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arXiv:1607.04173 · Physical review. E

Genuine localization transition in a long-range hopping model

Xiangyu Cao, Alberto Rosso, Jean‐Philippe Bouchaud, Pierre Le Doussal

Published Jun 14, 2017• 17 citations
arXivHTMLPDF

Abstract

We introduce and study a banded random matrix model describing sparse, long-range quantum hopping in one dimension. Using a series of analytic arguments, numerical simulations, and a mapping to a long-range epidemics model, we establish the phase diagram of the model. A genuine localization transition, with well defined mobility edges, appears as the hopping rate decreases slower than ${\ensuremath{\ell}}^{\ensuremath{-}2}$, where $\ensuremath{\ell}$ is the distance. Correspondingly, the decay of the localized states evolves from a standard exponential shape to a stretched exponential and finally to a $exp(\ensuremath{-}C{ln}^{\ensuremath{\kappa}}\ensuremath{\ell})$ behavior, with $\ensuremath{\kappa}>1$.

DOI
10.1103/physreve.95.062118

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We introduce and study a banded random matrix model describing sparse, long-range quantum hopping in one dimension. Using a series of analytic arguments, numerical simulations, and a mapping to a long-range epidemics model, we establish the phase diagram of the model. A genuine localization transition, with well defined mobility edges, appears as the hopping rate decreases slower than \ensuremathℓ\ensuremath−2{\ensuremath{\ell}}^{\ensuremath{-}2}\ensuremathℓ\ensuremath−2, where \ensuremathℓ\ensuremath{\ell}\ensuremathℓ is the distance. Correspondingly, the decay of the localized states evolves from a standard exponential shape to a stretched exponential and finally to a exp(\ensuremath−Cln\ensuremathκ\ensuremathℓ)exp(\ensuremath{-}C{ln}^{\ensuremath{\kappa}}\ensuremath{\ell})exp(\ensuremath−Cln\ensuremathκ\ensuremathℓ) behavior, with \ensuremathκ>1\ensuremath{\kappa}>1\ensuremathκ>1.
We introduce and study a new class of Banded Random Matrix model describing sparse, long range quantum hopping in one dimension. Using a series of analytic arguments, numerical simulations, and mappings to statistical physics models, we establish the phase diagram of the model. A genuine localisation transition, with well defined mobility edges, appears as the hopping rate decreases slower than ℓ−2\ell^{-2}ℓ−2, where ℓ\ellℓ is the distance. Correspondingly, the decay of the localised states evolves from a standard exponential shape to a stretched exponential and finally to a novel exp⁡(−Cln⁡κℓ)\exp(-C\ln^\kappa \ell)exp(−Clnκℓ) behaviour, with κ>1\kappa > 1κ>1.
We study Anderson localisation on high-dimensional graphs with spatial structure induced by long-ranged but distance-dependent hopping. To this end, we introduce a class of models that interpolate between the short-range Anderson model on a random regular graph and fully connected models with statistically uniform hopping, by embedding a random regular graph into a complete graph and allowing hopping amplitudes to decay exponentially with graph distance. The competition between the exponentially growing number of neighbours with graph distance and the exponentially decaying hopping amplitude positions our models effectively as power-law hopping generalisation of the Anderson model on random regular graphs. Using a combination of numerical exact diagonalisation and analytical renormalised perturbation theory, we establish the resulting localisation phase diagram emerging from the interplay of the lengthscale associated to the hopping range and the onsite disorder strength. We find that increasing the hopping range shifts the localisation transition to stronger disorder, and that beyond a critical range the localised phase ceases to exist even at arbitrarily strong disorder. Our results indicate a direct Anderson transition between delocalised and localised phases, with no evidence for an intervening multifractal phase, for both deterministic and random hopping models. A scaling analysis based on inverse participation ratios reveals behaviour consistent with a Kosterlitz-Thouless-like transition with two-parameter scaling, in line with Anderson transitions on high-dimensional graphs. We also observe distinct critical behaviour in average and typical correlation functions, reflecting the different scaling properties of generalised inverse participation ratios.
Long-range hoppings in quantum disordered systems are known to yield quantum multifractality, the features of which can go beyond the characteristic properties associated with an Anderson transition. Indeed, critical dynamics of long-range quantum systems can exhibit anomalous dynamical behaviors distinct from those at the Anderson transition in finite dimensions. In this paper, we propose a phenomenological model of wave packet expansion in long-range hopping systems. We consider both their multifractal properties and the algebraic fat tails induced by the long-range hoppings. Using this model, we analytically derive the dynamics of moments and inverse participation ratios of the time-evolving wave packets, in connection with the multifractal dimension of the system. To validate our predictions, we perform numerical simulations of a Floquet model that is analogous to the power law random banded matrix ensemble. Unlike the Anderson transition in finite dimensions, the dynamics of such systems cannot be adequately described by a single parameter scaling law that solely depends on time. Instead, it becomes crucial to establish scaling laws involving both the finite size and the time. Explicit scaling laws for the observables under consideration are presented. Our findings are of considerable interest towards applications in the fields of many-body localization and Anderson localization on random graphs, where long-range effects arise due to the inherent topology of the Hilbert space.
We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes t(r)∝r−at(r)\propto r^{-a}t(r)∝r−a. In contrast to the standard one-dimensional Anderson model (a→∞a\to\inftya→∞), in which all states are localized and the localization length is minimal at the band edge, the long-range model with $1
We investigate a one-dimensional tight-binding model in which onsite potentials [Formula: see text] exhibit power-law spatial correlations (with exponent α) and the hopping amplitudes decay as [Formula: see text]. This two-parameter family interpolates continuously between short-range Anderson-like disorder, correlated disorder with conventional hopping, and long-range hopping models with nontrivial delocalization tendencies. Using large-scale exact diagonalization, we construct a comprehensive phase map in the [Formula: see text] plane by combining spectral statistics, density-of-states analysis, and energy-resolved localization indicators such as the participation ratio, single-particle entanglement entropy, level-spacing ratio r, and the ratio of the geometric to arithmetic density of states. From these observables we define phase-indicator functions that compactly quantify localization behavior across the spectrum. Our analysis reveals robust mobility edges and multiple regimes of spectral coexistence between localized, extended, resonant, and critical states. Finite-size scaling, implemented via an explicit smoothness-based cost function, enables extraction of critical exponents and delineation of transition lines across the [Formula: see text] parameter space. To validate and complement these physics-based diagnostics, we employ a supervised autoencoder that learns high-level representations of eigenstate structure directly from raw features and reliably reproduces the phase classification defined by the indicator functions. Together, these approaches provide a coherent and internally consistent picture of the spectral transitions driven by correlated disorder and long-range hopping, establishing a unified framework for characterizing mobility edges in long-range one-dimensional systems.
Exponentially decaying long-range hoppings are ubiquitous in realistic tight-binding models and are often truncated to obtain a finite-range description. We show that this approximation can fail dramatically in non-Hermitian systems under open boundary conditions: an infinitesimal long-range hopping can nonperturbatively reconstruct the spectrum and eigenstates of a short-range non-Hermitian system. The mechanism is controlled by a competition between the decay length of infinitesimal long-range hoppings and the localization length of non-Hermitian skin modes, leading to a sharp transition as the decay rate is tuned. In one dimension, we show that a squeezed generalized Brillouin zone (GBZ) replaces the original GBZ of the short-ranged Hamiltonian, yielding the reconstructed open-boundary spectrum. In two or higher dimensions, we formulate a squeezed amoeba formulation describing the reconstructed spectral density. We further show that long-range hoppings can qualitatively reshape Green's function, which can be readily detected in experiments.