Large Block Properties of the Entanglement Entropy of Disordered Fermions

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 $S_{\Lambda }$ 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 $L\to \infty$ 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 $L\to \infty$. According to numerical results of [33] 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 $L\to \infty$, i.e., the entanglement entropy is selfaveraging.

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References in corpus (7)

• Quantum entanglement in condensed matter systems
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Entanglement in fermion systems and quantum metrology
• On the Area Law for Disordered Free Fermions
• Entanglement Typicality
• Entanglement Area Law in Disordered Free Fermion Anderson Model in One, Two, and Three Dimensions
• An Area Law for the Bipartite Entanglement of Disordered Oscillator Systems

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