TY - JOUR

T1 - Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

AU - Elgart, A.

AU - Pastur, L.

AU - Shcherbina, M.

N1 - Funding Information:
We wish to express our special thanks to J. Fillman for careful reading of a draft of this manuscript, numerous corrections and suggestions that markedly improved the final version. We are grateful to A. Sobolev for numerous discussions and for drawing our attention to his work [], which allowed us to make the paper more transparent and the results (especially Result and Theorem ) stronger. L.P. would like to thank the Isaac Newton Institute for Mathematical Sciences (Cambridge) for its hospitality during the program “Periodic and Ergodic Spectral Problems”, May 2015 supported by EPSRC Grant Number EP/K032208/1 and the Erwin Schrödinger Institute (Vienna) for its hospitality during the program “Quantum Many Body Systems, Random Matrices and Disorder”, July 2015. Financial support of grant 4/16-M of the National Academy of Sciences of Ukraine is also acknowledged. A.E. is supported in part by NSF under Grant DMS-1210982.
Publisher Copyright:
© 2016, Springer Science+Business Media New York.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - 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.

AB - 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.

KW - Anderson localization

KW - Entanglement entropy

KW - Free fermions

UR - http://www.scopus.com/inward/record.url?scp=84994453144&partnerID=8YFLogxK

U2 - 10.1007/s10955-016-1656-z

DO - 10.1007/s10955-016-1656-z

M3 - Article

AN - SCOPUS:84994453144

SN - 0022-4715

VL - 166

SP - 1092

EP - 1127

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3-4

ER -