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arXiv:2208.11409 · Modern Physics Letters A

Exploring entanglement characteristics in disordered free fermion systems through random bi-partitioning

Mohammad Pouranvari

Published Feb 20, 2024• 1 citation
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Abstract

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.

DOI
10.1142/s0217732324500032

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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 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.
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.
We study theoretically and numerically the entanglement entropy of the ddd-dimensional free fermions whose one-body Hamiltonian is the Anderson model. Using the basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩$ of the ddd dimension cube $\mathrm{\ensuremath{\Lambda}}$ of side length lll admits the area law scaling $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩\ensuremath{\sim}{l}^{(d\ensuremath{-}1)},l\ensuremath{\gg}1$, even in the gapless case, thereby manifesting the area law in the mean for our model. For d=1d=1d=1 and $l\ensuremath{\gg}1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not self-averaging, i.e., has nonvanishing random fluctuations even if $l\ensuremath{\gg}1$.
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.
We consider Anderson localization and the associated metal-insulator transition for non-interacting fermions in D = 1, 2 space dimensions in the presence of spatially correlated on-site random potentials. To assess the nature of the wavefunction, we follow a recent proposal to study momentum-space entanglement. For a D = 1 model with long-range disorder correlations, both the entanglement spectrum and the entanglement entropy allow us to clearly distinguish between extended and localized states based upon a single realization of disorder. However, for other models including the D = 2 case with long-range correlated disorder, we find that the method is not similarly successful. We analyze the reasons for its failure, concluding that the much desired generalization to higher dimensions may be problematic.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.