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Convexity of the Entanglement Entropy of SU( 2 N )-Symmetric Fermions with Attractive Interactions

William J. Porter, Joaquín E. Drut

Published Jul 2, 2021• 0 citations
Publisher

Abstract

The positivity of the probability measure of attractively interacting systems of $2N$-component fermions enables the derivation of an exact convexity property for the ground-state energy of such systems. Using analogous arguments, applied to path-integral expressions for the entanglement entropy derived recently, we prove non-perturbative analytic relations for the R\'enyi entropies of those systems. These relations are valid for all sub-system sizes, particle numbers and dimensions, and in arbitrary external trapping potentials.

DOI
10.17615/3xfy-d504

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The positivity of the probability measure of attractively interacting systems of 2N2N2N-component fermions enables the derivation of an exact convexity property for the ground-state energy of such systems. Using analogous arguments, applied to path-integral expressions for the entanglement entropy derived recently, we prove non-perturbative analytic relations for the R'enyi entropies of those systems. These relations are valid for all sub-system sizes, particle numbers and dimensions, and in arbitrary external trapping potentials.
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The positivity of the probability measure of attractively interacting systems of 2N-component fermions enables the derivation of an exact convexity property for the ground-state energy of such systems. Using analogous arguments, applied to path-integral expressions for the entanglement entropy derived recently, we prove nonperturbative analytic relations for the Rényi entropies of those systems. These relations are valid for all subsystem sizes, particle numbers, and dimensions, and in arbitrary external trapping potentials.
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.
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 investigate the entanglement entropy of a class of NNN-particle excited state of fermions confined in a two-dimensional harmonic trap rotating at an angular frequency ΩΩΩ. The excited state is constructed by filling a particular set of NNN single-particle energy levels. We analytically compute the Rényi entropies of order qqq in a disc of radius rrr around the centre of the trap, and the cumulants corresponding to number fluctuations of fermions within the disc. We found that the area law scaling of entanglement entropy holds even for a class of excited states. We also verified the well-known series expansion of entanglement entropy in terms of the particle number cumulants for non-interacting fermions. We further derive the centered cumulant generating function, demonstrating that the associated probability distribution function in the disc has identical scaling properties, up to a variable shift, to the known ground-state result. Finally, we extend our result to an annular region, showing that both the Rényi entropy and particle number cumulants of the annulus decompose into sums of the corresponding quantities of the two bounding discs. These additive relations hold as long as the width of the annulus is sufficiently large.
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with a fixed number parity yet not necessarily a fixed particle number. The mode entanglement between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space H\mathcal{H}H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of H\mathcal{H}H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined by J. Schliemann et al. [Phys. Rev. A 64, 022303 (2001)], implying that particle entanglement may be seen as minimum mode entanglement. It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that in the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy is evaluated analytically, extending the results in previous reference to general states.