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

Argha Debnath, Ayan Sahoo, Debraj Rakshit

TL;DR

This work demonstrates a disorder-free localization transition in a 1D lattice with power-law graded hopping controlled by α, showing that any infinitesimal grading induces localization in the thermodynamic limit. Through exact diagonalization and finite-size scaling of the localization length, IPR, energy gap, and fidelity susceptibility, it uncovers a distinct universality class with ν ≈ 0.49 and z ≈ 2.02. The fidelity susceptibility further connects to quantum Fisher information, suggesting metrological relevance, and a cost-function analysis confirms robust data collapse across observables. The nonequilibrium Kibble-Zurek analysis of driven dynamics across the transition shows consistent scaling governed by the static exponents, highlighting a coherent static-dynamic critical framework for this graded hopping localization. The results point to a tunable, disorder-free platform with implications for photonic lattices, ultracold atoms, and quantum sensing.

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.

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

TL;DR

This work demonstrates a disorder-free localization transition in a 1D lattice with power-law graded hopping controlled by α, showing that any infinitesimal grading induces localization in the thermodynamic limit. Through exact diagonalization and finite-size scaling of the localization length, IPR, energy gap, and fidelity susceptibility, it uncovers a distinct universality class with ν ≈ 0.49 and z ≈ 2.02. The fidelity susceptibility further connects to quantum Fisher information, suggesting metrological relevance, and a cost-function analysis confirms robust data collapse across observables. The nonequilibrium Kibble-Zurek analysis of driven dynamics across the transition shows consistent scaling governed by the static exponents, highlighting a coherent static-dynamic critical framework for this graded hopping localization. The results point to a tunable, disorder-free platform with implications for photonic lattices, ultracold atoms, and quantum sensing.

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, , which serves as the tuning parameter of the system. In the thermodynamic limit, the ground state becomes localized as , 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.

Paper Structure

This paper contains 10 sections, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Localization Transition And Quantum Criticality
  4. Localization Length
  5. Inverse Participation Ratio
  6. Energy Gap
  7. Fidelity Susceptibility
  8. Cost Function
  9. Kibble-Zurek SCALING OF DRIVEN DYNAMICS
  10. Discussion

Figures (4)

  • Figure 1: Variation of (a1) localization length($\xi$), (b1) IPR($\chi$), and (c1) energy gap($\Delta E$) with $\alpha$ for different system sizes, $L=500$ (yellow circle), 1000 (orange square), 1500 (green diamond), 2000 (pink up-triangle) for groundstate using exact diagonalization. Collapse plot with $(\nu,s,z)\sim(0.49(1),0.49(1),2.02(2))$ are shown for (a2) $\xi$, (b2) $\chi$, (c2) $\Delta E$ with their re-scaled axis. $\alpha_c\sim 10^{-9}$
  • Figure 2: (a) presents The FS, $\eta_Q$ versus $\alpha$ for different system sizes, $L=500$ (yellow circle), 1000 (orange square), 15000 (green diamond), 2000 (pink up-triangle) for groundstate using exact diagonalization. (b) shows a collapse plot for the $\eta_Q$ with $(\gamma, \nu)\sim(2.02(2), 0.49(1))$. In the inset we plot the QFI, $F_Q$ (brown dots), as a function of $L$ for the ground at $\alpha=10^{-8}$ and we fitted it with function $\propto L^\beta$ (blue dashed line). we found $\beta=3.98$.
  • Figure 3: Cost function to find $\nu$, $s$, $z$ and $\gamma$. A global minimum of $C_Q$ is at (a) $\nu=0.49(1)$, (b) $s=0.49(1)$, (c) $z=2.02(2)$, (d) $\gamma=1.96(4)$.
  • Figure 4: Driven dynamics with the initial state being the ground state. The curves of $\xi$ versus $\alpha$ for $R=0.001$ (blue circle), $R=0.0015$ (orange diamond), $R=0.002$ (green square), $R=0.0025$ (pink up-triangle), $R=0.003$ (yellow down-triangle) (a1) before and (a2) after rescaled with $R$. The curves of $\chi$ versus $\alpha$ (b1) before and (b2) after rescaled with $R$. We have plotted $E_D$ calculated from Eq. \ref{['eq:kz_an3']} versus $\alpha$ (c1) before and (c2) after rescaled with $R$. The arrows in (a1), (b1) and (c1) point the quench direction. The lattice size is $L=1000$.