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arXiv:1703.10587 · Journal of Statistical Mechanics Theory and Experiment

Particle partition entanglement of one dimensional spinless fermions

Hatem Barghathi, Emanuel Casiano-Diaz, Adrian Del Maestro

Published Aug 1, 2017• 15 citations
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Abstract

Abstract We investigate the scaling of the Rényi entanglement entropies for a particle bipartition of interacting spinless fermions in one spatial dimension. In the Tomonaga–Luttinger liquid regime, we calculate the second Rényi entanglement entropy and show that the leading order finite-size scaling is equal to a universal logarithm of the system size plus a non-universal constant. Higher-order corrections decay as power-laws in the system size with exponents that depend only on the Luttinger parameter. We confirm the universality of our results by investigating the one dimensional t − V model of interacting spinless fermions via exact-diagonalization techniques. The resulting sensitivity of the particle partition entanglement to boundary conditions and statistics supports its utility as a probe of quantum liquids.

DOI
10.1088/1742-5468/aa819a

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Abstract We investigate the scaling of the Rényi entanglement entropies for a particle bipartition of interacting spinless fermions in one spatial dimension. In the Tomonaga–Luttinger liquid regime, we calculate the second Rényi entanglement entropy and show that the leading order finite-size scaling is equal to a universal logarithm of the system size plus a non-universal constant. Higher-order corrections decay as power-laws in the system size with exponents that depend only on the Luttinger parameter. We confirm the universality of our results by investigating the one dimensional t − V model of interacting spinless fermions via exact-diagonalization techniques. The resulting sensitivity of the particle partition entanglement to boundary conditions and statistics supports its utility as a probe of quantum liquids.
We consider the R\'enyi entanglement entropy of bosonic Tomonaga-Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher-order corrections involve a microscopic length scale, the leading-order scaling depends only on this sole dimensionless parameter which characterizes the low-energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Tomonaga-Luttinger liquid theory for the Lieb-Liniger model of $\ensuremath{\delta}$-function interacting bosons in the one-dimensional spatial continuum.
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 introduce a systematic framework for calculating the bipartite entanglement entropy of a compact spatial subsystem in a one-dimensional quantum gas which can be mapped into a non-interacting fermion system. We show that when working with a finite number of particles N , the Rényi entanglement entropies grow as ln N , with a prefactor that is given by the central charge. We apply this novel technique to the ground state and to excited states of periodic systems. We also consider systems with boundaries. We derive universal formulas for the leading behavior and for subleading corrections to the scaling. The universality of the results allows us to make predictions for the finite size scaling forms of the corrections to the scaling.
Particle entanglement provides information on quantum correlations in systems of indistinguishable particles. Here, we study the one-particle entanglement entropy for an integrable model of spinless, interacting fermions both at equilibrium and after an interaction quantum quench. Using both large-scale exact diagonalization and time-dependent density matrix renormalization group calculations, we numerically compute the one-body reduced density matrix for the $J\ensuremath{-}V$ model, as well as its postquench dynamics. We include an analysis of the fermionic momentum distribution, showcasing its time evolution after a quantum quench. Our numerical results, extrapolated to the thermodynamic limit, can be compared with field theoretic bosonization in the Tomonaga-Luttinger liquid regime. Excellent agreement is obtained using an interaction cutoff that can be determined uniquely in the ground state.
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
For indistinguishable itinerant particles subject to a superselection rule fixing their total number, a portion of the entanglement entropy under a spatial bipartition of the ground state is due to particle fluctuations between subsystems and thus is inaccessible as a resource for quantum information processing. We quantify the remaining operationally accessible entanglement in a model of interacting spinless fermions on a one-dimensional lattice via exact diagonalization and the density matrix renormalization group. We find that the accessible entanglement exactly vanishes at the first-order phase transition between a Tomonaga-Luttinger liquid and phase separated solid for attractive interactions and is maximal at the transition to the charge density wave for repulsive interactions. Throughout the phase diagram, we discuss the connection between the accessible entanglement entropy and the variance of the probability distribution describing intrasubregion particle-number fluctuations.