Transfer tensor analysis of localization in the Anderson and Aubry-André-Harper models

TL;DR

The paper employs transfer tensor (TT) analysis to study memory effects in disorder-averaged dynamics for the 1D Anderson and Aubry–André–Harper models. It shows that averaging over static disorder generates memory terms $T(k)$, which remove fictitious cross-terms and create an eternal memory regime when $ extsigma>0$, linking memory to localization but not guaranteeing it. The authors introduce the outgoing-pseudoflux $F_m(k)$ as a robust indicator of localization versus diffusion, demonstrating its behavior across both models and under varying noise strengths and system parameters. The work establishes a bridge between dynamical-map theory and localization phenomena, offering TT-derived diagnostics that may relate to diffusion constants and inform studies of transport in realistic quantum systems.

Abstract

We use the transfer tensor method to analyze localization and transport in simple disordered systems, specifically the Anderson and Aubry-André-Harper models. Emphasis is placed on the memory effects that emerge when ensemble-averaging over disorder, even when individual trajectories are strictly Markovian. We find that transfer tensor memory effects arise to remove fictitious terms that would correspond to redrawing static disorder at each time step, which would create a temporally uncorrelated dynamic disorder. Our results show that while eternal memory is a necessary condition for localization, it is not sufficient. We determine that signatures of localization and transport can be found within the transfer tensors themselves by defining a metric called "outgoing-pseudoflux". This work establishes connections between theoretical research on dynamical maps and Markovianity and localization phenomena in physically realizable model systems.

Figures (3)

• Figure 1: (a) The Frobenius norm of the transfer tensor $||T(k)||$ as a function of $k$ for $\Gamma=0$ with different $\sigma$ values. (b) Population of site $m=8$ following initialization in a delta function localized on $m$ as a function of time step for several ensembles including a case in which all transfer tensor memory terms were artificially set to zero. The dashed black line is the equilibrium population. (c) The Frobenius norm of the transfer tensor $||T(k)||$ as a function of time step $k$ for several different $\Gamma$ values with $\sigma=V$. (d) The cumulative outgoing-pseudoflux as a function of time step for the same $\Gamma$ and $\sigma$ values as in (c) showing convergence to non-zero values at finite $\Gamma$ and convergence to zero at $\Gamma=0$. Note that, when $\Gamma=10V$, noise overpowers disorder and the system is effectively memoryless for the chosen $\Delta t$. All calculations in this figure involved a basis of size $15$, $V\Delta t = 0.1$ and $2000$ trajectories. In (a) and (c) the norm of the first transfer tensor is on the order of the basis size and not shown on these axes.
• Figure 2: (a) The Frobenius norm of the transfer tensor $||T(k)||$ as a function of time step $k$ for several different $\lambda$ strengths. (b) The cumulative outgoing-psuedoflux after $500$ time steps as a function of $\lambda$ for several different basis sizes of the AAH model, showing convergence towards the long-chain limit. The inset shows the convergence towards the final cumulative values as a function of time step for the varying basis sizes at a fixed $\lambda=1.5V$. (c) The Frobenius norm of the transfer tensor as a function of index given different $\Gamma$ for an AAH model with $\lambda=2V$. (d) The cumulative outgoing-pseudoflux as a function of time step for the same models as in (c).All calculations in this figure were run with $V\Delta t = 1/2$ and $2000$ trajectories and, unless otherwise specified, a basis of size $21$. Note that the norm of $T(1)$ is on the order of the basis size and not shown on the axes of (a) and (c).
• Figure 3: Systems with exponentially correlated noise ($\langle\epsilon_n(t)\epsilon_m(t')\rangle=\Gamma e^{-\gamma |t-t'|}\delta_{nm}$) with 9 sites, $V\Delta t=0.1$, $\Gamma/V=1$, and various $\gamma$ values. (a) Population at the central site as a function of time step and $\gamma$. The full thermalization limit $\rho_{m,m}(t\rightarrow\infty)=N^{-1}$ is indicated by the dashed line. (b) Tensor Frobenius norm. The plateauing for smaller $\gamma$ curves suggest more noise samples are needed for convergence. With more noise samples the plateau continues to decrease in value. (c) Outgoing-pseudoflux.