Beyond the imbalance: site-resolved dynamics probing resonances in many-body localization

TL;DR

This work shows that imbalance, a standard diagnostic for many-body localization, can obscure microscopic dynamics because spatial averaging hides local resonances. By analyzing site-resolved autocorrelators in the strongly disordered random-field XXZ chain, the authors reveal resonant structures (2-body and 3-body) that produce secondary peaks in local magnetization histograms, a feature absent in global imbalance. They develop a tractable few-site toy model that analytically captures these resonances and their finite-size scaling, corroborated by full ED and Krylov dynamics. The study demonstrates that the long-time imbalance depends strongly on the initial state and system size, and provides experimentally testable predictions for site-resolved measurements, including how finite sampling and finite time impact resonance visibility. Overall, the results refine the understanding of ergodicity-breaking dynamics in MBL by linking local resonances to macroscopic observables and finite-size effects, with clear implications for interpreting experiments in ultracold atoms and related platforms.

Abstract

We explore the limitations of using imbalance dynamics as a diagnostic tool for many-body localization (MBL) and show that spatial averaging can mask important microscopic features. Focusing on the strongly disordered regime of the random-field XXZ chain, we use state-of-the-art numerical techniques (Krylov time evolution and full diagonalization) to demonstrate that site-resolved spin autocorrelators reveal a rich and complex dynamical behavior that is obscured by the imbalance observable. By analyzing the time evolution and infinite-time limits of these local probes, we reveal resonant structures and rare local instabilities within the MBL phase. These numerical findings are supported by an analytical, few-site toy model that captures the emergence of a multiple-peak structure in local magnetization histograms, which is a hallmark of local resonances. These few-body local effects provide a more detailed understanding of ergodicity-breaking dynamics, and also allow us to explain the finite-size effects of long-time imbalance, and its sensitivity to the initial conditions in quench protocols. Overall, our experimentally testable predictions highlight the necessity of a refined, site-resolved approach to fully understand the complexities of MBL and its connection to rare-region effects.

Figures (10)

• Figure 1: Illustration of what spatial averaging of imbalance hides --- Local spin dynamics for two typical ($L=24$ sites) samples, at strong disorder ($h=10$) and interaction $\Delta=1$. Starting from an initial product state $\ket{m}$ (shown on each plots), the system follows unitary time evolution Eq. \ref{['eq:evolution']}. The main panels (a-b) show the (time-averaged) site-resolved spin autocorrelators $[Z^{(m)}_j(t)]_T$, see Eq. \ref{['eq:Zmtime']}, for two random field realizations $\{h_j\}$ displayed in panels (a'-b'). The (site and time-averaged) imbalance ${\cal{I}}^{m}(t)$, Eq. \ref{['eq:Im']}, is also shown (+) for comparison. While most sites exhibit an $\mathcal{O}(1)$ plateau at long times---consistent with the site-averaged saturation value of the imbalance in the MBL regime---a few sites (highlighted by colored circles) deviate from this behavior, signaling local resonances that are invisible in $\mathcal{I}^m(t)$.
• Figure 2: Illustration of what initial-state averaging hides --- Time-averaged autocorrelators of a single $L=20$ disordered sample having the same first 20 random fields as the one in Fig. \ref{['fig:samples']} (b'). Panel (a) shows the partial traces Eq. \ref{['eq:Ajp']} for all sites $j=1,\ldots, 20$, averaged over ${N_{s}}=50$ initial states. The other panels (b-f) focus each on particular sites ($j=1,\,3,\,4,\,18,\,19$), for which all $[Z^{(m)}_j(t)]_T$ are shown ($m=1,\ldots,\,50$ correspond to different colors).
• Figure 3: Histograms of the various quantities entering in the average imbalance: Eqs. (\ref{['eq:avgI']}, \ref{['eq:avg1']}, \ref{['eq:avg2']}), all yielding ${\cal{I}}_{\rm avg} \approx 0.83$ (vertical doted line). Long time data ($t \ge 5000$, obtained for $L=20$ sites, $N_{r}=500$ samples, and $N_{s}=50$ initial states) are shown for $h = 10$ and $\Delta = 1$, except for the full trace at $t=\infty$ (red symbols) for which $L=18$ and $N_{r}\approx 10^4$ samples. $P(Z_j^m)$ (dark thick line) is strongly peaked at 1 with a long tail, indicating a fast suppression of fluctuating sites. $P({\cal{I}}^m)$ (purple crosses) is more peaked close to (but not exactly at) the mean value, shows clear signs of self-averaging. $P({\cal{A}}_j)$ reveals additional structure, notably secondary peaks in ${\cal{A}}_j^{\rm Full}$ (red symbols), also visible but less marked in ${\cal{A}}_j^{\rm Partial}$ (orange squares): these features originate from local resonances, see Sec. \ref{['sec:toy']}.
• Figure 4: Full diagonalization results for the XXZ Hamiltonian [Eq. \ref{['eq:XXZ']}] at $h = 10$ and $\Delta = 1$. Histograms of the infinite-time, full-trace observable ${\cal{A}}_j^{\rm Full}(\infty)$ [Eq. \ref{['eq:Aj']}], obtained from $N_r = 10^4$–$10^5$ disorder realizations for various system sizes (as labeled). The distribution of ${\cal{A}}_j^{\rm Full}(\infty)$ differs markedly from that of the squared local magnetizations computed in each individual eigenstate $\langle\phi_n| \sigma_j^z |\phi_n\rangle^2$ (dark crosses, $L = 18$). Spectral averaging reveals a much richer structure (see text for details). The red vertical dotted line signals the 2-body resonance peak, analytically predicted at ${\cal{A}}\approx 0.603$ from Eq. \ref{['eq:2body_even']} (for $\Delta=1$, $L=18$), and the thick yellow line indicates the 3-body resonance at ${\cal{A}}\approx 0.46(1)$ (boundary and middle-site contributions from Eq. \ref{['eq:3b']} cannot be resolved, see also Fig. \ref{['fig:Histo_2-3br']}).
• Figure 5: Histograms of the (infinite-time, site-resolved) autocorrelators ${\cal A}_j(\infty)$, averaged over initial states (using full or partial traces), shown for strong disorder $h=10$. (a) Non-interacting case $\Delta=0$: free-fermion results are displayed for system sizes $L=8$–$128$. Data are collected over ${ N}_{r}$ disorder realizations, ranging from $4\times10^4$ to $10^6$ samples. The partial trace is performed over ${N}_{s}=10^4$ randomly drawn basis states $\ket m$. The vertical lines indicate the analytical predictions for 2-body Eq. \ref{['eq:2body_even']} (solid) and 3-body Eq. \ref{['eq:3b']} (dashed) contributions (see also Tab. \ref{['tab:3b']}). Inset $(i)$ demonstrates for $L=24$ how the resonance peaks build in the partial trace upon averaging over an increasing number $N_{ s}$ of initial states. (b) Finite interaction $\Delta=1$ (Heisenberg): same data as in Fig. \ref{['fig:Histo_h10']} for $L=18$, together with the comparison with analytical predictions (vertical lines) of Tab. \ref{['tab:3b']}. Inset $(ii)$ displays the positions of the few-body resonance peaks vs.$1/L$, again compared to analytical expressions (right column in Tab. \ref{['tab:3b']}). The middle vs. edge contributions are hard to resolve numerically.