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arXiv:1308.6257 · Physical Review E

Entanglement in ground and excited states of gapped free-fermion systems and their relationship with Fermi surface and thermodynamic equilibrium properties

Michelle Storms, Rajiv R. P. Singh

Published Jan 17, 2014• 60 citations
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Abstract

We study bipartite entanglement entropies in the ground and excited states of free-fermion models, where a staggered potential, μs, induces a gap in the spectrum. Ground-state entanglement entropies satisfy the "area law", and the "area-law" coefficient is found to diverge as a logarithm of the staggered potential, when the system has an extended Fermi surface at μs=0. On the square lattice, we show that the coefficient of the logarithmic divergence depends on the Fermi surface geometry and its orientation with respect to the real-space interface between subsystems and is related to the Widom conjecture as enunciated by Gioev and Klich [ Phys. Rev. Lett. 96 100503 (2006)]. For point Fermi surfaces in two-dimension, the "area-law" coefficient stays finite as μs→0. The von Neumann entanglement entropy associated with the excited states follows a "volume law" and allows us to calculate an entropy density function sV(e), which is substantially different from the thermodynamic entropy density function sT(e), when the lattice is bipartitioned into two equal subsystems but approaches the thermodynamic entropy density as the fraction of sites in the larger subsystem, that is integrated out, approaches unity.

DOI
10.1103/physreve.89.012125

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We study bipartite entanglement entropies in the ground and excited states of free-fermion models, where a staggered potential, μs, induces a gap in the spectrum. Ground-state entanglement entropies satisfy the "area law", and the "area-law" coefficient is found to diverge as a logarithm of the staggered potential, when the system has an extended Fermi surface at μs=0. On the square lattice, we show that the coefficient of the logarithmic divergence depends on the Fermi surface geometry and its orientation with respect to the real-space interface between subsystems and is related to the Widom conjecture as enunciated by Gioev and Klich [ Phys. Rev. Lett. 96 100503 (2006)]. For point Fermi surfaces in two-dimension, the "area-law" coefficient stays finite as μs→0. The von Neumann entanglement entropy associated with the excited states follows a "volume law" and allows us to calculate an entropy density function sV(e), which is substantially different from the thermodynamic entropy density function sT(e), when the lattice is bipartitioned into two equal subsystems but approaches the thermodynamic entropy density as the fraction of sites in the larger subsystem, that is integrated out, approaches unity.
We show how the area law for the entanglement entropy may be violated by free fermions on a lattice and look for conditions leading to the emergence of a volume law. We give an explicit construction of the states with maximal entanglement entropy based on the fact that, once a bipartition of the lattice in two complementary sets AAA and Aˉ\bar{A}Aˉ is given, the states with maximal entanglement entropy (volume law) may be factored into Bell-pairs (BP) formed by two states with support on AAA and Aˉ\bar{A}Aˉ. We then exhibit, for translational invariant fermionic systems on a lattice, an Hamiltonian whose ground state is such to yield an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy SAS_ASA​ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where SA(L)∝LβS_A (L) \propto L^\betaSA​(L)∝Lβ with β\betaβ being intermediate between β=0\beta = 0β=0 (area law) and β=1\beta = 1β=1 (BP-state inducing volume law). For this model, the dispersion relation has a zig-zag structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, β\betaβ. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a non-trivial topology of the Fermi surface.
Exactly solving a spinless fermionic system in two and three dimensions, we investigate the scaling behavior of the block entropy in critical and noncritical 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 and I. Klich [Phys. Rev. Lett. 96, 100503 (2006)].
We show how the area law for entanglement entropy may be violated by free fermions on a lattice, and we look for conditions leading to the emergence of a volume law. We give an explicit construction of the states with maximal entanglement entropy based on the fact that, once a bipartition of the lattice in two complementary sets AAA and A‾\overline{A}A is given, the states with maximal entanglement entropy (volume law) may be factored into Bell pairs (BPs) formed by two states with support on AAA and A‾\overline{A}A. We then exhibit, for translational invariant fermionic systems on a lattice, a Hamiltonian whose ground state is such that it yields an exact volume law. As expected, the corresponding Fermi surface has a fractal topology. We also provide some examples of fermionic models for which the ground state may have an entanglement entropy SA{S}_{A}SA​ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where ${S}_{A}(L)\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, with $\ensuremath{\beta}$ being intermediate between $\ensuremath{\beta}=0$ (area law) and $\ensuremath{\beta}=1$ (BP state inducing volume law). For this model, the dispersion relation has a ``zigzag'' structure leading to a fractal Fermi surface whose counting box dimension equals, for large lattices, $\ensuremath{\beta}$. Our analysis clearly relates the violation of the area law for the entanglement entropy of the ground state to the emergence of a nontrivial topology of the Fermi surface.
We show that entanglement entropy of free fermions scales faster than area law, as opposed to the scaling ${L}^{d\ensuremath{-}1}$ for the harmonic lattice, for example. We also suggest and provide evidence in support of an explicit formula for the entanglement entropy of free fermions in any dimension ddd, $S\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega}){L}^{d\ensuremath{-}1}\mathrm{log}L$ as the size of a subsystem $L\ensuremath{\rightarrow}\ensuremath{\infty}$, where $\ensuremath{\partial}\ensuremath{\Gamma}$ is the Fermi surface and $\ensuremath{\partial}\ensuremath{\Omega}$ is the boundary of the region in real space. The expression for the constant $c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega})$ is based on a conjecture due to Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimate on the entropy SSS.
We study general entanglement properties of the excited states of the one-dimensional translational invariant free fermions and coupled harmonic oscillators. In particular, using the integrals of motion, we prove that these Hamiltonians independent of the gap (mass) have infinite excited states that can be described by conformal field theories with integer or half-integer central charges. In the case of free fermions, we also show that because of the huge degeneracy in the spectrum, even a gapless Hamiltonian can have excited states with an area law. Finally, we study the universal average entanglement entropy over eigenstates of the Hamiltonian of the free fermions introduced recently in Vidmar et al. [Phys. Rev. Lett. 119, 020601 (2017)]. In particular, using a duality relation, we map the average bipartite entanglement entropy over all the exponential numbers of eigenstates of a generic free fermion Hamiltonian to the problem of the calculation of the average multibipartite entanglement entropy over a single eigenstate of the XX\mathit{XX}XX chain. This relation can be useful for experimental measurement of the universal average entanglement entropy. Part of our conclusions can be extended to the quantum spin chains that are associated with the free fermions via Jordan-Wigner transformation.
law is enhanced by a logarithmic factor. In Ref. [1], we use multi-dimensional bosonization to provide a simple derivation of this result, and show that the logarithimic factor has a 1D origin. More importantly the bosonization technique allows us to take into account the Fermi liquid interactions, and obtain the leading scaling behavior of the entanglement entropy of Fermi liquids. The central result of our work is that Fermi liquid interactions do not alter the leading scaling behavior of the entanglement entropy, and the logarithmic enhancement of area law is a robust property of the Fermi liquid phase. In sharp contrast to the fermioic systems with Fermi surfaces, quantum critical (or gapless) bosonic systems do not violate the area law above 1D (except for the case discussed below). The fundamental difference lies in the fact that gapless excitations live near a single point (usually origin of momentum space) in such bosonic systems, while they live around an (extended) Fermi surface in Fermi liquids. In Ref. [2], we studied entanglement properties of some specific examples of the so called Bose metal states, in which bosons neither condense (and become a superfluid) nor localize (and insulate) at T=0. The system supports gapless excitations around “Bose surfaces,” instead of isolated points in momentum space. We showed that similar to free Fermi gas and Fermi liquids, these states violate the entanglement area law in a logarithmic fashion. Compared to ground states, much less is known concretely about entanglement in (highly) excited states. Going back to free fermion systems, in Ref. [3] we show that there exists a duality relation between ground and excited states, and the area law obeyed by ground state turns into a volume law for excited states, something widely expected but hard to prove. Most importantly, we find in appropriate limits the reduced density matrix of a subsystem takes the form of thermal density matrix, providing an explicit example of the eigenstate thermalization hypothesis. Our work [3] explicitly demonstrates how statistical physics emerges from entanglement in a single eigenstate.