Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

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 S Λ is the entanglement...

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Bibliographic Details
Published in:Journal of statistical physics 2017-02, Vol.166 (3-4), p.1092-1127
Main Authors: Elgart, A., Pastur, L., Shcherbina, M.
Format: Article
Language:English
Subjects:
Entanglement
Entropy
Fermi surfaces
Fermions
Mathematical analysis
Mathematical and Computational Physics
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Theoretical
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Summary: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 S Λ 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 ) S Λ } 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 ) S Λ decays polynomially as L → ∞ , i.e., the entanglement entropy is selfaveraging.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-016-1656-z