Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

A. Elgart, L. Pastur, M. Shcherbina

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10 Scopus citations

Abstract

We consider a macroscopic disordered system of free d-dimensional lattice fermions whose one-body Hamiltonian is a Schrödinger operator H with ergodic potential. We assume that the Fermi energy lies in the exponentially localized part of the spectrum of H. We prove that if SΛ is the entanglement entropy of a lattice cube Λ of side length L of the system, then for any d≥ 1 the expectation E{L-(d-1)SΛ} has a finite limit as L→ ∞ and we identify the limit. Next, we prove that for d= 1 the entanglement entropy admits a well defined asymptotic form for all typical realizations (with probability 1) as L→ ∞. According to numerical results of Pastur and Slavin (Phys Rev Lett 113:150404, 2014) the limit is not selfaveraging even for an i.i.d. potential. On the other hand, we show that for d≥ 2 and an i.i.d. random potential the variance of L-(d-1)SΛ decays polynomially as L→ ∞, i.e., the entanglement entropy is selfaveraging.

Original languageEnglish
Pages (from-to)1092-1127
Number of pages36
JournalJournal of Statistical Physics
Volume166
Issue number3-4
DOIs
StatePublished - 1 Feb 2017
Externally publishedYes

Keywords

  • Anderson localization
  • Entanglement entropy
  • Free fermions

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