Quantum Avalanche Stability of Many-Body Localization with Power-Law Interactions

TL;DR

The paper addresses whether many-body localization remains stable against avalanche-induced thermalization in systems with power-law interactions. It combines exact diagonalization for static ETH–MBL diagnostics with Lindblad open-system simulations seeded at a boundary bath to probe avalanche propagation. A central finding is a unified scaling law $T_{r_{\mathrm{th}}} \sim \exp[\kappa(α)LW]$ for the avalanche-induced thermalization time, together with an interaction-dependent threshold $W_{\mathrm{stab}}(α) = \frac{2\ln 2}{\kappa(α)}$ that delineates asymptotic stability in the thermodynamic limit. These results resolve aspects of the long-range MBL debate and provide concrete guidance for observing avalanche dynamics in experimental platforms such as Rydberg atom arrays.

Abstract

We investigate the stability of the many-body localized phase against quantum avalanche instabilities in a one-dimensional Heisenberg spin chain with long-range power-law interactions ($V\propto r^{-α}$). By combining exact diagonalization of static properties with Lindblad master equation simulations of open-system dynamics, we systematically map the interplay between interaction range and disorder strength. Our finite-size scaling analysis of entanglement entropy identifies a critical interaction exponent $α_c \approx 2$, which separates a fragile regime, characterized by an exponentially diverging critical disorder, from a robust short-range regime. To rigorously test the system's resistance to avalanches, we couple the boundary to an infinite-temperature bath and track the propagation of the thermalization front into the localized bulk. We find that the characteristic thermalization time follows a unified scaling law, $T_{r_{\text{th}}} \sim \exp[κ(α) LW]$ (herein, $L$ is the system size, and $W$ is the disorder intensity), which diverges exponentially with the product of system size and disorder strength. This suppression enables the derivation of a quantitative stability criterion, $W_{\text{stab}}(α)$, representing the minimum critical disorder strength required to maintain avalanche stability. Our results confirm that the MBL phase remains asymptotically stable in the thermodynamic limit when disorder exceeds an interaction-dependent threshold, bridging theoretical debates on long-range MBL and providing a roadmap for observing these dynamics in experimental platforms such as Rydberg atom arrays.

Figures (10)

• Figure 1: Phase diagram of the one-dimensional Heisenberg chain in the $(\alpha, W)$ plane with power-law interactions. The colormap represents the normalized half-chain entanglement entropy $\langle S \rangle /S_T$ and the normalized level statistics ratio $r$. The data reveal two distinct phases: an ergodic phase (orange/red regions), characterized by volume-law entanglement ($\langle S \rangle \approx S_T$), and a many-body localized (MBL) phase (green regions), characterized by area-law entanglement ($\langle S \rangle \ll S_T$). Circular markers with error bars indicate the estimated critical disorder strength $W_c(\alpha)$, which delineates the ETH--MBL transition boundary. The phase boundary indicates that as the interaction range increases (decreasing $\alpha$), the MBL phase requires substantially stronger disorder to remain stable.
• Figure 2: Finite-size scaling analysis of the normalized half-chain entanglement entropy $\langle S \rangle / S_T$ as a function of disorder strength $W$ for four representative interaction exponents: (a) $\alpha = 0.5$, (b) $\alpha = 1.5$, (c) $\alpha = 2.5$, and (d) $\alpha = 3.5$. The main panels show the entropy ratio for system sizes $L=8, 10, 12, 14, 16$ (represented by different colors and symbols). The crossing of the curves signifies the transition from the volume-law ergodic phase to the area-law MBL phase. The insets display the data collapse obtained using the scaling ansatz $f[(W - W_c)L^{1/\nu}]$, which allows for the extraction of the critical disorder $W_c$ and the critical exponent $\nu$ listed in each panel Yousefjani2023a. The excellent collapse confirms the validity of the estimated critical points used in the phase diagram.
• Figure 3: Finite-size scaling results for the MBL transition parameters as a function of the interaction exponent $\alpha$. (a) The critical disorder strength $W_c$. (b) The critical exponent $\nu$. In both panels, black circles with error bars represent values extracted from the scaling analysis of the normalized entanglement entropy $\langle S \rangle/S_T$. The data reveals two distinct regimes separated by a critical interaction exponent $\alpha_c \approx 2$. The blue solid lines denote power-law fits [Eq. (\ref{['Eq.5']})] in the short-range regime ($\alpha > 2$), illustrating the divergence near $\alpha_c$. The red dashed lines indicate exponential fits in the long-range regime ($\alpha < 2$), highlighting the rapid growth of both the critical disorder and the critical exponent as interactions become longer-ranged.
• Figure 4: Dynamics of the magnetization imbalance $\mathcal{I}(t)$ for system sizes $L \in \{8, 10, 12, 14,16\}$. The columns correspond to interaction exponents $\alpha = 0.5, 1.5, 2.5, 3.5$ from left to right. (a)-(d) Weak disorder regime ($W=2$). (e)-(h) Strong disorder regime ($W=15$). In all panels, solid lines represent the isolated system ($\gamma=0$), while dotted lines represent the open system coupled to a thermal bath ($\gamma=1$). In the ergodic regime (top row), the system thermalizes rapidly regardless of $\alpha$. In the strong disorder regime (bottom row), a sharp contrast emerges: for isolated systems, stable MBL plateaus appear for large $\alpha$. When coupled to the bath, the decay rate exhibits distinct size dependence; for $\alpha < 1.5$, the curves collapse, indicating size-independent instability, whereas for $\alpha > 1.5$, the decay significantly slows down with increasing $L$, signaling robustness against avalanches.
• Figure 5: Finite-size scaling of the relative magnetization imbalance $\mathcal{I}_r(t)$ for system sizes $L \in \{8, 10, 12, 14, 16\}$. The panels are organized by disorder strength and interaction range: (a)–(d) the weak disorder regime ($W=2$), with $\alpha \in \{0.5, 1.5, 2.5, 3.5\}$ from left to right; (e)–(h) the strong disorder regime ($W=15$), with corresponding $\alpha$ values.