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arXiv:2210.13502 · Physical review. B./Physical review. B

Area-law entanglement from quantum geometry

Nisarga Paul

Published Feb 28, 2024• 8 citations
arXivHTMLPDF

Abstract

Quantum geometry, which encompasses both Berry curvature and the quantum metric, plays a key role in multiband interacting electron systems. We study the entanglement entropy of a region of linear size $\ensuremath{\ell}$ in fermion systems with nontrivial quantum geometry, i.e., whose Bloch states have nontrivial $k$ dependence. We show it scales as $S=\ensuremath{\alpha}{\ensuremath{\ell}}^{d\ensuremath{-}1}ln\ensuremath{\ell}+\ensuremath{\beta}{\ensuremath{\ell}}^{d\ensuremath{-}1}+\ensuremath{\cdots}$, where the first term is the well-known area-law violating term from the Fermi surface and $\ensuremath{\beta}$ contains the leading contribution from quantum geometry. We compute this for the case of uniform quantum geometry and cubic domains and provide numerical results for the Su-Schrieffer-Heeger model, two-dimensional (2D) massive Dirac cone, and 2D Chern bands. An experimental probe of the quantum geometric entanglement entropy is proposed using particle number fluctuations. We offer an intuitive account of the area-law entanglement related to the spread of maximally localized Wannier functions.

DOI
10.1103/physrevb.109.085146

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Quantum geometry, which encompasses both Berry curvature and the quantum metric, plays a key role in multiband interacting electron systems. We study the entanglement entropy of a region of linear size \ensuremathℓ\ensuremath{\ell}\ensuremathℓ in fermion systems with nontrivial quantum geometry, i.e., whose Bloch states have nontrivial kkk dependence. We show it scales as S=\ensuremathα\ensuremathℓd\ensuremath−1ln\ensuremathℓ+\ensuremathβ\ensuremathℓd\ensuremath−1+\ensuremath⋯S=\ensuremath{\alpha}{\ensuremath{\ell}}^{d\ensuremath{-}1}ln\ensuremath{\ell}+\ensuremath{\beta}{\ensuremath{\ell}}^{d\ensuremath{-}1}+\ensuremath{\cdots}S=\ensuremathα\ensuremathℓd\ensuremath−1ln\ensuremathℓ+\ensuremathβ\ensuremathℓd\ensuremath−1+\ensuremath⋯, where the first term is the well-known area-law violating term from the Fermi surface and \ensuremathβ\ensuremath{\beta}\ensuremathβ contains the leading contribution from quantum geometry. We compute this for the case of uniform quantum geometry and cubic domains and provide numerical results for the Su-Schrieffer-Heeger model, two-dimensional (2D) massive Dirac cone, and 2D Chern bands. An experimental probe of the quantum geometric entanglement entropy is proposed using particle number fluctuations. We offer an intuitive account of the area-law entanglement related to the spread of maximally localized Wannier functions.
Free fermions with a finite Fermi surface are known to exhibit an anomalously large entanglement entropy. The leading contribution to the entanglement entropy of a region of linear size L in d spatial dimensions is S∼L(d-1)logL, a result that should be contrasted with the usual boundary law S∼L(d-1). This term depends only on the geometry of the Fermi surface and on the boundary of the region in question. I give an intuitive account of this anomalous scaling based on a low energy description of the Fermi surface as a collection of one-dimensional gapless modes. Using this picture, I predict a violation of the boundary law in a number of other strongly correlated systems.
The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement in its pure ground state. Here we establish scaling laws for this entanglement in critical quasifree fermionic and bosonic lattice systems, without resorting to numerical means. We consider the setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic-where an area law is proven to hold-or fermionic where we determine a logarithmic correction to the area law, which depends on the topology of the Fermi surface. We find Lifshitz quantum phase transitions accompanied with a nonanalyticity in the prefactor of the leading order term.
The geometry of quantum states could offer indispensable insights for characterizing the topological properties, phase transitions, and entanglement nature of many-body systems. In this work, we reveal the quantum geometry and the associated entanglement entropy (EE) of Floquet topological states in one-dimensional periodically driven systems. The quantum metric tensors of Floquet states are found to show nonanalytic signatures at topological phase transition points. Away from the transition points, the bipartite geometric EE of Floquet states exhibits an area-law scaling vs the system size, which holds for a Floquet band at any filling fractions. For a uniformly filled Floquet band, the EE further becomes purely quantum geometric. At phase transition points, the geometric EE scales logarithmically with the system size and displays cusps in the nearby parameter ranges. These discoveries are demonstrated by investigating typical Floquet models including periodically driven spin chains, Floquet topological insulators, and superconductors. Our findings uncover the rich quantum geometries of Floquet states, unveiling the geometric origin of EE for gapped Floquet topological phases, and introducing information-theoretic means of depicting topological transitions in Floquet systems.
Recently, tight-binding models on hyperbolic lattices (discretized anti–de Sitter space) have gained significant attention, leading to hyperbolic band theory and non-Abelian Bloch states. In this paper, we investigate these quantum systems from the perspective of quantum information, focusing particularly on the scaling of entanglement entropy (EE) that has been regarded as a powerful quantum-information probe into exotic phases of matter. It is known that on a d -dimensional translation-invariant Euclidean lattice, the EE of band insulators scales as an area law ( ∼ L d − 1 , where L is the linear size of the boundary between two subsystems). Meanwhile, the EE of metals [with a finite density of state (DOS)] scales as the renowned Gioev-Klich-Widom scaling law ( ∼ L d − 1 ln L ). The appearance of logarithmic divergence, as well as the analytic form of the coefficient c , is mathematically controlled by the Widom conjecture of asymptotic behavior of Toeplitz matrices and can be physically understood via the Swingle's argument. However, the hyperbolic lattice, which generalizes translational symmetry, results in the inapplicability of these analytic approaches and the potential nontrivial behavior of the EE. Here we make an initial attempt through numerical simulation. Remarkably, we find that both cases adhere to the area law, indicating the effect of background hyperbolic geometry that influences quantum entanglement. To achieve the results, we first apply the vertex-inflation method to generate a hyperbolic lattice on the Poincaré disk, and then apply the Haydock recursion method to compute the DOS. Finally, we study the scaling of the EE for different bipartitions via exact diagonalization and perform finite-size scaling. We also investigate how the coefficient of the area law is correlated to the bulk gap in the gapped case and to the DOS in the gapless case, respectively. Future directions are discussed.
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.
Abstract We study the entanglement properties of non-Hermitian free fermionic models with translation symmetry using the correlation matrix technique. Our results show that the entanglement entropy has a logarithmic correction to the area law in both one-dimensional and two-dimensional systems. For any one-dimensional one-band system, we prove that each Fermi point of the system contributes exactly 1/2 to the coefficient c of the logarithmic correction. Moreover, this relation between c and Fermi point is verified for more general one-dimensional and two-dimensional cases by numerical calculations and finite-size scaling analysis. In addition, we also study the single-particle and density–density correlation functions.