Area Law Scaling for the Entropy of Disordered Quasifree Fermions

Abstract

We study theoretically and numerically the entanglement entropy of the d-dimensional free fermions whose one-body Hamiltonian is the Anderson model. Using the basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy ⟨S(Λ)⟩ of the d dimension cube Λ of side length l admits the area law scaling ⟨S(Λ)⟩ ∼ l((d-1)),l ≫ 1, even in the gapless case, thereby manifesting the area law in the mean for our model. For d = 1 and l ≫ 1 we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not self-averaging, i.e., has nonvanishing random fluctuations even if l ≫ 1.

How Much Delocalisation is Needed for an Enhanced Area Law of the Entanglement Entropy?

“…An area law also holds if H models a particle in a constant magnetic field [LSS20]. Area laws are proven to occur for random Schrödinger operators and Fermi energies in the region of dynamical localisation [PS14,EPS17,PS18a]. The proofs rely on the exponential decay in space of the Fermi projection for E in the region of complete localisation.…”

Full Szegő-Type Trace Asymptotics for Ergodic Operators on Large Boxes

“…Moreover, they exemplified that spectral localization can suppress the logarithmic enhancement (1.4). The latter point was generalized in [PS14,EPS17] to the random Anderson model on the lattice in arbitrary dimension and a larger classes of functions g, h. On the other hand, in [PS] it was proved that the logarithmic enhancemet (1.4) does occur for one-dimensional periodic continuum Schrödinger operators. Those findings are in line with the heuristics that for a logarithmically enhanced subleading term to pop up, a function g with a discontinuity within a conducting energy region of the Hamiltonian H is needed.…”

Anomalous Enhancement of Entanglement Entropy in Nonequilibrium Steady States Driven by Zero-Temperature Reservoirs

“…In this case, S L quantifies the entanglement between a subsystem of length L and the rest of the system. In many systems with natural Hamiltonians, the area law S L = O(1) [29,30] and the logarithmic law S L = O(ln L) [11][12][13][14][31][32][33][34] were found, in consistency with thermodynamics (i.e., S L = o(L) at zero temperature). Larger S L was found only in artificial toy models [35][36][37][38][39].…”