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Bell-Paired States Inducing Volume Law for Entanglement Entropy in Fermionic Lattices

Giacomo Gori, Simone Paganelli, Auditya Sharma, Pasquale Sodano, Andrea Trombettoni

Published May 14, 2014• 2 citations
OpenAlex

Abstract

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 $A$ and $\bar{A}$ is given, the states with maximal entanglement entropy (volume law) may be factored into Bell-pairs (BP) formed by two states with support on $A$ and $\bar{A}$. 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 $S_A$ between the area and the volume law, building an explicit example of a one-dimensional free fermion model where $S_A (L) \propto L^\beta$ with $\beta$ being intermediate between $\beta = 0$ (area law) and $\beta = 1$ (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, $\beta$. 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.

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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.
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 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 investigate the scaling of the entanglement entropy in an infinite translational invariant fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. Thus, in contrast with analogous bosonic systems, the entropic area law is violated for fermions. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven that the presented scaling law holds whenever the Fermi surface is finite. This is, in particular, true for all ground states of Hamiltonians with finite range interactions.
We study the scaling of the ground-state entanglement entropy of various free fermionic models on one-dimensional lattices, where the hopping and pairing terms decay as a power law. We seek to understand the scaling of entanglement entropy in generic models as the exponent of the power law $\ensuremath{\alpha}$ is varied. We ask if a common ${\ensuremath{\alpha}}_{c}$ exists across different systems governing the transition to area-law scaling found in local systems. We explore several examples numerically and argue that when applicable, the scaling of entanglement entropy in long-range models is constrained by predictions from the low-energy theory. In contrast, disordered models and models without a continuum limit show fractal scaling of entanglement approaching volume-law behavior as $\ensuremath{\alpha}$ approaches zero. These general features are expected to persist on turning on interactions.
We investigate the scaling of the entanglement entropy in an infinite translational invariant Fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. The latter can be traced back to the criticality of the models. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven, that the presented scaling law holds whenever the Fermi surface is finite. This is in particular true for all ground states of Hamiltonians with finite range interactions.
To produce a fermionic model exhibiting an entanglement entropy volume law, we propose a particular version of nonlocality in which the energy-momentum dispersion relation is effectively randomized at the shortest length scales while preserving translation invariance. In contrast to the ground state of local fermions, exhibiting an entanglement entropy area law with logarithmic corrections, the entropy of nonlocal fermions is extensive, scaling as the volume of the subregion and crossing over to the anomalous fermion area law at scales larger than the locality scale, $\ensuremath{\alpha}$. In the 1D case, we are able to show that the central charge appearing in the universal entropy expressions for large subregions is simply related to the locality scale. These results are demonstrated by exact diagonalizations of the corresponding discrete lattice fermion models. Within the Ryu-Takayanagi holographic picture, the relation between the central charge and the locality scale suggest a dual spacetime in which the size of the flat UV portion and the radius of anti--de Sitter in the IR are both proportional to the locality scale, $\ensuremath{\alpha}$.