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arXiv:1911.04189 · Physical review. B./Physical review. B

Entanglement conductance as a characterization of a delocalized-localized phase transition in free fermion models

Mohammad Pouranvari, Jahanfar Abouie

Published Nov 7, 2019• 9 citations
arXivHTMLPDF

Abstract

We study the entanglement Hamiltonian (EH) associated with the reduced density matrix of free fermion models in a delocalized-localized Anderson phase transition. We show numerically that the structure of the EH matrix differentiates the delocalized from the localized phase. In the delocalized phase, EH becomes a long-range Hamiltonian but is short-range in the localized phase, no matter what the configuration of the system's Hamiltonian is (whether it is long- or short-range). With this view, we introduce the entanglement conductance (EC), which quantifies how much EH is long-range, and we propose it as an alternative quantity to measure entanglement in the Anderson phase transition, by which we locate the phase transition point of some one-dimensional free fermion models. In addition, by applying the finite-size method to the EC, we find three-dimensional Anderson phase transition critical disorder strength.

DOI
10.1103/physrevb.100.195109

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We study the entanglement Hamiltonian (EH) associated with the reduced density matrix of free fermion models in a delocalized-localized Anderson phase transition. We show numerically that the structure of the EH matrix differentiates the delocalized from the localized phase. In the delocalized phase, EH becomes a long-range Hamiltonian but is short-range in the localized phase, no matter what the configuration of the system's Hamiltonian is (whether it is long- or short-range). With this view, we introduce the entanglement conductance (EC), which quantifies how much EH is long-range, and we propose it as an alternative quantity to measure entanglement in the Anderson phase transition, by which we locate the phase transition point of some one-dimensional free fermion models. In addition, by applying the finite-size method to the EC, we find three-dimensional Anderson phase transition critical disorder strength.
This study investigates the entanglement properties of disordered free fermion systems undergoing an Anderson phase transition from a delocalized to a localized phase. The entanglement entropy is employed to quantify the degree of entanglement, with the system randomly divided into two subsystems. To explore this phenomenon, one-dimensional tight-binding fermion models and Anderson models in one, two, and three dimensions are utilized. Comprehensive numerical calculations reveal that the entanglement entropy, determined using random bi-partitioning, follows a volume-law scaling in both the delocalized and localized phases, expressed as [Formula: see text], where D represents the dimension of the system. Furthermore, the role of short and long-range correlations in the entanglement entropy and the impact of the distribution of subsystem sites are analyzed.
We study the entanglement entropy(EE) of disordered one-dimensional spinless fermions with attractive interactions. With intensive numerical calculation of the EE using the density matrix renormalization group method, we find clear signatures of the transition between the localized and delocalized phase. In the delocalized phase, the fluctuations of the EE becomes minimum and independent of the system size. Meanwhile the EE's logarithmic scaling behavior is found to recover to that of a clean system. We present a general scheme of finite size scaling of the EE at the critical regime of the Anderson transition, from which we extract the critical parameters of the transition with good accuracy, including the critical exponent, critical point and a power-law divergent localization length.
The use of quantum entanglement to study condensed matter systems has been flourishing in critical systems and topological phases. Additionally, using real-space entanglement one can characterize localized and delocalized phases of disordered fermion systems. Here we instead propose the momentum-space entanglement spectrum as a means of characterizing disordered models. We show that localization in one dimension can be characterized by the momentum space entanglement between left and right movers and illustrate our methods using explicit models with spatially correlated disorder that exhibit phases which avoid complete Anderson localization. The momentum space entanglement spectrum clearly reveals the location of delocalized states in the energy spectrum, can be used as a signature of the phase transition between a delocalized and localized phase, and only requires a single numerical diagonalization to yield clear results.
The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying nonequilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its nonequilibrium steady-state density matrix. We show that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.
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.
Characterizing the delocalization transition in closed quantum systems with a many-body localized phase is a key open question in the field of nonequilibrium physics. We exploit the fact that localization of particles as realized in Anderson localization and standard many-body localization (MBL) implies Fock-space localization in single-particle basis sets characterized by a real-space index. Using a recently introduced quantitative measure for Fock-space localization computed from the density distributions, the occupation distance, we systematically study its scaling behavior across delocalization transitions and identify critical points from scaling collapses of numerical data. Excellent agreement with literature results is found for the critical disorder strengths of noninteracting fermions, such as the one-dimensional Aubry-Andr\'e model and the three-dimensional Anderson model. We observe a distinctively different scaling behavior in the case of interacting fermions with random disorder consistent with a Kosterlitz-Thouless transition. Finally, we use our measure to extract the transition point as a function of filling for interacting fermions.