Hierarchy of timescales in a disordered spin-$1/2$ XX ladder

TL;DR

This work links spectral signatures to transport in a disordered spin-1/2 XX ladder by contrasting the onset of random-matrix theory (RMT) statistics, via the spectral form factor, with diffusion-based Thouless times for spin and energy. The authors extract $t_ ext{RMT}$ from SFF data and compute diffusion constants using linear-response theory at infinite temperature, revealing an $L^2$ scaling for $t_ ext{RMT}$ that upper-bounds the Thouless times. They find a persistent hierarchy $t_ ext{Th}^{(\mathrm{S})}<t_ ext{Th}^{(\mathrm{E})}$, i.e., spin diffusion is faster than energy diffusion in this ladder, attributed to the absence of diagonal terms in the spin-transport picture. Comparisons with a disordered XXZ chain show that the diffusion-order reverses in some models, but the $L^2$ scaling and the relation $t_ ext{Th}<t_ ext{RMT}$ remain robust, highlighting a general connection between hydrodynamic transport and spectral universality in chaotic many-body systems.

Abstract

Understanding the timescales associated with relaxation to equilibrium in closed quantum many-body systems is one of the central focuses in the study of their non-equilibrium dynamics. At late times, these relaxation processes exhibit universal behavior, emerging from the inherent randomness of chaotic Hamiltonians. In this work, we investigate a disordered spin-$1/2$ XX ladder - an experimentally realizable model known for its diffusive dynamics - to explore the connection between transport properties and spectral measures derived solely from the system's energy levels via these relaxation timescales. We begin by analyzing the spectral form factor, which yields the time when the system begins to follow the random matrix theory (RMT) statistics, known as the RMT time. We then determine the Thouless times - the average times for a local excitation to diffuse across the entire finite system - through the linear-response theory for both spin and energy transport. Our numerical results confirm that the RMT time scales quadratically with system size and upper bounds the Thouless times. Interestingly, we also find that, unlike other non-integrable models, spin diffusion proceeds faster than energy diffusion.

Figures (10)

• Figure 1: Sequence of regimes for a diffusive nonintegrable many-body quantum spin system with a finite system size $L$. The diffusion of a conserved quantity for such a system is governed by the $L^2$-scaling timescale called Thouless time $t_\mathrm{Th}$, which represents the average time for a transport quantity to spread through the finite system. Often, energy diffusion is expected to occur more rapidly than spin diffusion, i.e., $t_\mathrm{Th}^{(\mathrm{E})} < t_\mathrm{Th}^{(\mathrm{S})}$. The RMT time $t_\mathrm{RMT}$ indicates when the system begins to behave like a random matrix, following universal RMT statistics. This time is expected to be identical (or related) to the longest physical timescale. On top of all of these $L^2$-scaling relaxation timescales, the Heisenberg time $t_\mathrm{H}$, determined by the mean-level spacing $\overline{\delta E}$, is the longest timescale and exhibits exponential scaling with respect to the system size.
• Figure 2: Sketch of a spin-$1/2$ XX model on a two-leg ladder lattice with periodic boundary conditions along the legs. The double-sided arrows along the legs and rungs represent the exchange coupling constants with strengths $J_\parallel$ and $J_\perp$, respectively.
• Figure 3: Comparison between a typical ensemble-averaged SFF of a time-reversal symmetric Hamiltonian (white solid line) $\overline{K}{(\tau)}$ and the SFF of the Gaussian orthogonal ensemble (black dashed line) $K_\mathrm{GOE}{(\tau)}$ as a function of normalized time $\tau$ at disorder strength $W/J_\parallel = 1.5$ and coupling ratio $J_\perp/J_\parallel = 1.0$. $\overline{K}{(\tau)}$ exhibits four regimes: quantum, hydrodynamic, universal RMT, and finite size; in contrast, $K_\mathrm{GOE}{(\tau)}$ displays only the RMT and finite-size regimes haakeQuantumSignaturesChaos1991stockmannQuantumChaosIntroduction1999mehtaRandomMatrices2004.
• Figure 4: Comparison between disorder-averaged time-dependent diffusion constants of a disordered spin-$1/2$ XX ladder as a function of time at disorder strength $W/J_\parallel = 1.5$ and coupling ratio $J_\perp/J_\parallel = 1.0$ for various numbers of rungs $L=7,8,9,10$. The spin diffusion (purple lines) occurs at a faster rate than the energy diffusion (green lines). The opacity of each line denotes the result for a chosen $L$---lower opacity corresponds to a smaller system, while higher opacity indicates a larger one. The arrows specify how the time-dependent diffusion constant converges with an increase in $L$.
• Figure 5: Running-and-disorder-averaged SFF $\overline{K}_\mathrm{RA}{(t)}$ of a spin-$1/2$ XX ladder as a function of time normalized by $L^2$ at disorder strengths (a) $W/J_\parallel = 1.25$ and (b) $W/J_\parallel = 1.5$ and coupling ratio $J_\perp/J_\parallel = 1.0$. The (real) time is obtained from the normalized time $\tau$ via $t = \frac{\tau}{2\pi} t_\mathrm{H}$, where $t_\mathrm{H}$ denotes the Heisenberg time. The color of each line corresponds to a specific number of rungs, $L$. Solid lines represent $\overline{K}_\mathrm{RA}{(t)}$, while black dashed lines denote the SFF of the GOE $K_\mathrm{GOE}{(t)}$. Each SFF is computed with a discrete time step in $\log_{10}$ scale $\Delta(\log_{10}\tau)$ of $0.001$ using a running average window in $\log_{10}$ scale $\delta(\log_{10}\tau)$ of $0.025$ and a Gaussian filter width factor $\eta$ of $0.5$. All SFFs are averaged over $M = 5000$ disorder realizations. The RMT time $t_\mathrm{RMT}$ (red dots with error bars) is determined for each $L$ by varying the threshold value for the criterion function $\Delta K{(t)}$, $\epsilon_{\Delta K}$, within the range $[0.035, 0.2]$.