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arXiv:2411.16851 · Physical review. B./Physical review. B

Reentrant localization induced by short-range hopping in the fractal Rosenzweig-Porter model

Roopayan Ghosh, Madhumita Sarkar, Ivan M. Khaymovich

Published Jun 10, 2025• 5 citations
arXivHTMLPDF

Abstract

Typically, metallic systems localized under strong disorder exhibit a transition to delocalization as the kinetic terms increase. In this Letter, we reveal the opposite effect—increasing kinetic terms leads to an unexpected reduction of mobility, enhancing localization of the system, and even leading to reentrant delocalization transitions. Specifically, we add a nearest-neighbor hopping with amplitude κ to the Rosenzweig-Porter (RP) model with fractal on-site disorder and surprisingly see that, as κ grows, the system initially tends to localization from the fractal phase, but then reenters the ergodic phase. We build an analytical framework to explain this re-entrant behavior, supported by exact diagonalization results. The interplay between the spatially local κ term, insensitive to fractal disorder, and the energy-local RP coupling, sensitive to fine-level spacing structures, drives the observed reentrant behavior. This mechanism offers a distinct pathway to reentrant localization phenomena in many-body quantum systems.

DOI
10.1103/k957-fcmh

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Typically, metallic systems localized under strong disorder exhibit a transition to delocalization as the kinetic terms increase. In this Letter, we reveal the opposite effect—increasing kinetic terms leads to an unexpected reduction of mobility, enhancing localization of the system, and even leading to reentrant delocalization transitions. Specifically, we add a nearest-neighbor hopping with amplitude κ to the Rosenzweig-Porter (RP) model with fractal on-site disorder and surprisingly see that, as κ grows, the system initially tends to localization from the fractal phase, but then reenters the ergodic phase. We build an analytical framework to explain this re-entrant behavior, supported by exact diagonalization results. The interplay between the spatially local κ term, insensitive to fractal disorder, and the energy-local RP coupling, sensitive to fine-level spacing structures, drives the observed reentrant behavior. This mechanism offers a distinct pathway to reentrant localization phenomena in many-body quantum systems.
The Rosenzweig-Porter (RP) model has garnered much attention in the last decade, as it is a simple analytically tractable model showing both ergodic-nonergodic extended and Anderson localization transitions. Thus, it is a good toy model to understand the Hilbert-space structure of many-body localization phenomenon. In our Letter, we present analytical evidence, supported by exact numerics, that demonstrates the controllable tuning of the phase diagram in the RP model by employing on-site potentials with a nontrivial fractal dimension instead of the conventional random disorder. We demonstrate that such disorder extends the fractal phase and creates an unusual dependence of fractal dimensions of the eigenfunctions. Furthermore, we study the fate of level statistics in such a system to understand how these changes are reflected in the eigenvalue statistics.
Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions D q > 0 . In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions D 2 and D ∞ , relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of nonergodic phases in the entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
A reentrant localization transition was predicted recently in a one-dimensional quasiperiodic lattice with dimerized hopping between the nearest-neighbor sites [Roy et al., Phys. Rev. Lett. 126, 106803 (2021)]. It was shown that the interplay between the hopping dimerization and a staggered quasiperiodic disorder manifests two localization transitions through two intermediate phases, resulting in four critical points as a function of the quasiperiodic potential. In this paper, we study the critical nature of the states across the localization transitions by computing the mass exponents and the corresponding fractal dimensions of the states through a multifractal analysis. Moreover, we analyze the phenomenon of this reentrant localization transition by examining the spectral properties of the eigenstates. By performing a systematic finite-size scaling analysis for a fixed value of the hopping dimerization, we obtain accurate critical disorder strengths for different transitions and the associated critical exponents. Further, we complement the critical nature of the energy spectrum by computing the Hausdorff dimensions.
The combined effects of disorder and long-range hopping play an important role in determining localization and transport in low-dimensional quantum systems. We study transport in a one-dimensional disordered chain with nearest-neighbor hopping supplemented by a long-range hopping term. The long-range coupling is characterized by site-dependent amplitudes that may be either uniform or random. We investigate the interplay between diagonal disorder and long-range hopping, characterizing the resulting transport efficiency. We find that nearest-neighbor hopping in the presence of uniform long-range hopping amplitudes strongly suppresses delocalization over a range of weak disorder. The disorder-enhanced transport regime, established in previous studies, can be understood here as the recovery of delocalization from this strongly suppressed regime. An analytical solution in the Anderson eigenstate basis reveals the microscopic origin of this behavior. In particular, we show that the low-disorder regime of strongly suppressed delocalization can be eliminated by introducing randomness into the long-range hopping amplitudes, providing an additional mechanism for enhancing transport and qualitatively changing the response of the system to disorder.
The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy ETE_TET​, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around ETE_TET​, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as 10−4010^{-40}10−40. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.
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.