We derive exact relations between the Rényi entanglement entropies and the particle-number fluctuations of (connected and disjoint) spatial regions in systems of N non-interacting fermions in arbitrary dimension. We prove that the asymptotic large- N behavior of the entanglement entropies is proportional to the variance of the particle number. We also consider 1D Fermi gases with a localized impurity, where all particle cumulants contribute to the asymptotic large- N behavior of the entanglement entropies. The particle cumulant expansion turns out to be convergent for all integer-order Rényi entropies (except for the von Neumann entropy) and the first few cumulants provide already a good approximation. Since the particle cumulants are accessible to experiments, these relations may provide a measure of entanglement in these systems.
We derive exact relations between the Rényi entanglement entropies and the particle-number fluctuations of (connected and disjoint) spatial regions in systems of N non-interacting fermions in arbitrary dimension. We prove that the asymptotic large- N behavior of the entanglement entropies is proportional to the variance of the particle number. We also consider 1D Fermi gases with a localized impurity, where all particle cumulants contribute to the asymptotic large- N behavior of the entanglement entropies. The particle cumulant expansion turns out to be convergent for all integer-order Rényi entropies (except for the von Neumann entropy) and the first few cumulants provide already a good approximation. Since the particle cumulants are accessible to experiments, these relations may provide a measure of entanglement in these systems.
We study the entanglement entropy of connected bipartitions in free-fermion gases of N particles in arbitrary dimension d . We show that the von Neumann and Rényi entanglement entropies grow asymptotically as N ( d −1)/ d ln N , with a prefactor that is analytically computed using the Widom conjecture both for periodic and open boundary conditions. The logarithmic correction to the power-law behavior is related to the area-law violation in lattice free fermions. These asymptotic large- N behaviors are checked against exact numerical calculations for N -particle systems.
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 consider the entanglement entropy for a line segment in the system of noninteracting one-dimensional fermions at zero temperature. In the limit of a large segment length L , the leading asymptotic behavior of this entropy is known to be logarithmic in L . We study finite-size corrections to this asymptotic behavior. Based on an earlier conjecture of the asymptotic expansion for full counting statistics in the same system, we derive a full asymptotic expansion for the von Neumann entropy and obtain first several corrections for the Rényi entropies. Our corrections for the Rényi entropies reproduce earlier results. We also discuss the entanglement spectrum in this problem in terms of single-particle occupation numbers.
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.
Abstract Entanglement entropy under a particle bipartition generates complementary information to mode entanglement because it is sensitive to interactions and particle statistics at leading order and does not depend on any externally imposed length scale. In this paper, we investigate the particle entanglement entropy in a system of N interacting spinless lattice fermions in one spatial dimension by combining bosonization techniques with exact and approximate numerical methods. We introduce a general scaling form for the fermionic particle entanglement entropy captured by a shape function that enters as an extensive interaction-induced correction to a known free fermion result. A general asymptotic expansion in the total number of particles demonstrates that its form is robust for different values of the Rényi index and highlights how quantum correlations are encoded in the n-particle density matrix of a pure many-body quantum state.
We investigate in detail the behavior of the bipartite fluctuations of particle number $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$ and spin ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}$ in many-body quantum systems, focusing on systems where such U(1) charges are both conserved and fluctuate within subsystems due to exchange of charges between subsystems. We propose that the bipartite fluctuations are an effective tool for studying many-body physics, particularly its entanglement properties, in the same way that noise and full counting statistics have been used in mesoscopic transport and cold-atomic gases. For systems that can be mapped to a problem of noninteracting fermions, we show that the fluctuations and higher-order cumulants fully encode the information needed to determine the entanglement entropy as well as the full entanglement spectrum through the R\'enyi entropies. In this connection, we derive a simple formula that explicitly relates the eigenvalues of the reduced density matrix to the R\'enyi entropies of integer order for any finite density matrix. In other systems, particularly in one dimension, the fluctuations are in many ways similar but not equivalent to the entanglement entropy. Fluctuations are tractable analytically, computable numerically in both density matrix renormalization-group and quantum Monte Carlo calculations, and in principle accessible in condensed-matter and cold-atom experiments. In the context of quantum point contacts, measurement of the second charge cumulant showing a logarithmic dependence on time would constitute a strong indication of many-body entanglement.