Advances in Condensed Matter Physics

Volume 2015, Article ID 397630, 4 pages

http://dx.doi.org/10.1155/2015/397630

## Entanglement Area Law in Disordered Free Fermion Anderson Model in One, Two, and Three Dimensions

National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, FL 32306, USA

Received 12 October 2014; Revised 11 February 2015; Accepted 12 February 2015

Academic Editor: Jan A. Jung

Copyright © 2015 Mohammad Pouranvari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We calculate numerically the entanglement entropy of free fermion ground states in one-, two-, and three-dimensional Anderson models and find that it obeys the area law as long as the linear size of the subsystem is sufficiently larger than the mean free path. This result holds in the metallic phase of the three-dimensional Anderson model, where the mean free path is finite although the localization length is infinite. Relation between the present results and earlier ones on area law violation in special one-dimensional models that support metallic phases is discussed.

#### 1. Introduction

Recent years have witnessed tremendous progress in the study of entanglement in condensed matter/many-body physics. Among these studies, free fermion systems play a very special role [1]. Simple as they may seem, fermions are intrinsically nonlocal, due to the anticommutation relation fermion operators satisfy, no matter how far apart they are. Such nonlocality shows up as* enhanced* entanglement in the ground state; for example, for many years Fermi sea states were the only known ground states whose block entanglement entropy (EE) violates the area law satisfied by most ground states above 1D [2–4]. It is only recently shown that a similar violation occurs in interacting fermion systems in the Fermi liquid phase [5] and bosonic models with excitation spectra that vanish on (extended) Bose surfaces [6]. The existence of sharp Fermi or Bose surfaces is crucial for the area law violation in translationally invariant systems.

Comparatively much less effort has been devoted to studies of fermions in the presence of disorder potential. In a recent work [7] we studied two very special (one-dimensional) 1D models that exhibit free fermion metal-insulator transition (MIT) and found area law violation in the metallic phase,* despite* the presence of disorder, and thus absence of sharp Fermi surface (actually points in 1D). It was conjectured [7] that as long as the system is metallic, namely, states are delocalized at the Fermi energy, there will be area law violation. In the present work we test this conjecture by performing detailed numerical studies of the Anderson model [8] in one, two, and three dimensions. We find that the area law is actually respected in all cases, including the metallic phase in 3D. We do observe an enhancement (beyond area law) as systems sizes increase while below the mean free path; such enhancement disappears once the system size becomes sufficiently bigger than the mean free path. The origin of the difference between the Anderson model studied here and the special models studied earlier [7] will be discussed.

The remainder of the paper is organized as follows. In Section 2 we introduce our model and numerical method for calculating EE. Results of our calculations are presented in Section 3. Section 4 offers a summary and discussions on our results.

#### 2. Model and Basic Considerations

Anderson model in dimension is a model with constant nearest neighbor hopping term and random on-site energy : where summation is over all sites in dimensional hyper cubic lattice (with lattice constant set to be 1) and is a vector connecting a site to its nearest neighbor. ’s are uniformly distributed between and [9]. The Fermi energy is set to be 0 (so the lattice is half-filled) in all cases, while in 2D we also study to avoid the van Hove singularity at the band center. We consider cubic-shaped finite-size systems with linear size and open-boundary conditions. We then divide them into two equal subsystems and with size and calculate the disorder-averaged entanglement entropy as detailed below.

For a system in a pure state , the density matrix is . Reduced density matrix of each subsystem ( or ) is obtained by tracing over degrees of freedom of the other subsystem: . Block EE between the two subsystems is . For a single Slater-determinant ground state,
are characterized by free fermion* entanglement* Hamiltonians
where is determined by the normalization condition . We calculate EE using the method of [10] by diagonalizing correlation matrix of subsystem
and find its eigenvalues ’s. Then EE takes the form
where is number of sites in subsystem .

In one and two dimensions, all states are localized with any finite disorder. However there is an important difference between them: in 1D the localization length is of the same order as mean free path , while in 2D we have for weak disorder. In 3D there is a metal-insulator transition (MIT) at a critical value of disorder strength [11], where diverges. The focus of our numerical calculation is the interplay of the three different length scales, mean free path (calculated perturbatively in the Appendix), localization length , and (sub)system size , and their effects on entanglement.

#### 3. Results

##### 3.1. Anderson Model in One and Two Dimensions

In these two cases, all states are localized as long as .

Figure 1 shows 1D EE as a function of system size for different values of . As size increases, EE grows logarithmically for as expected. For , EE grows with in a manner similar to the disorder free case up to some point and then saturates, indicating area law is obeyed for sufficiently large system sizes. We find substantial deviation (from case) starts when the system size reaches the mean free path , and saturation occurs around . We note in 1D that we have the localization length ; it is thus not immediately clear at this point which of the two controls is the crossover.