Quasiconservation Laws and Suppressed Transport in Weakly Interacting Localized Models

TL;DR

This work analyzes the robustness of one-dimensional localization against weak interactions by perturbatively dressing noninteracting LIOMs via the adiabatic gauge potential (AGP) at first order and introducing a transport-based diagnostic, the charge transport capacity. It shows that an $O(1)$ fraction of the original LIOMs remain approximately conserved to $O(\lambda^2)$, giving rise to extensive quasi-conserved quantities and long relaxation times, while a subset experiences local resonances that hinder but do not immediately destroy localization. The authors demonstrate that the first-order correction to charge transport remains bounded in both the Anderson and Aubry–André models, indicating perturbative localization persists, with bottlenecks preventing global transport. They also find that tails in the perturbative corrections arise from four-energy resonances and may signal a need for higher-order or nonperturbative treatments to fully resolve avalanche-type instabilities. Overall, the results support perturbative localization robustness at weak interactions in strongly disordered regimes, while leaving open the fate of higher-order corrections and potential nonperturbative effects.

Abstract

The stability of localization in the presence of interactions remains an open problem, with finite-size effects posing significant challenges to numerical studies. In this work, we investigate the perturbative stability of noninteracting localization under weak interactions, which allows us to analyze much larger system sizes. Focusing on disordered Anderson and quasiperiodic Aubry-André models in one dimension, and using the adiabatic gauge potential (AGP) at first order in perturbation theory, we compute first-order corrections to noninteracting local integrals of motion (LIOMs). We find that for at least an $O(1)$ fraction of the LIOMs, the corrections are well-controlled and converge at large system sizes, while others suffer from resonances. Additionally, we introduce and study the charge-transport capacity of this weakly interacting model. To first order, we find that the charge transport capacity remains bounded in the presence of interactions. Taken together, these results demonstrate that localization is perturbatively stable to weak interactions at first order, implying that, at the very least, localization persists for parametrically long times in the inverse interaction strength. We expect this perturbative stability to extend to all orders at sufficiently strong disorder, where the localization length is short, representing the true localized phase. Conversely, our findings suggest that the previously proposed interaction-induced avalanche instability, namely in the weakly localized regime of the Anderson and Aubry-André models, is a more subtle phenomenon arising only at higher orders in perturbation theory or through nonperturbative effects.

Figures (27)

• Figure 1: Analysis of the AGP Frobenius norm in the interacting Anderson Model. All three figures were generated based on 500 disorder realizations and open boundary conditions. (a) Calculated probability distributions of $\ln\|\hat{X}\|_F$ with disorder strength $W=2.1$. Each colored curve represents varying system sizes $L$, from $L=20$ to $L=100$ in steps of $10$. (b) Same data as in (a) but with the vertical scale shown on a log scale. The red line represents a polynomial $\|\hat{X}\|_F^{-2}$ fit to the tail of the probability distribution of $\|\hat{X}\|_F$---see text around Eq. (\ref{['eq:tail_behavior']}) for details on the form plotted. (c) Median of the AGP norm as a function of system size $L$. Each colored curve represents a different disorder strength $W$, from $W=0.1$ to $W=3.1$ in step sizes of 0.5. Here, dashed lines represent power-law fits to the data, $\|\hat{X}\|_F = A L^D + C$ with power-law exponents $D$. In order of increasing disorder strength, we find $D=2.33, 2.33, 1.99, 1.71, 1.45, 1.38, 1.34$, consistent with expected eventual linear in $L$ scaling for a strictly localized system (see text in Sec. \ref{['sec:agp_resulsts_disordered']} for discussion). The error bars on the median curves, estimated using a bootstrapping method, are very small and are therefore omitted in this and subsequent plots for clarity.
• Figure 2: Analysis of the AGP Frobenius norm in the interacting quasiperiodic model. All three figures were generated based on 500 uniformly sampled phase shifts $\delta$ and open boundary conditions. We refer the reader to Sec. \ref{['sec:methods']} for details on sampling for the quasiperiodic model. (a) Calculated probability distributions of $\ln\|\hat{X}\|_F$ with potential depth $h=3.1$. Each colored distribution represents varying system sizes $L$, from $L=20$ to $L=100$, in steps of 10. (b) Same data as in (a) but with the vertical scale shown on a log scale. The red curve represents a polynomial $\|\hat{X}\|_F^{-1.75}$ fit to the tail of the probability distribution of $\|\hat{X}\|_F$---see text around Eq. (\ref{['eq:tail_behavior']}) for details on the form plotted. (c) Median of $\|\hat{X}\|_F$ as a function of system size $L$. Each colored curve represents a different potential depth $h$, from $h=0.1$ to $h=3.1$ in steps of 0.5. Here, we observe polynomial growth with $L$ of $\mathrm{med}\|\hat{X}\|_F$ and an anomalous peak at $L=40$ in the regime $h<2$. For a full discussion of these results, see Sec. \ref{['sec:agp_results_quasiperiodic']}.
• Figure 3: Analysis of the Frobenius norm of the first-order corrections to the noninteracting LIOMs, $\|\tilde{n}_\alpha^{(1)}\|_F$, in the weakly interacting Anderson model. Each plot displays all orbitals $\alpha = 1, \dots, L$ across the 500 disorder realizations used to generate the data shown in this figure. (a)-(d): Distributions of $\ln(\|\tilde{n}_\alpha^{(1)}\|_F)$ at $W=0.6, 1.6, 2.6, 3.1$ and for $\alpha = 1,\dots, L$. Within each panel, each colored curve represents a different system size. We show calculations between $L=20$ and $L=100$ in steps of $10$. The inset in (d) shows the same data as in the main (d) panel, but with the vertical scale shown on log-scale. The red line in the inset represents a polynomial $\|\tilde{n}_\alpha^{(1)}\|_F^{-2}$ fit to the tail of the probability distribution of $\|\tilde{n}_\alpha^{(1)}\|_F$--- see the text surrounding Eq. (\ref{['eq:tail_behavior']}) for details on the form plotted. The convergence of the distributions for large enough $W$ and $L$ indicates the presence of an extensive number of quasiconservation laws after interactions are added, even in the thermodynamic limit. (e): Median $\mathrm{med}\|\tilde{n}_\alpha^{(1)}\|_F$ as a function of $L$ on a log-log scale. Each colored curve represents a different disorder strength $W$, from $W=0.1$ to $W=3.1$ in steps of $0.5$. One observes that the curves appear to saturate for large $L$, indicating that at least half of the LIOMs are conserved to second order when interactions are added. See Sec. \ref{['sec:liom_results_disordered']} for a full discussion of the results.
• Figure 4: Same data as in Fig. \ref{['fig:liom_disordered_figure']}(e), but with the system size on the x-axis scaled by the participation ratio, where $\mathrm{PR}\sim W^{-1.78}$ (see Appendix \ref{['app:localization_length']} for details), and the y-axis scaled by a phenomenological scaling of $W^{-3}$. The collapse of the curves suggests that the increase of $\mathrm{med}\|\tilde{n}_\alpha^{(1)}\|_F$ at small $L$ or low disorder strength $W$ is a finite-size effect.
• Figure 5: A density scatter plot of the natural log of the LIOM corrections $\ln\|\tilde{n}^{(1)}_\alpha\|_F$ at $W=3.1$ and $L=80$ against (a) the energy $E$ of its corresponding orbital $\alpha$ and (b) the participation ratio (PR) of the same orbital. Here, we use 500 different disorder realizations. For each disorder realization, we keep corrections to all orbitals $\alpha = 1, \dots, L$. Yellower colors indicate a higher density of points.