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arXiv:2512.15795 · Physica Scripta

Localization from Infinitesimal Kinetic Grading: Critical Scaling and Kibble-Zurek Universality

Argha Debnath, Ayan Sahoo, Debraj Rakshit

cond-mat.quant-gasquant-phSubmitted Dec 16, 2025• Announced Dec 17, 2025• 0 citations
v1 · Dec 16, 2025v2 · Apr 17, 2026v3 · Aug 23, 2026
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Abstract

We study a one-dimensional lattice model with site-dependent nearest-neighbor hopping amplitudes that follow a power-law profile. The hopping variation is controlled by a grading exponent, $α$, which serves as the tuning parameter of the system. In the thermodynamic limit, the ground state becomes localized as $|α| \to 0$, signaling the presence of a critical point characterized by a diverging localization length. Using exact diagonalization, we perform finite-size scaling analysis and extract the associated critical exponent governing this divergence, revealing a universality class distinct from well-known Anderson, Aubry-Andre, and Stark localization. To further characterize the critical behavior, we analyze the inverse participation ratio, the energy gap between the ground and first excited states, and the fidelity susceptibility. We also investigate nonequilibrium dynamics by linearly ramping the hopping profile at various rates and tracking the evolution of the localization length and the inverse participation ratio. The Kibble-Zurek mechanism successfully captures the resulting dynamics using the critical exponents obtained from the static scaling analysis. Our results demonstrate a clean, disorder-free route to localization and provide a tunable platform relevant to photonic lattices and ultracold atom arrays with engineered hopping profiles.

Journal reference
Phys. Scr. 101, 305107 (2026)
DOI
10.1088/1402-4896/ae8ed2
Comments
8 pages, 4 figures

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We study a one-dimensional lattice model with site-dependent nearest-neighbor hopping amplitudes that follow a power-law profile. The hopping variation is controlled by a grading exponent, ααα, which serves as the tuning parameter of the system. In the thermodynamic limit, the ground state becomes localized as ∣α∣→0|α| \to 0∣α∣→0, signaling the presence of a critical point characterized by a diverging localization length. Using exact diagonalization, we perform finite-size scaling analysis and extract the associated critical exponent governing this divergence, revealing a universality class distinct from well-known Anderson, Aubry-Andre, and Stark localization. To further characterize the critical behavior, we analyze the inverse participation ratio, the energy gap between the ground and first excited states, and the fidelity susceptibility. We also investigate nonequilibrium dynamics by linearly ramping the hopping profile at various rates and tracking the evolution of the localization length and the inverse participation ratio. The Kibble-Zurek mechanism successfully captures the resulting dynamics using the critical exponents obtained from the static scaling analysis. Our results demonstrate a clean, disorder-free route to localization and provide a tunable platform relevant to photonic lattices and ultracold atom arrays with engineered hopping profiles.
We explore quantum criticality and Kibble-Zurek scaling (KZS) in the Aubry-Andr\'e-Stark (AAS) model, where the Stark field of strength $\ensuremath{\varepsilon}$ is added onto the one-dimensional quasiperiodic lattice. We perform scaling analysis and numerical calculations of the localization length, the inverse participation ratio (IPR), and the energy gap between the ground and first excited states to characterize critical properties of the delocalization-localization transition. Remarkably, our scaling analysis shows that, near the critical point, the localization length $\ensuremath{\xi}$ scales with $\ensuremath{\varepsilon}$ as $\ensuremath{\xi}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.3$, a new critical exponent for the AAS model, which is different from the counterparts for both the pure Aubry-Andr\'e (AA) model and the pure Stark model. The IPR I\mathcal{I}I scales as $\mathcal{I}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{s}$ with the critical exponent $s\ensuremath{\approx}0.098$, which is also different from both of the pure models. The energy gap $\mathrm{\ensuremath{\Delta}}E$ scales as $\mathrm{\ensuremath{\Delta}}E\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{\nu}z}$ with the same critical exponent $z\ensuremath{\approx}2.374$ as that for the pure AA model. We further reveal hybrid scaling functions in the overlap between the critical regions of the Anderson and Stark localizations. Moreover, we investigate the driven dynamics of the localization transitions in the AAS model. By linearly changing the Stark (quasiperiodic) potential, we calculate the evolution of the localization length and the IPR, and we study their dependence on the driving rate. We find that the driven dynamics from the ground state is well described by the KZS with the critical exponents obtained from the static scaling analysis. When both the Stark and the quasiperiodic potentials are relevant, the KZS form includes the two scaling variables. This work extends our understanding of critical phenomena on localization transitions and generalizes the application of the KZS to hybrid models.
The Aubry-Andr\'e one-dimensional lattice model describes a particle hopping in a pseudorandom potential. Depending on its strength $\ensuremath{\lambda}$, all eigenstates are either localized $(\ensuremath{\lambda}>1)$ or delocalized $(\ensuremath{\lambda}<1)$. Near the transition, the localization length diverges like $\ensuremath{\xi}\ensuremath{\sim}{(\ensuremath{\lambda}\ensuremath{-}1)}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=1$. We show that when the particle is initially prepared in a localized ground state and the potential strength is slowly ramped down across the transition, then---in analogy with the Kibble-Zurek mechanism---it enters the delocalized phase having finite localization length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+z\ensuremath{\nu})}$. Here ${\ensuremath{\tau}}_{Q}$ is the ramp/quench time and zzz is a dynamical exponent. At $\ensuremath{\lambda}=1$ we determine $z\ensuremath{\simeq}2.37$ from the power-law scaling of an energy gap with a lattice size LLL. Even though for infinite LLL the model is gapless, we show that the gap that is relevant for excitation during the ramp remains finite. Close to the critical point it scales like ${\ensuremath{\xi}}^{\ensuremath{-}z}$ with the value of zzz determined by the finite-size scaling. It is the gap between the ground state and the lowest of those excited states that overlaps with the ground state enough to be accessible for excitation. We propose an experiment with a noninteracting BEC to test our prediction. Our hypothesis is further supported by considering a generalized version of the Aubry-Andr\'e model possessing an energy-dependent mobility edge.
In this work, we explore the driven dynamics of the one-dimensional localization transitions. By linearly changing the strength of disorder potential, we calculate the evolution of the localization length $\ensuremath{\xi}$ and the inverse participation ratio I\mathcal{I}I in a disordered Aubry-Andr\'e (AA) model, and investigate the dependence of these quantities on the driving rate for a single typical realization of the disorder configuration. At first, we focus on the limit in the absence of the quasiperiodic potential. We find that the driven dynamics from both ground state and excited state can be described by the Kibble-Zurek scaling (KZS). Then, the driven dynamics near the critical point of the AA model is studied. Here, since both the disorder and the quasiperiodic potential are relevant directions, the KZS should include both scaling variables. Our present work not only extends our understanding of the localization transitions but also generalizes the application of the KZS.
In this work, we investigate the Stark localization near the Aubry-Andr\'e (AA) critical point. We perform careful studies for reporting system-dependent parameters, such as localization length, inverse participation ratio (IPR), and energy gap between the ground and first excited state, for characterizing the localization-delocalization transition. We show that the scaling exponents possessed by these key descriptors of localization are quite different from that of a pure AA model or Stark model. Near the critical point of the AA model, in the presence of Stark field of strength hhh, the localization length $\ensuremath{\zeta}$ scales as $\ensuremath{\zeta}\ensuremath{\propto}{h}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.29$, which is different from both the pure AA model $(\ensuremath{\nu}=1)$ and Stark model $(\ensuremath{\nu}\ensuremath{\approx}0.33)$. The IPR in this case scales as IPR $\ensuremath{\propto}{h}^{s}$ with $s\ensuremath{\approx}0.096$, which is again significantly different from both the pure AA model $(s\ensuremath{\approx}0.33)$ and Stark model $(s\ensuremath{\approx}0.33)$. The energy gap, $\mathrm{\ensuremath{\Delta}}$, scales as $E\ensuremath{\propto}{h}^{\ensuremath{\nu}z}$, where $z\ensuremath{\approx}2.37$, which is however the same as the pure AA model. Finally, we discuss how invoking a criticality-inducing additional control parameter may help in designing better many-body quantum sensors. Quantum critical sensors exploit the venerability of the wave function near the quantum critical point against small parameter shifts. By incorporating a control parameter in the form of the quasiperiodic field, i.e., the AA potential, we show a significant advantage can be drawn in estimating an unknown parameter, which is considered here to be the Stark weak field strength, with high precision. We compute and extract the scaling exponent associated with the quantum Fisher information, FQ{F}_{Q}FQ​, which provides a bound on the uncertainty associated with measuring an unknown parameter. We demonstrate that by a control field in the form of the quasiperiodic potential, and setting it at AA criticality, FQ{F}_{Q}FQ​ scales with the system size as ${F}_{Q}\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, where $\ensuremath{\beta}\ensuremath{\approx}6.7$, which is greater in comparison to the pure Stark case, for which $\ensuremath{\beta}$ is known to be 5.8. We, thus, pitch a generic idea of invoking quantum-criticality-inducing additional control parameters for obtaining enhanced quantum advantage in many-body critical sensors.
The critical exponents and universality classes of localization transitions in quasiperiodic systems are of fundamental importance for understanding critical phenomena in aperiodic systems. Here we show that the dynamical critical behavior can be tuned without adding new terms or changing the form of the Hamiltonian, but solely by varying the incommensurate frequency of the quasiperiodic onsite potential. We construct a family of incommensurate frequencies from the limiting ratios of generalized Fibonacci sequences controlled by the parameters (m,n)(m,n)(m,n), and use them to define the quasiperiodic onsite potential. By combining generalized fidelity susceptibility, localization-length scaling, and finite-size gap analysis, we find that the correlation-length exponent is insensitive to the choice of the incommensurate frequency and remains consistent with the correlation-length critical exponent, ν≃1ν\simeq 1ν≃1, in the localization transition of the standard Aubry--Andr'e--Harper model. In contrast, the dynamical exponent extracted from the low-energy gap scaling varies systematically with the incommensurate frequency. Our results show that changing the incommensurate frequency provides a simple way to tune dynamical critical scaling in deterministic aperiodic systems. Our results suggest instead that the arithmetic structure of an irrational number can serve as a control parameter for nonequilibrium quantum dynamics, enabling the tuning of dynamical critical behavior without changing the microscopic Hamiltonian or the physical spatial dimension.
Recently, the driven dynamics of localization phase transition have garnered growing interest. However, studies so far have mainly considered initial localized states, whose driven dynamics follow the Kibble-Zurek mechanism (KZM). In this study, we investigate the driven dynamics of the localization phase transition in the Aubry-André model starting from a gapless extended state, which violates the adiabatic-impulse scenario of KZM. By linearly driving the quasiperiodic potential strength across the critical point, we numerically simulate the driven dynamics and analyze the scaling behavior of both the inverse participation ratio ( I ) and the dynamic deviation from the instantaneous ground-state energy ( D ) . We demonstrate that the driven dynamics starting from initially extended states satisfies the criterion for the applicability of KZM and its extension, finite-time scaling (FTS). The scaling functions governing the driven dynamics of both I and D have been derived based on FTS and numerically validated. We found that the scaling functions exhibit significant differences at large R and small R , and also differ considerably from the scaling functions when the initial state is localized, highlighting the crucial role of initial-state behavior. The established scaling laws remain robust across a wide range of system sizes and driving rates, providing testable predictions for experimental realizations.