Area Law for the entanglement entropy of free fermions in nonrandom ergodic field
Leonid Pastur, Mira Shamis
TL;DR
This work addresses the entanglement entropy scaling, $S_\Lambda(\varepsilon_F)$, for free lattice fermions in nonrandom ergodic fields by connecting spectral localisation properties of the one-body Hamiltonian $H_\omega$ to area-law behavior. Employing a two-step approach—spectral analysis to bound the Fermi projection or eigenfunction correlator and nonlinear entropy analysis—the authors prove Area Law scaling in broad deterministic ergodic models, including quasi-periodic, limit-periodic, and subshift-of-finite-type potentials, across various dimensions. Central to the methodology are exponential dynamical localisation and exponential decay of the eigenfunction correlator $Q_I(m,n)$, which translate into $S_\Lambda(\varepsilon_F) \lesssim C L^{d-1}$ with finite $C$; the bottom of the spectrum is specifically treated for 1D subshift models. By leveraging Lyapunov exponent positivity and large deviation estimates, the results extend area-law entanglement from random disordered systems to a wide class of deterministic ergodic operators, linking spectral localization properties to quantum information characteristics.
Abstract
The paper deals with the asymptotic behavior of one of the widely used characteristics of correlations in large quantum systems. The correlations are known as quantum entanglement, the characteristic is called the entanglement entropy, and as large systems we consider an ideal gas of spinless lattice fermions. The system is determined by its one-body Hamiltonian. As shown in \cite{EPS}, if the Hamiltonian is an ergodic finite difference operator with exponentially decaying spectral projection, then the asymptotic form of the entanglement entropy is the so-called Area Law. However, the only one-body Hamiltonian for which this spectral condition is verified is the $d$-dimensional discrete Schrödinger operators with random potential. In the present paper, we prove that the same asymptotic form of the entanglement entropy holds for a wide class of Schrödinger operators whose potentials are ergodic but nonrandom. We start with the quasiperiodic and limit periodic operators, and then pass to the interesting and highly non-trivial case of the potentials generated by subshifts of finite type. They arose in the theory of dynamical systems in the study of non-random chaotic phenomena. As it turns out, obtaining the asymptotics of the entanglement entropy of free fermions requires a quite involved spectral analysis of the corresponding Schrödinger operator. Specifically, we prove for this class two important and interesting in itself spectral properties, known as exponential dynamic localisation in expectation and the exponential decay of the eigenfunction correlator, implying the Area Law for the entanglement entropy.