Journal of Statistical Physics

, Volume 166, Issue 3–4, pp 1092–1127 | Cite as

Large Block Properties of the Entanglement Entropy of Free Disordered Fermions

  • A. Elgart
  • L. Pastur
  • M. Shcherbina


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 \(S_\Lambda \) is the entanglement entropy of a lattice cube \(\Lambda \) of side length L of the system, then for any \(d \ge 1\) the expectation \(\mathbf { E}\{L^{-(d-1)}S_\Lambda \}\) has a finite limit as \(L \rightarrow \infty \) 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 \( L \rightarrow \infty \). 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 \(d \ge 2\) and an i.i.d. random potential the variance of \(L^{-(d-1)}S_\Lambda \) decays polynomially as \(L \rightarrow \infty \), i.e., the entanglement entropy is selfaveraging.


Entanglement entropy Free fermions Anderson localization 



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 [41], which allowed us to make the paper more transparent and the results (especially Result 4 and Theorem 3.7) 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.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsVirginia TechBlacksburgUSA
  2. 2.Mathematical DivisionB.Verkin Institute for Low Temperature Physics and EngineeringKharkivUkraine

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