Scaling behavior of entanglement in two- and three-dimensional free-fermion systems

Abstract

Exactly solving a spinless fermionic system in two and three dimensions, we investigate the scaling behavior of the block entropy in critical and non-critical phases. The scaling of the block entropy crucially depends on the nature of the excitation spectrum of the system and on the topology of the Fermi surface. Noticeably, in the critical phases the scaling violates the area law and acquires a logarithmic correction only when a well defined Fermi surface exists in the system. When the area law is violated, we accurately verify a conjecture for the prefactor of the logarithmic correction, proposed by D. Gioev The nature of many-body entanglement in various solid-state models has been the focus of recent interest. The motivation for this effort is two-fold. On the one side, these systems are of interest for the purpose of quantum information processing and quantum computation [1]. At a fundamental level, the study of entanglement represents a purely quantum way of understanding and characterizing quantum phases and quantum phase transitions in many-body physics [2,3,4,5].A striking feature of entangled states |Ψ is that a local accurate description of such states is impossible, namely each subsystem A of the total system U can have a finite entropy, quantified as the von-Neumann entropy S A = −Trρ A log 2 ρ A of its reduced density matrix ρ A = T r U\A |Ψ Ψ|, whereas the total system clearly has zero entropy. The entropy of entanglement S A of the subsystem is a reliable estimate of the entanglement between the subsystem A and the rest, U \ A. Assuming that the system U corresponds to the whole universe in d dimensions, a fundamental question concerns the scaling behavior of the entropy of entanglement S L of an hypercubic subsystem L d (hereafter denoted as a block ) with its size L. Indeed, such scaling probes directly the spatial range of entanglement: when the block size exceeds the characteristic length over which two sites are entangled, the block entropy should become subadditive, and scale at most as the area of the block boundaries, following a so-called area law :A crucial question is then if and how the scaling of the block entropy changes when the nature of the quantum many-body state evolves in a critical way by passing through a quantum phase transition, and how the characteristic spatial extent of entanglement relates to the correlation length of the system. This question has been extensively addressed in the case of one-dimensional spin systems [5,6,7,8], in chains of harmonic oscillators [9,10] and in related conformal field theories (CFT) [11,12]. Here it is found unambiguosly that in states with exponentially decaying (connected) correlators S L follows the area law S L ∼ L 0 , saturating to a finite value, whereas for critical states, displaying power-law decaying correlations, a logarithmic correction to the area is always present: S L = [(c +c)/6] log 2 L, where c is the central charge of the related CFT. The asymptotic value of the block entropy is found to diverge logarithmically with t...

Entanglement in many-body systems

“…For regular block and Fermi surface, numerical analysis has confirmed the modified area-law for critical two-dimensional (Barthel et al, 2006a;Li et al, 2006) and three-dimensional models. Barthel et al, 2006a study the tight binding model as an example for a two-dimensional model with a connected Fermi surface as well as the model…”

Bipartite fluctuations as a probe of many-body entanglement

“…8, a numerical fit shows that the prefactor of the leading L ln L term is almost certainly exactly 1/3 as predicted by a formula based on the Widom conjecture [96,97]. Of course, unlike the one-dimensional case the prefactor is not universal and depends, for example, on the chemical potential.…”

Tensor Network States and Geometry

“…In D > 1, the relation between the scaling of entanglement entropy and the existence of a gap in H is less clear-cut. For instance, the study of possible scalings of entanglement entropy in the ground state of systems of free fermions shows that the boundary law is obeyed by gapped systems, as expected, but also for a class of critical systems (namely, systems with a Fermi surface of dimension Γ smaller than D − 1), whereas a second class of critical systems (with a Fermi surface of dimension Γ = D − 1) display logarithmic multiplicative corrections to the boundary law [75][76][77][78] , see Table I. Such logarithmic corrections are also believed to be present in other gapless systems in D > 1 dimensions, such as Fermi Liquids and spin Bose metals [79][80][81][82][83] .…”

“…However, there are also ground states that display logarithmic corrections to the above boundary law [71][72][73][74][75][76][77][78][79][80][81][82][83] ,…”