_{r_j}^{s_i}$ are defined as $M_{r_j}^{s_i} = \\frac{1}{2^n}\\bigotimes\\limits_{k=1}^n \\Big(\\frac{\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}}{3}+(-1)^{r_j(k)}\\sigma_{s_i(k)}\\Big)$~\\cite{James,PhDNikolai}, where $\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}$ denotes the $2\\times2$ identity matrix and $r_{j(k)}$ is the k-th entry in the string $r_j$.\nThen, the fidelity $F_{\\ket{\\psi}}$ with respect to a pure target state $\\ket{\\psi}$ can be calculated as\n\\begin{equation}\nF_{\\ket{\\psi}} = \\bra{\\psi}\\rho\\ket{\\psi} = \\sum\\limits_{i=1}^{3^n}\\sum\\limits_{j=1}^{2^n} \\frac{c_{r_j}^{s_i}}{N_{s_i}} \\bra{\\psi} M_{r_j}^{s_i} \\ket{\\psi}.\n\\end{equation}\nFor Poissonian measurement statistics, i.e., $\\Delta c_{r_j}^{s_i} = \\sqrt{c_{r_j}^{s_i}}$, the error to the fidelity $\\Delta F_{\\ket{\\psi}} = \\sqrt{\\Delta^2 F_{\\ket{\\psi}}}$ can be deduced via Gaussian error propagation as $\\Delta^2 F_{\\ket{\\psi}} = \\sum\\limits_{i=1}^{3^n}\\sum\\limits_{j=1}^{2^n} (\\frac{1}{N_{s_i}} - \\frac{1}{N_{s_i}^2})^2\\bra{\\psi} M_{r_j}^{s_i} \\ket{\\psi}^2 c_{r_j}^{s_i}$\nwhich is approximately\n\\begin{equation}\n\\Delta^2 F_{\\ket{\\psi}} = \\sum\\limits_{i=1}^{3^n}\\Delta^2 F_{\\ket{\\psi}}^{s_i} = \\sum\\limits_{i=1}^{3^n}\\sum\\limits_{j=1}^{2^n} \\frac{c_{r_j}^{s_i}}{N_{s_i}^2}\\bra{\\psi} M_{r_j}^{s_i} \\ket{\\psi}^2\n\\label{eq:approxerr}\n\\end{equation}\nfor large number of counts per setting as in our experiment.\nAs an example, in table \\ref{tab:zzz} we give the corresponding values for $c_{r_j}^{s_i}$ and $|\\langle \\psi | M_{r_j}^{s_i}| \\psi \\rangle |$ for the $2^3=8$ possible results of the $zzz$ measurement of the three qubit $\\ket{W}$ state to get an impression of the size of the $3^3=27$ terms in Eq.~(\\ref{eq:approxerr}).\\\\\n\n\\renewcommand{\\arraystretch}{1.4}\n\n\\begin{table*}[!ht]\n\t \\begin{tabular*}{129mm}{l|c|r|r|r|r|r|r|r|r}\n \\hline\\hline\n & $r_j$ & $000$ & $001$ & $001$ & $011$ & $100$ & $101$ & $110$ & $111$ \\\\\\cline{1-10}\n $zzz$& $ |\\langle \\psi | M_{r_j}^{zzz} | \\psi \\rangle |$ & 1.48e-01 & 1.48e-01 & 1.48e-01 & 1.11e-01 & 1.48e-01 & 1.11e-01 & 1.11e-01 & 7.41e-02 \\\\\\cline{2-10}\n & counts $c_{r_j}^{zzz}$ & 14 & 309 & 250 & 8.71 & 283 & 8 & 7.07 & 0 \\\\\\cline{2-10}\n \\hline\\hline\t\t\t\n \\end{tabular*}\n\\caption{\\label{tab:zzz} The values of $c_{r_j}^{s_i}$ and $|\\langle \\psi | M_{r_j}^{s_i}| \\psi \\rangle |$ for the measurement of the setting $zzz$ of the experimentally observed state $\\ket{W}^{{exp}}$.\n The first row shows all possible results $r_j$ associated with the eigenvectors on which projection measurements are performed, labeled in binary representation.\n Please note that the observed counts $c_{r_j}^{s_i}$ are not integers since the slightly differing relative detection efficiencies of the single photon counters were included.\n\tFrom these data we obtain for $s_i=zzz$ a contribution for Eq.~(\\ref{eq:approxerr}) of $\\Delta^2 F_{\\ket{W}}^{zzz} = 2.46$e-05.}\n \\end{table*}\n\n\n\n\nSimilarly, also the error of the $4^3=64$ correlations of the given state are evaluated. For example, we obtain for the correlation value $T_{zzz} = -0.914 \\pm 0.034$.\nThe error for the maximum likelihood estimate was determined by non-parametric bootstrapping, for details see~\\cite{bootstrapping}.\n\n\\subsection{Hypothesis testing}\n\\label{sec:hypotheses}\n\\subsubsection*{Vanishing correlations}\nAfter having calculated the experimental error of the $zzz$ correlation, we find that the measurements of the remaining 26 full correlations have similar errors.\nWe test our hypothesis of vanishing full correlations by comparing our measured correlation values with a normal distribution with mean $\\mu=0$ and standard deviation $\\sigma=0.0135$, which corresponds to the average experimental standard deviation.\nIf our data are in agreement with this distribution, we can retain the hypothesis of vanishing full correlations. \\\\\n\\begin{figure}[!ht]\n\\includegraphics[width=0.49\\textwidth]{hypothesis_cdf_KS_test.pdf}\n\\caption{The cumulative distribution of the experimentally determined correlations is compared to the cumulative distribution of the expected correlations ($\\mu=0$, $\\sigma=0.0135$).\nThe shaded blue region contains points that would be sampled from the normal distribution with probability smaller than $5\\%$.\nSince the empirical function lies in between the shaded regions, our hypothesis of vanishing correlations can be retained with significance level of $0.05$.}\n\\label{HypothesisKS}\n\\end{figure}\n\nTo test the hypothesis\n\\begin{quote}\n$H_0^{(nc,3)}:$ all full correlations of the state $\\rho_W^{{nc,exp}}$ vanish,\n\\end{quote}\naccording to the Kolmogorov-Smirnov method, the cumulative distribution of the $27$ measured full correlations is compared with the cumulative probability distribution of the assumed normal distribution, see Fig.~\\ref{HypothesisKS}, quantifying the hypothesis of vanishing full correlations.\nWe can directly see that the data do not enter the region of rejection given by a significance level of $0.05$.\nThis clearly indicates that the hypothesis of normal distribution with mean $\\mu=0$ and $\\sigma=0.135$ cannot be rejected.\nWhile this test (Kolmogorov-Smirnov hypothesis test) is demonstrative, the Anderson-Darling test is considered to be more powerful, i.e., to decrease the probability of errors of second kind.\nSince the Anderson-Darling test gives a $p$-value of $0.44$ far above a $0.05$ significance level, we can retain the claim that our measured data indeed correspond to vanishing full correlations, while their scatter can be fully explained by the experimental error.\n\n\\subsubsection*{Testing for genuine multipartite entanglement}\nFurthermore, we also check our hypotheses of the main text that the tripartite and five-partite states are genuinely multipartite entangled.\nFor that purpose, we calculate the probability that a state without genuine multipartite entanglement achieves values comparable to the measured value based on the assumption that the measurement errors are normally distributed.\nLet us formulate for the tripartite state the null hypothesis\n\\begin{quote}\n$H_0^{(3)}:$ state $\\rho_W^{{nc,exp}}$ is not genuinely tripartite entangled.\n\\end{quote}\nTo show the genuine tripartite entanglement of that state, we want to reject the null hypothesis $H_0^{(3)}$.\nIn order to estimate the error of first kind, i.e., the probability that $H_0^{(3)}$ is \\textit{true}, we calculate the probability that a state without tripartite entanglement achieves the measured value of $\\left(T,T_W^{{nc,exp}}\\right)=3.858$.\nThe calculation is based on the assumption of a normal distributed result of the indicator with mean $\\mu=\\frac{10}{3}$, i.e., the bi-separable bound, and with standard deviation given by our experimental error of $\\sigma=0.079$.\nThe probability of the error of first kind is then at most\n\\begin{eqnarray}\np&=&\\operatorname{Pr}\\left[\\left(T,T_W^{{nc,exp}}\\right)\\geq3.858\\Big|H_0^{(3)}\\right] \\\\\n&<&\\frac{1}{\\sqrt{2\\pi}\\sigma} \\int_{3.858}^{\\infty} {\\rm d}x \\exp\\left({-\\frac{\\left(x-\\mu\\right)^2}{2\\sigma^2}}\\right) \\nonumber \\\\\n&=& 1.55\\times10^{-11} \\ll 0.05. \\nonumber\n\\end{eqnarray}\nSince $p$ is far below the significance level of $0.05$, our experimentally implemented state $\\rho^{nc}_{W}$ is genuine tripartite entangled. \\\\\n\nAnalogously, we test if the state $\\rho_{D_5^{(2)}}^{{nc,exp}}$ is indeed genuinely five-partite entangled.\nFor that purpose, we formulate the null hypothesis\n\\begin{quote}\n$H_0^{(5)}:$ state $\\rho_{D_5^{(2)}}^{{nc,exp}}$ is not genuinely five-partite entangled.\n\\end{quote}\nIn order to test the probability that a bi-separable state can achieve $\\left(T,T_{D_5^{(2)}}^{{nc,exp}}\\right)=13.663$, we now use a normal distribution centered around the bi-separable bound of $\\mu=12.8$.\nThe standard deviation is chosen according to the experimental error of $\\sigma=0.340$, such that the probability for a false rejection of the null hypothesis $H_0^{(5)}$ is estimated to be at most\n\\begin{eqnarray}\np&=&\\operatorname{Pr}\\left[\\left(T,T_{D_5^{(2)}}^{{nc,exp}}\\right)\\geq13.663\\Big|H_0^{(5)}\\right] \\\\\n&<&\\frac{1}{\\sqrt{2\\pi}\\sigma} \\int_{13.663}^{\\infty} {\\rm d}x \\exp\\left({-\\frac{\\left(x-\\mu\\right)^2}{2\\sigma^2}}\\right) \\nonumber\\\\\n&=&5.6\\times10^{-3} \\ll 0.05, \\nonumber\n\\end{eqnarray}\nclearly indicating the five-partite entanglement of our state with high significance.\n\n\\subsubsection*{Bell inequality}\nFinally, we test whether we can retain our claim that the five-partite state is non-classical due to its violation of the Bell inequality.\nIn order to show the violation, we formulate the null hypothesis\n\\begin{quote}\n$H_0^{B}:$ violation of the Bell inequality can be explained by LHV model (finite statistics loophole).\n\\end{quote}\nFor the considered Bell inequality~\\cite{PhysRevA.86.032105}\n\\begin{eqnarray}\n{\\cal B}&=&E_{\\mathcal{P}\\left(11110\\right)}+E_{\\mathcal{P}\\left(22220\\right)}+E_{\\mathcal{P}\\left(12220\\right)} \\\\\n&-&E_{\\mathcal{P}\\left(21110\\right)}-E_{\\mathcal{P}\\left(11000\\right)}-E_{\\mathcal{P}\\left(22000\\right)}\\leq6 \\nonumber\n\\end{eqnarray}\nwith $\\mathcal{P}$ denoting the summation over all permutations, e.g. $E_{\\mathcal{P}\\left(11110\\right)}=E_{11110}+E_{11101}+E_{11011}+E_{10111}+E_{01111}$, we calculate the probability that an LHV model can achieve the measured value of ${\\cal B}=6.358$, which was estimated with a standard deviation of $\\Delta {\\cal B}=0.149$.\nFollowing Ref.~\\cite{arxiv14070363} we assume that the LHV model gives the maximal allowed expectation value of our Bell parameter, equal to $\\mu = 6$, and that the standard deviation of a normal distribution about this mean value is equal to our experimental standard deviation $\\Delta {\\cal B}$.\nTherefore, the probability that the LHV model gives values at least as high as observed is found to be\n\\begin{eqnarray}\np&=&\\operatorname{Pr}\\left[{\\cal B}\\geq6.358\\Big|H_0^{B}\\right] \\\\\n&<& \\frac{1}{\\sqrt{2\\pi}\\sigma} \\int_{6.358}^{\\infty} {\\rm d}x \\exp\\left({-\\frac{\\left(x-\\mu\\right)^2}{2\\sigma^2}}\\right) = 0.0083 \\ll 0.05. \\nonumber\n\\end{eqnarray}\nThis small $p$-value clearly indicates that the null hypothesis $H_0^{B}$ is to be rejected and thus the non-classicality of the no-correlation state is confirmed.\n\n\n\\subsection{Vanishing full correlations with arbitrary measurement directions}\nThe measurements presented in the main text show not only vanishing full correlations for measurements in $x$, $y$, $z$ directions, but also for measurements of one qubit rotated in the $yz$-plane. \nHere, we show that full correlations have to vanish for arbitrary measurement directions.\nSince the $2$-norm of the correlation tensor is invariant under local rotations, its entries vanish in all local coordinate systems if they do in one.\nMoreover, $l$-fold correlations in one set of local coordinate system only depend on $l$-fold correlations of another set.\nAs an example, we explicitly show this for the case of three qubits.\n\\begin{equation}\nT_{(\\theta_1,\\phi_1)\\,(\\theta_2,\\phi_2)\\,(\\theta_3,\\phi_3)}={\\rm Tr}(\\rho ~\\sigma_{(\\theta_1,\\phi_1)} \\otimes \\sigma_{(\\theta_2,\\phi_2)} \\otimes \\sigma_{(\\theta_3,\\phi_3)}) \n\\end{equation}\nwith\n\\begin{equation}\n\\sigma_{(\\theta_i,\\phi_i)} = \\sin(\\theta_i)\\cos(\\phi_i)\\sigma_x+\\sin(\\theta_i)\\sin(\\phi_i)\\sigma_y+\\cos(\\theta_i)\\sigma_z.\n\\end{equation}\nConsequently,\n\\begin{eqnarray}\n&&T_{(\\theta_1,\\phi_1)\\,(\\theta_2,\\phi_2)\\,(\\theta_3,\\phi_3)}\\\\\n&&=\\sin(\\theta_1)\\cos(\\phi_1)\\sin(\\theta_2)\\cos(\\phi_2)\\sin(\\theta_3)\\cos(\\phi_3)T_{xxx}\\nonumber\\\\\n&&+\\sin(\\theta_1)\\cos(\\phi_1)\\sin(\\theta_2)\\cos(\\phi_2)\\sin(\\theta_3)\\sin(\\phi_3)T_{xxy}\\nonumber\\\\\n&&+\\dots\\nonumber\\\\\n&&+\\cos(\\theta_1)\\cos(\\theta_2)\\cos(\\theta_3)T_{zzz}, \\nonumber\n\\end{eqnarray}\nwhich has to vanish since all full correlations along Pauli directions vanish.\n\n\n\\bibliographystyle{apsrev4-1}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} |
|
{"text":"\\section{Introduction}\n\nCa$_{2-x}$Sr$_x$RuO$_4$~ has attracted interest for its complex and puzzling phase diagram including\nmetallic as well as Mott-insulating magnetic phases \\cite{N00} which depend in a subtle way \non structural properties. \nRecent magnetostriction experiments \\cite{B05,B06} in connection with the metamagnetic transition (MMT) for $ x \\approx 0.2 $ underline the strong coupling of the electronic properties to the lattice. \nThis may be taken as a hint for the relevance of localized electronic orbital and spin degrees of freedom\nas can be found in a number of transition metal oxides. \nIn particular, the mutual influence of spin and orbital\ndegrees of freedom plays an important role in the behavior of\nmanganites, ruthenates or titanates \\cite{Science-Nagaosa}.\n\nThe result of the apparent interplay between orbital and magnetic\ncorrelations in Ca$_{2-x}$Sr$_x$RuO$_4$ can be seen in the complex $T$-$x$ phase\ndiagram shown in Fig.~\\ref{fig:PD}. Both Sr$^{2+}$ and Ca$^{2+}$ are isovalent ions so that the substitution\nof one by the other does not change the number of conductance electrons.\nSr$_2$RuO$_4$ is a good metal and the \\emph{a priori} expected change due to\nthe substitution of Sr by Ca should be increasing metallicity, because the\ndoping with a smaller ion (Ca) would imply a widening of the\nband. This is not the case because the smaller Ca-ion induces lattice distortions which alter\nthe overlap integrals and the crystal fields of the relevant electronic orbitals. \nActually, Ca$_2$RuO$_4$ behaves as a Mott insulator, and the\nevolution between these end-members builds a very rich phase diagram where\ndifferent structural and magnetic phases appear (Fig.\\ref{fig:PD}). For $x=2$, corresponding to Sr$_2$RuO$_4$, the system has\ntetragonal symmetry with the RuO$_6$-octahedra slightly elongated along the\n$c$-axis. Ca substitution initially induces the rotation of RuO$_6$ octahedra around the c-axis in order to accommodate the smaller ions. The system behaves still as a paramagnetic metal with tetragonal\nsymmetry. \nFurther doping with Ca reduces the conductivity and\nthe susceptibility increases when approaching $x\\rightarrow 0.5$,\nbecoming a Curie-like susceptibility \\cite{NM00}. At $x=0.5$ there is a structural phase transition, where the crystallographic structure of the Ca$_{2-x}$Sr$_x$RuO$_4$~ series changes from tetragonal to orthorhombic through a second-order phase transition. For $x<0.5$ there is, besides the c-axis rotation, a tilting of the\nRuO$_6$ octahedra leading to a reduction of the symmetry and a reduction of the c-axis lattice constant. Moreover, these distortions are responsible for a narrowing of the conduction bands which in turn enhances the correlation effects. In the region $0.2\\leq x \\leq 0.5$ the experiments suggest the appearance of a low-temperature antiferromagnetic (AFM) order. Furthermore, the application of a magnetic field leads to a metamagnetic transition which influences the tilting of the\nRuO$_6$-octahedra \\cite{B05,B06}. Within the region $x\\leq 0.2$ the system is\na Mott-insulator with true long-range AFM order and a total\nspin of $S=1$. \n\n\n\\begin{figure}[t]\n{\\includegraphics[width=0.8\\linewidth]{figs\/PD}} \\caption{Sketch of\nthe temperature-doping phase diagram of Ca$_{2-x}$Sr$_x$RuO$_4$.\nRegion I corresponds to a Mott insulator with long range\nantiferromagnetic order. Region II is characterized by a metallic\nbehavior with orthorhombic lattice symmetry and antiferromagnetic\ncorrelations at low temperatures. Region III corresponds to a\nparamagnetic metal with tetragonal symmetry in the lattice. For\n$x\\rightarrow 0.5$ there are strong ferromagnetic correlations at\nlow temperatures.} \\label{fig:PD}\n\\end{figure}\n\nIn general, a MMT can occur for a material which, under the\napplication of an external field, undergoes a first order\ntransition or is close to the critical endpoint of such a transition \nto a phase with strong ferromagnetic correlations. This is\nexperimentally observed by a very rapid increase of the magnetization\nover a narrow range of applied magnetic field. The field dependence\nof the magnetization and magnetoresistance of Ca$_{2-x}$Sr$_x$RuO$_4$~ has been analyzed in\nRef.~\\cite{N03}. There, a MMT to a\nhighly polarized state with a local moment of $S=1\/2$ is found. These\nmeasurements are interestingly supplemented by the magnetostriction\nexperiments published in Ref.~\\cite{B05,B06} where the\nchange of the lattice constants as a function of the magnetic field\nfor different temperatures is shown. \nNamely, these results demonstrate that\ncrossing the MMT leads to \nan elongation of the $c$ axis accompanied by a shrinking\nalong both in-plane directions. Apparently, the application of a high\nmagnetic field at low temperatures can reverse the structural distortion that\noccurs for $0.2\\leq x \\leq 0.5$ upon cooling in zero field. \nFurthermore, apart from different energy scales, the qualitative effects associated with the MMT are independent of the magnetic field direction.\n\n\nAll the members of the Ca$_{2-x}$Sr$_x$RuO$_4$~ family have 4 electrons in the $t_{2g}$ orbitals\nof the Ru 4$d$ shell. What is different is the occupation of these orbitals in\nthe two end-members of the phase diagram: while for $x=0$ there is an average\noccupation of two electrons in the $d_{xy}$ orbital and the other two in the\n$d_{yz}$ and $d_{zx}$ orbitals, for $x=2$ there is a fractional occupation of\n4\/3 in the $d_{xy}$ band and 8\/3 in the $d_{yz}$-$d_{zx}$-bands. LDA calculations for this concentration give three Fermi surface\nsheets, one with essentially $xy$ and two with mixed $\\{ xz, yz\\}$ character\n\\cite{O95}. The first one is usually called $\\gamma$ band while the others are\nlabeled by $\\alpha$ and $\\beta$ bands. For intermediate values of $x$ the orbital occupation is still a matter of debate \\cite{ANKRS02,FANG04,OKA04,KO07,LIEB07}.\n\n\nIn this work we focus on the region II and the Ca-rich part of region III in the schematic $T-x$ phase diagram. For the microscopic description we follow the scenario of an orbital-selective Mott insulator. This scenario was put forward by Anisimov \\emph{et al.} \\cite{ANKRS02} in order to explain the unexpected effective magnetic moment close to a $S=1\/2$ spin for $0.2 \\lesssim x \\lesssim 1.5$ \\cite{N03}. \nAssuming that the orbital occupation in this region is $(n_{\\alpha,\\beta},n_{\\gamma})\\approx(3,1)$, they proposed that the electrons in the $\\{\\alpha,\\beta\\}$ bands undergo an orbital-selective Mott transition (OSMT) while the $\\gamma$ band remains metallic. In this scenario the experimental observation of the $1\/2$ effective spin is assigned to the localized hole in the $\\{\\alpha,\\beta\\}$ bands. Angular magnetoresistance oscillations measurements \\cite{BS05} indeed show a strong dependence of the Fermi surface on the Ca concentration which is consistent with the scenario of coexisting itinerant and localized $d$-electronic states. Furthermore, from the theoretical point of view, there is by now a consensus that an OSMT can in principle occur in multi-band Hubbard models under rather general conditions \\cite{Liebsch:03,Koga:04,Ruegg:05,Ferrero:05,Medici:05,Arita:05,Knecht:05,Costi:07,Inaba:06}. However, it is still unclear to which extend this concept is applicable in the Ca$_{2-x}$Sr$_x$RuO$_4$~ system \\cite{Lee:2006,Wang:2004}. Despite of these uncertainties, there is little doubt that the localized degrees of freedom play an important role for the understanding of the puzzling physics of this material, and in the following we will assume that the concept of the OSMT is valid as a lowest order picture for Ca concentrations corresponding to region II and at the boundary of region III of the $T-x$ phase diagram.\n\nUsing a mean-field description we focus on the interplay between structural distortion and magnetic and orbital ordering. We find a theoretical phase diagram which can be related to the experimental one. Furthermore, our calculations qualitatively reproduce the metamagnetic transition accompanied by a structural transition observed in the system. The paper is organized as follows: in Sec.~\\ref{sec:model} we introduce the microscopic model. A mean-field analysis is performed in Sec.~\\ref{sec:mf}. The results for zero and finite magnetic field are given in Sec.~\\ref{sec:res}. In Sec.~\\ref{sec:exp} we relate our results to the experimental measurements. We summarize the main conclusions of this work in Sec.~\\ref{sec:con}.\n\n\n\n\n\\section{The model}\n\\label{sec:model}\nFollowing the scenario of the OSMT we assume one hole in the \\{$\\alpha,\\beta$\\} bands and focus on the localized orbital and spin degrees of freedom \\cite{ANKRS02,ST04}. Neglecting for the our discussion the itinerant $\\gamma$ band it is natural to consider a two-dimensional extended Hubbard model of the form\n\n\\begin{eqnarray}\n\\label{ExtendedH}\n{\\cal H}_{\\alpha,\\beta}& = &-t \\sum_{i,{\\vec{\\bf a}},s}\n\\left(c_{i+a_y,yz, s}^{\\dag} c_{i,yz,s} + c_{i+a_x,zx,s}^{\\dag}\nc_{i,zx,s} + h.c.\\right) \\nonumber\\\\\n&& - \\mu \\sum_{i,s,\\nu}\nc_{i,\\nu,s}^{\\dag} c_{i,\\nu,s} \\nonumber\\\\\n&&+ U\n\\sum_{i} \\sum_{\\nu} n_{i\\nu \\uparrow} n_{i \\nu \\downarrow} + U'\n\\sum_{i} n_{i,zx} n_{i,yz}\\nonumber \\\\\n & &+J_H\\sum_{i,s,s'} c_{i,yz,s}^{\\dag} c_{i,zx,s'}^{\\dag} c_{i,zx,s}\nc_{i,yz,s'}\n\\end{eqnarray}\nwhere $ c_{i,\\nu,s}^{\\dag} $ ($ c_{i,\\nu,s} $) creates (annihilates) an\nelectron on site $ i $ with orbital index $ \\nu $ ($ = yz,zx $) and\nspin $ s $ ($ n_{i,\\nu,s} = c_{i,\\nu,s}^{\\dag} c_{i,\\nu,s} $, $\nn_{i,\\nu} = n_{i, \\nu, \\uparrow} + n_{i, \\nu, \\downarrow} $; $ {\\vec{\\bf a}}\n= (a_x , a_y ) = (1,0) $ or $ (0,1) $ basis lattice vector). With\nthis Hamiltonian we restrict ourselves to nearest-neighbor hopping\nand on-site interaction for the intra- and inter-orbital Coulomb\nrepulsion, $U$ and $U'$, respectively, and the Hund's rule coupling\n$J_H$. The hopping terms considered in this model come from the\n$\\pi$-hybridization between the Ru-$d$ and O-$p$-orbitals and lead\nto the formation of two independent quasi-one-dimensional bands: the\nband associated to the $d_{yz}$-orbital disperses only in the\n$y$-direction while the band associated to the $d_{zx}$-orbital\ndisperses in the $x$-direction.\n\nIn the strongly interacting limit it was proposed that the $ \\alpha $-$\\beta$-bands absorb 3 of the four electrons available per site and form a Mott-insulating state with localized degrees of freedom, spin 1\/2 and orbital \\cite{ANKRS02}. The local orbital degree of freedom can be represented as an isospin\nconfiguration $ | + \\rangle $ and $ | - \\rangle $ corresponding to the\nsingly occupied $ d_{zx} $ and $ d_{yz} $ orbitals, respectively. The isospin operators therefore may be defined as:\n\\[\nI^z | \\pm \\rangle = \\pm \\frac{1}{2} | \\pm \\rangle, \\qquad I^+ |-\n\\rangle = | + \\rangle, \\qquad I^- | + \\rangle = | - \\rangle \\; .\n\\]\nTaking into account the additional spin 1\/2 degree of freedom ($ |\\uparrow \\rangle $ and\n$|\\downarrow \\rangle$) leads to four possible\nconfigurations at each site, represented by the states \n\\[\n\\{ | \\uparrow + \\rangle, \\; | \\uparrow - \\rangle , \\; | \\downarrow +\n\\rangle , \\; |\\downarrow - \\rangle \\}.\n\\]\nWithin second order perturbation in $t\/U$ it is possible to derive from ${\\cal H}_{\\alpha,\\beta}$ an effective model describing the interaction between the localized degrees of freedom. One finds the following Kugel-Khomskii-type\nmodel \\cite{ANKRS02}:\n\\begin{eqnarray}\\label{Heff}\n{\\cal H}_{eff} = J \\sum_{i,{\\vec{\\bf a}}} &&\\Big\\{ \\left[A\n(I^z_{i+{\\vec{\\bf a}}} +\n \\eta_{{\\vec{\\bf a}}})(I^z_{i} +\n\\eta_{{\\vec{\\bf a}}}) +B \\right]\n{\\bf S}_{i+{\\vec{\\bf a}}} \\cdot {\\bf S}_{i} \\nonumber \\\\\n& & + [C (I^z_{i+{\\vec{\\bf a}}} + \\eta'_{{\\vec{\\bf a}}})(I^z_i +\n\\eta'_{{\\vec{\\bf a}}}) + D] \\Big\\}\n\\end{eqnarray}\nwhere $J=4t^2\/U$. We have imposed the approximatively valid\nrelation $U=U^{\\prime}+2J_H$ and have assumed that $\\alpha=U^{\\prime}\/U>1\/3$. The parameters $A,B,C,D,\\eta_{\\vec{\\bf a}}$ and $\\eta_{\\vec{\\bf a}}^{\\prime}$ are functions of $\\alpha$ alone and have been given elsewhere \\cite{ANKRS02,ST04}. The energy scale $JC>0$ of the isospin\ncoupling is the largest in the present Hamiltonian. Therefore, in a mean-field approximation, one expects antiferro-orbital (AFO) order below a critical temperature $T_{AFO}\\sim JC$ ($C>0$). On the other hand, the value of the spin-spin\ninteraction depends on the orbital order and lies between $J_1=J[A(\\eta_{\\mathbf{a}}^2-1\/4)+B]<0$\nand $J_2=J[A\\eta_{\\mathbf{a}}^2+B]$. Thus, in the presence of AFO order the spin will align\nferromagnetically (FM) below a critical temperature\n$T_{FM}\\sim-J_1$. If, however, AFO order is suppressed, as in the\ncase of an orthorhombic distortion (see below) the spin-spin\ncoupling is given by $J_2$. We mention here that the sign of $J_2$\ndepends on the value of $\\alpha$. In particular, $J_2<0$ for $\\alpha<\\alpha_c=0.535$ and consequently we expect FM order at low temperatures whereas\nfor $\\alpha>\\alpha_c$ we have $J_2>0$ and antiferromagnetic (AFM)\norder sets in at sufficiently low temperatures. To be consistent with experiments we will choose throughout this article a value $\\alpha=0.75$.\n\n\nThe Sr substitution for Ca acts as an effective negative pressure. In order to account for the orthorhombic distortion due to the tilting of the\nRuO$_6$ octahedra, we introduce a new term in the Hamiltonian ${\\cal\n H}_{dist}$, defined as\n\\begin{equation} {\\cal\n H}_{dist}=\\frac{1}{2}GN(\\varepsilon-\\varepsilon_0)^2+K\\varepsilon\\sum_iI_i^x,\n\\label{eq:Hdis}\n\\end{equation}\nwhere $G$ is the elastic constant, $N$ is the number of Ru atoms and $K$ is a coupling constant. $\\varepsilon$ is a strain-field which accounts for an orthorombic\ndistortion. Any orthorombic distortion\nyields a uniform bias for the local orbital configuration which suppresses AFO\nordering. This is modeled by the coupling of $\\varepsilon$ to the orbital degrees of freedom. In other words, the orthorombic distortion introduces a transverse\nfield which aims to align the isospins. \nThe first term in Eq.~(\\ref{eq:Hdis}) is a measure of the lattice elastic\nenergy. In addition to the strain $\\varepsilon$ driven by orbital correlations we assume a constant contribution $\\varepsilon_0$. We do not specify further the origin of this contribution but it might include effects of the $\\gamma$ band or other, non-electronic, mechanisms. For actual calculations we fix the value at $\\varepsilon_0=0.1$. We will discuss later to which extend we can relate the elasticity $G$ in the theoretical model to the Sr concentration $x$ in Ca$_{2-x}$Sr$_x$RuO$_4$.\n\n\n\nFinally, in order to study the metamagnetic transition, we introduce a\ncoupling of the system to a magnetic field, by the inclusion of\nthe term ${\\cal H}_{mag}$,\n\\begin{equation}\\label{Hmag}\n{\\cal H}_{mag}=-g\\mu_BH\\sum_iS_i^x,\n\\end{equation}\nwhere $g$ is the electron gyromagnetic factor, $\\mu_B$ is the Bohr\nmagneton, $\\mu_B=\\frac{e\\hbar}{2m_e}$ and $H$ is the magnetic field strength. With all these\ningredients, the full Hamiltonian can be written as\n\n\n\\begin{equation}\\label{FullHamil}\n{\\cal H}={\\cal H}_{eff}+{\\cal H}_{dist}+{\\cal H}_{mag}.\n\\end{equation}\nIn the next section we treat this model in a mean-field approximation.\n\n\n\\section{Mean-field analysis.}\n\\label{sec:mf}\nThe mean-field decoupling for ${\\cal H}_{eff}$\nreproduces well some of the experimentally observed features of\nCa$_{2-x}$Sr$_x$RuO$_4$~ in the $x$ region where\nthe band filling corresponds to the $(n_{(\\alpha,\\beta)},n_{\\gamma})=(3,1)$ orbital occupation, as shown in Ref. \\cite{ANKRS02}. Here we extend this analysis to the full Hamiltonian\nEq.~(\\ref{FullHamil}) and obtain the zero and finite magnetic field\nphase diagrams where the different competing orders of the system\nare represented. In the presence of a transverse magnetic field, there appears an uniform component of the magnetization in the\n$x$-direction, \n\\begin{equation}\n\\langle S_i^x \\rangle=m_0,\n\\end{equation}\nas well as a staggered component in the $z$-direction, \n\\begin{equation}\n\\langle S_i^z \\rangle=\\left\\{\n\\begin{array}{ccc}\n m_s &\\,\\,\\,\\, \\mathrm{if} &\\,\\,\\,\\, i\\in A; \\\\\n -m_s &\\,\\,\\,\\, \\mathrm{if} &\\,\\,\\,\\, i \\in B. \\\\\n\\end{array}\n \\right.\n\\end{equation}\nHere, we have made use of the bipartite structure of the Hamiltonian Eq.~(\\ref{FullHamil}): $A$ and $B$ label the two sublattices. In addition, we introduce the staggered isospin component in the $z$-direction\n\\begin{equation}\n\\langle I_i^z \\rangle=\\left\\{\n\\begin{array}{ccc}\n t_s &\\,\\,\\,\\, \\mathrm{if} &\\,\\,\\,\\, i\\in A; \\\\\n -t_s &\\,\\,\\,\\, \\mathrm{if} &\\,\\,\\,\\, i \\in B. \\\\\n\\end{array}\n \\right.\n\\end{equation}\n\n\n\nThe partition function of the system can be calculated by ${\\cal\nZ}(\\beta)=\\mathrm{Tr} e^{-\\beta {\\cal H}}$, where $\\beta=1\/k_BT$. On the mean-field level, the\nbipartite nature of the lattice splits the system into two subsystems, so we can express the partition function as\n\\begin{equation}\n{\\cal Z}(\\beta)=\\left(e^{-\\beta E_0}\\right)^N\\prod_{\\substack{i\\in A,B \\\\ \\alpha \\in t,s}}\n{\\cal Z}_i^{\\alpha}\n\\end{equation}\nwhere $E_0$ denotes the energy density (per hole) corresponding to\nthe term in the mean-field Hamiltonian that does not couple to any\nspin or isospin operator,\n\\begin{eqnarray}\nE_0&=&J\\Big[6A\\,t_s^2(m_0^2-m_s^2)-2(A\\,\\eta_{\\vec{\\bf a}}^2+B)(m_0^2-m_s^2)\\nonumber\\\\\n&&+2C\\,t_s^2+2C\\,{\\eta_{\\vec{\\bf a}}^{\\prime}}^2+2D\\Big]+\\frac{G}{2}(\\varepsilon-\\varepsilon_0)^2.\n\\end{eqnarray}\n$N=N_A+N_B$ is the total number of sites.\nNow we can define the orbital ${\\cal Z}_i^t$ and magnetic ${\\cal\nZ}_i^s$ one-particle partition functions as \\cite{BZ74}:\n\\begin{eqnarray}\n{\\cal\nZ}_{i\\in A,B}^t&=&\\mathrm{Tr}_ie^{-\\beta \\vec{H}_i^t\\cdot\\vec{I}_i}\\\\\n{\\cal Z}_{i \\in A,B}^s&=&\\mathrm{Tr}_ie^{-\\beta \\vec{H}_i^s\\cdot\\vec{S}_i}\n\\end{eqnarray}\nwhere $\\vec{H}_i^t$ and $\\vec{H}_i^s$ are the molecular\nfield vectors that couple to the isospin and spin degrees of\nfreedom. If we denote the total free energy of the system by $F$,\nthe free energy per site (per hole) ${\\cal F}=F\/N=-\\ln {\\cal Z}\/\\beta N$ is\n\\begin{equation}\n{\\cal F}=E_0-\\frac{1}{2\\beta}\\left(\\ln\n{\\cal Z}_A^t +\\ln {\\cal Z}_B^t+\\ln {\\cal Z}_A^s+\\ln {\\cal\nZ}_B^s\\right)\n\\end{equation}\nEventually, the free\nenergy of the system for the given mean fields is found to be\n\\begin{eqnarray}\\label{Fh}\n{\\cal\nF}&=&J\\Big[6A\\,t_s^2(m_0^2-m_s^2)-2(A\\,\\eta_{\\vec{\\bf a}}^2+B)(m_0^2-m_s^2)\\nonumber\\\\\n&&+2C\\,t_s^2+2C\\,{\\eta_{\\vec{\\bf a}}^{\\prime}}^2+2D\\Big]+\\frac{G}{2}(\\varepsilon-\\varepsilon_0)^2\\nonumber\\\\\n&&-\\frac{1}{\\beta}\\ln\\left[2\\cosh\\left(\\frac{\\beta}{2}\\sqrt{f_1}\\right)\\right]\\nonumber\\\\\n&&-\\frac{1}{\\beta}\\ln\\left[2\\cosh\\left(\\frac{\\beta}{2}\\sqrt{f_2}\\right)\\right],\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nf_1&=&K^2\\varepsilon^2+16J^2\\left[A\\,t_s(m_0^2-m_s^2)+C\\,t_s\\right]^2,\\nonumber\\\\\nf_2&=&\\left\\{-g\\mu_BH+4Jm_0\\left[A(\\eta_{\\vec{\\bf a}}^2-t_s^2)+B\\right]m_0\\right\\}^2\\nonumber\\\\\n&&+16J^2m_s^2\\left[A(\\eta_{\\vec{\\bf a}}^2-t_s^2)+B\\right]^2.\n\\end{eqnarray}\nThe values of the mean fields are determined by the solution of the self-consistency equations \n\\begin{equation}\n0=\\frac{\\partial\\cal{F}}{\\partial\\phi}\n\\end{equation}\nwhere $\\phi=\\varepsilon, t_s, m_0, m_s$. These equations are solved numerically. Because\n\\begin{equation}\\label{I^x}\n\\langle I^x\\rangle=\\frac{G}{K}(\\varepsilon-\\varepsilon_0)\n\\end{equation}\nthe ferro-orbital order parameter is directly related to the difference $\\varepsilon-\\varepsilon_0$.\n\n\n\n\\section{Results}\\label{sec:res}\n\\begin{figure}[t]\n \\includegraphics[width=0.44\\linewidth]{figs\/AFMFO}\n \\includegraphics[width=0.42\\linewidth]{figs\/FMAFO}\n\\caption{Sketch of the low-temperature competing orders of the\nsystem. On the left hand side it is represented a lattice with\northorhombic symmetry and ferro-orbital order plus\nantiferromagnetism (FO \\& AFM). On the right hand side, a lattice with\ntetragonal symmetry with antiferro-orbital order plus\nferromagnetism (AFO \\& FM).}\n\\label{fig:CompetingOrders}\n\\end{figure}\nThe competing low temperature orders are schematically shown in Fig.~\\ref{fig:CompetingOrders}. For small values of $G$, the free energy minimization gives a ground state\nwith a strong lattice distortion breaking the tetragonal symmetry, as shown on the left\nhand side of Fig.~\\ref{fig:CompetingOrders}. In this case, the symmetry of the lattice is orthorhombic and ferro-orbital order (FO) coexists with antiferromagnetic order (AFM).\nOn the other hand, for large values of the elastic constant $G$, anti-ferro orbital (AFO) and ferromagnetic (FM) spin order are simultaneously realized. This is sketched on the right hand side of Fig.~\\ref{fig:CompetingOrders}.\n\n\n\\subsection{Absence of magnetic field.}\nThe $G$-$T$ phase diagram for $H=0$ is shown in Fig.~\\ref{fig:phaseH0}. The elastic constant $G$ of the lattice controls the distortion. Apart from the high-temperature disordered phase (PO) we can distinguish two main regions in the phase diagram, as discussed below. \n\n\\subsubsection{Soft lattice}\nLowering the temperature from the disordered phase we find for a soft lattice (small $G$) a crossover to a ferro-orbital ordered (FO) state. This crossover takes place at a temperature in the range of $k_BT_{FO}=\\frac{K^2}{4G}$ indicated by the diffuse line in Fig.~\\ref{fig:phaseH0}. The FO order is accompanied by a substantial orthorhombic distortion $\\varepsilon\\gg\\varepsilon_0$ which is driven by the coupling of the strain field to the orbital degrees of freedom as described by Eq.~(\\ref{eq:Hdis}). In addition, below a critical temperature $k_BT_{AFM}=J\\left(B+A\\eta_{\\vec{\\bf a}}^2\\right)$, antiferromagnetic (AFM) order sets in.\n\n\\subsubsection{Hard lattice}\nFor a harder lattice (large $G$), the gain in energy by polarizing the orbital degrees of freedom is not sufficient to drive a substantial orthorhombic distortion and $\\varepsilon\\approx\\varepsilon_0$. Instead, there is a staggered orbital order (AFO) which sets in at temperatures slightly below $k_BT_{AFO}=JC$. The AFO order drives a ferromagnetic (FM) ordered phase roughly below $k_BT_{FM}=-J\\left[B+A\\left(\\eta_{\\vec{\\bf a}}^2-\\frac{1}{4}\\right)\\right]$. For smaller values of $G$, this transition is weakly first-order but changes its character at the tricritical point (TP) to second-order. \n\n\\subsubsection{Structural transition}\nThere is a structural transition between the soft and the hard lattice. Below $T_{AFM}$ this transition is of first-order and is characterized by a simultaneous discontinuity in all the order parameters. The first-order line splits into two second-order lines at a bicritical point (BP). For temperatures above $T_{AFM}$ the transition is characterized by the onset of a staggered orbital and a gradual suppression of the ferro-orbital order with a concurrent reduction of the lattice distortion. \n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth]{figs\/phasee0H0}\n\\caption{$T-G$ phase diagram for $H=0$ involving antiferromagnetic (AFM), ferromagnetic (FM), ferroorbital (FO) and antiferroorbital (AFO) order. Dashed\nlines indicate second order phase transitions while full lines represent first order transitions. TP is a tricritical point and BP is a bicritical point. The diffuse line between FO and paraorbital (PO) region indicates a crossover.} \n\\label{fig:phaseH0}\n\\end{figure}\n\n\n\\subsection{Applied magnetic field}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\linewidth]{figs\/phasee0H025}\n\\caption{$T-G$ phase diagram for $g\\mu_BH=0.25J$. CE is a critical endpoint. The other abbreviations have the same meaning as in Fig.~\\ref{fig:phaseH0}. } \n\\label{fig:phaseH025}\n\\end{figure}\n\nNow we introduce in our analysis the effect of an external\nmagnetic field applied in the $x$-direction, that enters in the\nHamiltonian by the term ${\\cal H}_{mag}$ given in Eq.~(\\ref{Hmag}). For $g\\mu_BH=0.25J$ we obtain the phase diagram shown in Fig.~\\ref{fig:phaseH025}. Comparing Fig.~\\ref{fig:phaseH0} and \\ref{fig:phaseH025} we see that the main effect of the application of the magnetic field is the displacement of the first order structural transition towards smaller values of $G$ and, consequently, the reduction of the region of the phase diagram with AFM order, as expected. Therefore, the lattice effectively becomes harder in the presence of a finite magnetic field. The first order nature of the structural transition is now present for a larger range of temperatures, up to the tricritical point TP shown in Fig.~\\ref{fig:phaseH025}. In addition, a critical end-point CE is now defined when the second order $k_BT_{AFM}$ line meets the first order structural transition line. \n\n\n\n\n\\subsection{Metamagnetic transition}\n\\begin{figure}\n\\centering\n{\\includegraphics[width=1\\linewidth]{figs\/meanfieldsitoiii}}\n\\caption{Evolution of the transverse magnetization $m_0$, the staggered magnetization $m_s$, the uniform orbital order $\\langle I^x\\rangle$ and the staggered orbital order parameter $t_s$\nas a function of the applied magnetic field for three different sets of parameters $(k_BT\/J,JG\/K^2)$ given by $(i)$ (0.15,0.55), $(ii)$ (0.15,0.45) and $(iii)$ (0.25,0.45). The points $(i)-(iii)$ are also shown in the phase diagrams Figs.~\\ref{fig:phaseH0} and \\ref{fig:phaseH025}.} \n\\label{fig:metamag}\n\\end{figure}\nThe results of our calculations include a first order metamagnetic transition (MMT). The characteristics of this transition are summarized in Fig.~\\ref{fig:metamag}.\nHere we show, from panels a) to d), how the magnetic and orbital order parameters change as a function of the\nfield strength, for different values of temperature $T$ and elasticity $G$. The different $(T,G)$ points are labeled by $(i)-(iii)$ and are also indicated in the phase diagrams Figs.~\\ref{fig:phaseH0} and \\ref{fig:phaseH025}. The curves $(ii)$ and $(iii)$ of Fig.~\\ref{fig:metamag}a) show the discontinuous evolution of the magnetization towards a strongly polarized magnetic state by the application of a magnetic field. This magnetic\ntransition is accompanied in our system by a structural\ntransition where at the same\nmetamagnetic critical field $H_c$ an orthorhombic FO phase changes discontinuously\ntowards an AFO phase with tetragonal symmetry, as shown in Fig.~\\ref{fig:metamag}c)-d). For the MMT to be observed it is not stringent that the zero-field phase has antiferromagnetic order, as long as it is close enough to such a phase. \nNotice however, that a MMT is only possible for a soft lattice (small $G$) at low temperatures, as it is the case for the $(T,G)$ points $(ii)$ and $(iii)$ of Fig.~\\ref{fig:phaseH0}. For larger values of the elasticity, such as for $(i)$, we find a continuos evolution of the order parameters. \nIn summary, for the critical field $H_c(T,G)$, we find the general behavior that both rising $T$ or lowering $G$ increases the critical field.\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=0.9\\linewidth]{figs\/phasee0T0}}\n\\caption{$H-G$ phase diagram for $T=0$. The values $G_<$ and $G_>$\nbound the $G$-axis region where a metamagnetic transition is\nallowed. The first order line (full) meets the second order line\n(dashed) at a QCP.} \\label{fig:phasediagramT0}\n\\end{figure}\n\nFurther insights concerning the conditions for the MMT to occur can be obtained from the $H$-$G$-phase\ndiagram at zero temperature shown in Fig.~\\ref{fig:phasediagramT0}. It is worth noting that a first\norder MMT (accompanied by a structural\ntransition) can only occur at $T=0$ if we apply a magnetic field\nin the FO \\& AFM zone, and only for elasticity values belonging to\nthe region $G_<<G<G_>$. For $\\varepsilon_0\\ll K\/J$, $G_<$ and $G_>$ are given by\n\\begin{eqnarray}\nG_<&=&\\frac{K^2}{J(A+4C)-4K\\varepsilon_0},\\nonumber\\\\\nG_>&=&\\frac{K^2}{4J\\left(A\\left[1\/4-2\\eta_{\\vec{\\bf a}}^2\\right]-2B+C-\\frac{K\\varepsilon_0}{J}\\right)}.\n\\end{eqnarray}\nNotice that at $G_>$ there is a first order structural phase\ntransition for $T,H=0$. In Fig.~\\ref{fig:phasediagramT0} it can be\nseen that the metamagnetic critical field decreases as $G$ is\nstrengthened. This qualitative behavior remains valid at finite but low temperatures. \n\n\\subsection{Magnetostriction and thermal expansion}\nSince there is a close relation between structural and metamagnetic transition the temperature dependence of the lattice parameters show a qualitative different low-temperature behavior for magnetic fields below and above the metamagnetic transition. In our model, changes of the lattice parameters are considered by the orthorhombic distortion field $\\varepsilon$, related to the ferro-orbital order $\\langle I^x\\rangle$ by Eq.~(\\ref{I^x}). An increase of the c-axis is assumed to be proportional to $\\varepsilon_0-\\varepsilon$. Therefore, we show in Fig.~\\ref{fig:epsilonT} the temperature dependence of $\\varepsilon_0-\\varepsilon$ for different magnetic fields at $G=0.45K^2\/J$. The $T=0$ critical field for this elasticity corresponds to $H_c\\approx 0.22J\/g\\mu_B$ as it can be deduced, for example, from the zero temperature $G-H$ phase diagram of Fig.~\\ref{fig:phasediagramT0}. For fields lower than $H_c$, a metamagnetic transition will never be reached and the heating of the system by increasing the temperature drives the lattice towards a disordered PO phase through the crossover region (diffuse line in Fig.~\\ref{fig:phaseH0} and Fig.~\\ref{fig:phaseH025}). \n\nHowever, for fields $H>H_c$ the system is in the metamagnetic region. There is a low temperature AFO order and consequently, the distortion $\\varepsilon$ is small. By heating the system, metamagnetism is destroyed and at the same time the lattice undergoes a first order structural transition. This is seen in the $\\varepsilon$ vs. $T$ plot (Fig.~\\ref{fig:epsilonT}) by a jump of the distortion to a large negative value of $\\varepsilon_0-\\varepsilon$. If we keep heating the system, we reach again the PO region by passing the crossover zone. Obviously, the temperature of the 1$^{st}$-order AFO\/FO transition is larger for higher magnetic fields. This is reflected in the evolution of the $\\varepsilon_0-\\varepsilon$ discontinuity in Fig.~\\ref{fig:epsilonT} from $H=0.25J\/g\\mu_B$ to $0.5J\/g\\mu_B$.\n \\begin{figure}\n \\centering\n{\\includegraphics[width=0.9\\linewidth]{figs\/epsilonT}}\n\\caption{The orthorhombic distortion order $\\varepsilon_0-\\varepsilon = - \\frac{K}{G}\\langle I^x \\rangle$ as function of the temperature for different values of the magnetic field between $H=0.2$ and $H=0.5$ (in units of $J\/g\\mu_B$).} \\label{fig:epsilonT}\n\\end{figure}\n\n\n\n\\section{Comparison to experiments}\n\\label{sec:exp}\nEventually, we will motivate our microscopic model in view of the experimental results. In particular, we will consider the phase diagram at zero-field as well as at the magnetostriction and magnetization\nmeasurements that characterize the metamagnetic transition in Ca$_{2-x}$Sr$_x$RuO$_4$.\n\n\\subsection{Phase diagram}\n\nThe theoretical zero-field phase diagram\n(Fig.\\ref{fig:phaseH0}) can be compared to the\nexperimental $T$-$x$-phase diagram obtained in Ref.~\\cite{NM00} and schematized in Fig.~\\ref{fig:PD}. We relate the phases on the left-hand side of Fig.~\\ref{fig:phaseH0} (soft lattice) to the phases\nthat are observed in region II of the experimental phase diagram. In both cases the symmetry of the lattice is reduced. The tilting of the\nRuO$_6$-octahedra observed in the real material is modeled by a distortion of the\n2D lattice where the orbital and magnetic modes live. This reduction of symmetry is characterized by the strain field $\\varepsilon$ or the ferro-orbital order $\\langle I^x \\rangle$. For larger $G$ \n$\\langle I^x \\rangle$ is reduced and we find a transition to an AFO (predominantly) tetragonal phase where, in addition,\nFM order sets in at small temperatures. This is in fact the characteristics found in\nregion III of the experimental phase diagram near $x=0.5$. Therefore, it is not unfounded to relate the elasticity of the lattice (as modeled by $G$) to the Sr concentration $x$: both control the amount of distortion in their respective phase diagrams. Note that a Curie-like behavior of the orbital order $\\langle I^x \\rangle \\approx K\\varepsilon\/4k_BT$ is shown in Fig.~\\ref{fig:epsilonT}, which accounts for the temperature dependence of experimentally measured anisotropy of the spin susceptibility in the distorted (orthorhombic) region \\cite{NM00,ST04}.\n\n\n\\subsection{Metamagnetic transition}\n\nThe metamagnetic transition shown in Fig.~\\ref{fig:metamag} can be\nrelated to the experimental MMT \\cite{B05,B06}\nin the following way: in zero magnetic field, at a\ntemperature and doping ($G$ in our language) leading us into\nthe low-temperature FO zone of the phase diagram (region II of\nFig.\\ref{fig:PD}), we find a large strain $\\varepsilon \\gg \\varepsilon_0$ and a zero-component of the magnetization along the $x$-direction. If we now\nturn on the transverse magnetic field, a finite component of\n$m_0$ appears, although for small enough fields the strain is\nstill present in the system. For some critical field $H_{c}(T,G)$\nwe find a first order transition in the magnetization $m_0$ which\njumps discontinuously to some larger value, while the strain drops\nsimultaneously to $\\varepsilon \\sim \\varepsilon_0$. The transcription of this to\nthe experiments is that the octahedra returns to the structure it had\nbefore tilting. Also the $c$-axes adapts to the initial\ndirection it had in region III of the phase diagram. The tetragonal symmetry needs, however,\nnot to be restored. This behavior explains now the reversal of the\nstructural distortion which occurs upon cooling at zero field, since applying a \nhigh magnetic field at low enough\ntemperatures leads back to the old structure, \nas shown in Fig.~\\ref{fig:epsilonT} and seen experimentally \\cite{B05}.\n\nThis first order transition in the magnetization corresponds to\nthe MMT observed in the experiments. Note that inhomogeneity \nis ignored in our description. What occurs as a discontinous first order transition here,\nwould be a smooth crossover (MMT) when disorder, e.g. in alloying Ca and Sr, is \nincluded \\cite{ST04}. \n\n\nThe critical elastic constants $ G_{<} $ and $ G_{>} $ found in\nFig.~\\ref{fig:phasediagramT0}, bounding the segment on the $G$-axis\nwhere the first order magnetic and structural transitions are\npossible at $T=0$, can be mapped to the concentration\nvalues $x$ of the experimental phase diagram that define the region\nwhere the MMT can be observed. Therefore we may identify\n$G_<$ with $x=0.2$ and $G_>$ with $x=0.5$. This relation is\nconsistent with the experimental results that show a smaller\nenergy scale for the transition at $x=0.5$ compare to the one at\n$x=0.2$. The MMT is shifted towards lower\nfields when the Sr content grows from $x=0.2$ to $0.5$. On\nthe other hand, the first order structural transition driven at\nzero temperature for $G_>$ in the theoretical model can be related\nto the structure quantum phase transition of Ca$_{2-x}$Sr$_x$RuO$_4$~ at $x=0.5$. At $G_>$, the metamagnetic transition may be considered as occurring at zero-temperature and zero magnetic field, in agreement with the experimental measurements that shows that the MMT seems to be shifted towards a field close to zero for $x=0.5$.\n\n\n\\subsection{Magnetostriction and thermal expansion}\n\nThe dependence of the first order magnetic transition on the\ntemperature shown in Fig.~\\ref{fig:metamag} for points $(ii)$ and $(iii)$ of the phase diagram, and the temperature dependence of the distortion of Fig.~\\ref{fig:epsilonT} can be compared to the\nexperimental measurements, too. This is actually the expected\nbehavior and reproduces some of the results shown in\nRef.~\\cite{B05,B06}. If we look for example at the magnetostriction measurements of\nRef.~\\cite{B05,B06}, where they show $\\Delta L(H)\/L_0$ along the $c$\naxis as a function of the applied magnetic field, these results\ncan be interpreted as the response of the lattice structure to the\nmetamagnetic transition. The jump of the magnetization is coupled\nto an increase of the $c$-lattice constant $L_0$, since the structure of the lattice before the octahedra\ntilting is restored (see Fig.~\\ref{fig:metamag}c)). \n\n\nIn addition, the thermal expansion coefficients and the integrated length changes measured below and above the critical field show that the pronounced shrinking of the octahedra along the $c$-direction in zero field is successively suppressed by the field and turns into a low-temperature elongation at fields larger than $H_c$~\\cite{B05,B06}. The experimental temperature dependence of $\\Delta L\/L_0$ for various magnetic fields, as a measure of the lattice distortion, follows a temperature dependence of $\\varepsilon$ similar to the case of fields above and below $H_c$ and shown in Fig.~\\ref{fig:epsilonT}. In fact, for fields below the MMT the lattice evolves from the low-temperature FO order to the high-temperature PO region. On the other hand, for fields above the MMT the system shows a tetragonal AFO symmetry within the MMT- region which is suppressed by increasing the temperature, switching to an orthorhombic FO order. For higher temperatures the system looses orbital order and reaches the disordered PO region.\n\n\n\\section{Conclusion}\n\\label{sec:con}\nIn summary, we have analyzed a microscopic model for the description of\nmagnetic and structural properties in Ca$_{2-x}$Sr$_x$RuO$_4$ which is based on the assumption that\ntwo of the three electron bands known in Sr$_2$RuO$_4$ are Mott localized in \nregions II and Ca-rich zone of region III\n(near $x=0.5$) of the phase diagram \\cite{ANKRS02}. The mean-field treatment of this model reproduces the basic magnetic and structural properties, as well as\nthe metamagnetic transition of this material. The elastic properties have been introduced assuming\nthat the elastic constant depends on the Ca-concentration. In this way we draw the connection\nbetween our phase diagrams based on model parameters and the physical phase diagram\nand find a good qualitative and in parts quantitative agreement. Our most important result is\nthe magnetostriction effect in connection with the metamagnetic transition which agrees well\non a qualitative level with recent experimental findings. \n\n\nR.R. thanks M.P. L\\'opez-Sancho for many useful discussions. R.R. acknowledges the hospitality of the ETH-Z\\\,:{:}} |
|
{: and ``cooking range\,:{:}} |
|
|
