Area Law for the entanglement entropy of free fermions in nonrandom ergodic field

Abstract

This paper deals with the asymptotic behaviour of a widely used correlation characteristic in large quantum systems. The correlations are known as quantum entanglement, the characteristic is called entanglement entropy, and the system is an ideal gas of spinless lattice fermions. The system is determined by its one-body Hamiltonian. It is shown in EPS [18] that if the Hamiltonian is an ergodic finite difference operator with an exponentially decaying spectral projection, then the asymptotic form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified is that consisting of discrete Schr\"odinger operators with random potential. In this paper, we prove the validity of the area law for several classes of Schr\"odinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move on to the interesting and highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying non-random chaotic phenomena. The corresponding asymptotic study requires quite an involved spectral analysis. Consequently, the majority of the paper is devoted to the proof and application of a variety of spectral properties of the operators in question, in particular we prove uniform localisation of the ejgenfunctions for the Maryland model and the exponential decay of the eihgenfunction correlator for a variety of models . We believe that these properties are of considerable independent interest.