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

Technical Summary: Beyond the Imbalance: Site-Resolved Dynamics Probing Resonances in Many-Body Localization

Problem Statement
Many-Body Localization (MBL) is a mechanism preventing quantum thermalization in interacting disordered systems. While the standard diagnostic for MBL is the "imbalance" ($I_m(t)$)—a spatially averaged measure of the memory of an initial density-wave state—this paper argues that such spatial averaging obscures crucial microscopic features. Specifically, the authors investigate whether the standard imbalance observable hides local dynamical phenomena, such as resonances, which are critical for understanding the transition between ergodic and localized phases. The work addresses the limitations of interpreting long-time imbalance decay solely through spatial averages, noting that ultra-slow power-law decays in numerics have been controversially interpreted as signs of eventual thermalization.

Methodology
The authors focus on the strongly disordered regime of the random-field XXZ spin chain ($L$ sites, disorder strength $h$, interaction $\Delta$). They employ two primary numerical techniques:

1. Full Diagonalization (ED): Used for systems up to $L=18$ (in the zero magnetization sector) to compute exact eigenstates and infinite-time limits.
2. Krylov Time Evolution: Used for larger systems ($L$ up to 24) to simulate unitary time evolution up to $t \approx 10^4$.

Instead of relying solely on the standard imbalance $I_m(t) = \frac{1}{L} \sum_j Z^m_j(t)$, the study centers on the site-resolved local autocorrelator $Z^m_j(t) = \langle m | \sigma^z_j(0) \sigma^z_j(t) | m \rangle$. The authors systematically analyze the distributions of these local quantities under different averaging protocols:

• Spatial averaging: Summing over sites $j$.
• Initial-state averaging: Summing over $N_s$ distinct initial product states (partial trace) or the full Hilbert space (full trace/infinite temperature).
• Disorder averaging: Summing over $N_r$ random field realizations.

To interpret the numerical findings, the authors construct an analytical few-site toy model. This model treats local resonances as effective few-body clusters (2-site and 3-site) embedded in a background of frozen (pinned) spins, allowing for the derivation of analytical predictions for the long-time limits of local observables.

Key Contributions and Results

1. Masking of Local Resonances by Spatial Averaging:
   The study demonstrates that the standard imbalance $I_m(t)$ exhibits a stable plateau consistent with MBL, but this average completely hides the behavior of individual sites. While most sites remain polarized (retaining memory of the initial state), a small fraction of sites exhibit strong fluctuations and decay to zero. These "fluctuating" sites correspond to local regions where the random fields are nearly identical (e.g., $h_j \approx h_{j+1}$), creating local resonances.

2. Multi-Peak Structure in Local Distributions:
   By analyzing the histograms of the long-time local autocorrelators ($P(Z^m_j)$ and $P(\mathcal{A}_j)$), the authors reveal a rich structure obscured by averages.

   • The distribution of site-resolved observables shows a dominant peak near 1 (frozen spins) and distinct secondary peaks at lower values (e.g., $\approx 0.6$ for 2-body resonances and $\approx 0.46$ for 3-body resonances in the Heisenberg case).
   • These secondary peaks are signatures of few-body resonances where spins flip coherently due to near-degenerate energy levels.

3. Analytical Validation via Toy Model:
   The authors derive closed-form expressions for the long-time limits of these resonances using a 2-site and 3-site effective Hamiltonian.

   • For 2-body resonances with $\Delta=1$, the analytical prediction for the peak position is $A \approx \frac{5}{8} - \frac{3}{8(L-1)}$.
   • For 3-body resonances, distinct peaks for middle and edge sites are predicted.
   • These analytical predictions align quantitatively with the numerical histograms, confirming that the secondary peaks arise from specific few-body resonant clusters.

4. Finite-Size Scaling and Initial State Dependence:
   A major finding is the strong dependence of the long-time imbalance on the initial state and system size, driven by these resonances:

   • Generic Random Initial States: The average imbalance approaches the thermodynamic limit from below (negative $1/L$ corrections). This is driven by the rare occurrence of resonant sites which lower the average.
   • Néel Initial State ($|\uparrow\downarrow\uparrow\downarrow\dots\rangle$): The imbalance approaches the thermodynamic limit from above (positive $1/L$ corrections). Because the Néel state consists entirely of flippable bonds ($\uparrow\downarrow$), it maximizes the coupling to resonant modes, leading to a different finite-size scaling behavior compared to random states.
   • Domain-Wall State: Shows intermediate behavior with specific finite-size corrections.

5. Experimental Detectability:
   The paper establishes that these resonance effects are detectable under realistic experimental conditions. The secondary peaks in the distribution of local autocorrelators can be resolved with a relatively small number of initial states ($N_s \sim 20-50$) and moderate time windows ($t \approx 500$), provided site-resolved imaging is available.

Significance and Claims
The paper claims that the "imbalance" is not a universal probe of MBL but is highly sensitive to the specific initial conditions and the method of averaging. The authors argue that:

• Refined Probes are Necessary: To fully understand MBL and its connection to rare-region effects, one must move beyond spatially averaged quantities to site-resolved dynamics.
• Reinterpretation of Slow Dynamics: The observed slow decay of imbalance in finite-time numerical studies (often starting from Néel states) may not necessarily indicate eventual thermalization. Instead, it could represent a transient crossover from a finite-size value (approached from above) to a smaller, non-zero thermodynamic asymptotic value.
• Role of Rare Resonances: Short-range few-body resonances, occurring with probability scaling as $1/h$ (for 2-body) and $1/h^2$ (for 3-body), govern the finite-size scaling of the imbalance in the strong-disorder regime.
• Experimental Relevance: With the advent of high-precision site-resolved imaging in ultracold atoms and superconducting circuits, the predicted multi-peak structures and initial-state-dependent finite-size scalings are directly testable.

The work concludes that a "refined, site-resolved approach" is essential to disentangle the complexities of MBL dynamics, particularly the interplay between localized behavior and rare resonant instabilities.