We exploit and clarify the use of random matrix theory for the calculation of the entanglement entropy of free Fermi gases. We apply this method to obtain analytic predictions for R\'enyi entanglement entropies of a one-dimensional gas trapped by a harmonic potential in all the relevant scaling regimes. We confirm our findings with accurate numerical calculations obtained by means of an ingenious discretization of the reduced correlation matrix.
We exploit and clarify the use of random matrix theory for the calculation of the entanglement entropy of free Fermi gases. We apply this method to obtain analytic predictions for R'enyi entanglement entropies of a one-dimensional gas trapped by a harmonic potential in all the relevant scaling regimes. We confirm our findings with accurate numerical calculations obtained by means of an ingenious discretization of the reduced correlation matrix.
We investigate the entanglement properties of the nonequilibrium dynamics of one-dimensional noninteracting Fermi gases released from a trap. The gas of N particles is initially in the ground state within a hard-wall or a harmonic trap, and then it expands after dropping the trap. We compute the time dependence of the von Neumann and R\'enyi entanglement entropies and the particle fluctuations of spatial intervals around the original trap in the limit of a large number N of particles. The results for these observables apply to one-dimensional gases of impenetrable bosons as well. We identify different dynamical regimes at small and large times, depending also on the initial condition, i.e., whether it is that of a hard-wall or a harmonic trap. In particular, we show analytically that the expansion from hard-wall traps is characterized by the asymptotic small-time behavior $S\ensuremath{\approx}(1/3)\mathrm{ln}(1/t)$ of the von Neumann entanglement entropy and the relation $S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3$ (where V is the particle variance), which are analogous to the equilibrium behaviors whose leading logarithms are essentially determined by the corresponding conformal field theory with central charge c=1. The time dependence of the entanglement entropy of extended regions during the expansion from harmonic traps has a remarkable property in that it can be expressed as a global time-dependent rescaling of the space dependence of the initial equilibrium entanglement entropy.
We establish an exact mapping between the positions of N noninteracting fermions in a two-dimensional rotating harmonic trap in its ground state and the eigenvalues of the $N\ifmmode\times\else\texttimes\fi{}N$ complex Ginibre ensemble of random matrix theory (RMT). Using RMT techniques, we make precise predictions for the statistics of the positions of the fermions, both in the bulk as well as at the edge of the trapped Fermi gas. In addition, we compute exactly, for any finite N, the R\'enyi entanglement entropy and the number variance of a disk of radius r in the ground state. We show that while these two quantities are proportional to each other in the (extended) bulk, this is no longer the case very close to the trap center nor at the edge. Near the edge, and for large N, we provide exact expressions for the scaling functions associated with these two observables.
We investigate quantum correlations in the ground state of noninteracting Fermi gases of N particles trapped by an external space-dependent harmonic potential, in any dimension. For this purpose we compute one-particle correlations, particle fluctuations, and bipartite entanglement entropies of extended space regions, and study their large-N scaling behaviors. The half-space von Neumann entanglement entropy is computed for any dimension, obtaining ${S}_{\mathrm{HS}}\ensuremath{\approx}{c}_{l}{N}^{(d\ensuremath{-}1)/d}\mathrm{ln}N$ analogously to homogenous systems, with cl=1/6,1/(62),1/(66) in one, two, and three dimensions, respectively. We show that the asymptotic large-N relation ${S}_{A}\ensuremath{\approx}{\ensuremath{\pi}}^{2}{V}_{A}/3$, between the von Neumann entanglement entropy SA and particle variance VA of an extended space region A, holds for any subsystem A and in any dimension, analogously to homogeneous noninteracting Fermi gases.
We study the ground-state entanglement entropy of free fermions on random graphs. We first establish a general criterion for the volume law on random graphs, based on the spectral characteristics of the Hamiltonian. We then apply this criterion to the Erdős--Rényi random graph, where each of the possible edges is present independently with some probability. Using random matrix theory and asymptotic freeness, we show that the ground-state entanglement entropy obeys an exact volume law in Erdős--Rényi graphs in the thermodynamic limit, with a universal coefficient that is independent of the edge probability of the graph. This coefficient is confirmed numerically to take the value approximately 0.3863 nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.
We study the ground-state entanglement entropy of a finite subsystem of size L of an infinite system of noninteracting fermions scattered by a potential of finite range a. We derive a general relation between the scattering matrix and the overlap matrix and use it to prove that for a one-dimensional symmetric potential the von Neumann entropy, the Rényi entropies, and the full counting statistics are robust against potential scattering, provided that L/a≫1. The results of numerical calculations support the validity of this conclusion for a generic potential.
Abstract In this work, the entropy functions of ideal quantum gases in a three-dimensional harmonic trap are analytically calculated using temperature as an explicit variable. Afterward, the applicability of the analytical formulas is validated by comparison with the numerical calculation. The results illustrate that the obtained functions could be applied for the whole temperature regime with a maximum relative deviation of less than 7.5% in the vicinity of the critical temperature T c in the case of Bose gases. Meanwhile, for Fermi gases, although the analytical formula fits well at very low- and high-temperature regimes, it cannot be applied at temperature in the range [ 0.3 - 0.5 ] T F , where T F is the Fermi temperature. In addition, the consistency between our formulas and classical ones at significantly high temperatures is also discussed.