Thermal avalanches in isolated many-body localized systems

TL;DR

This work investigates how a tunable weak-disorder region embedded in a strongly disordered, isolated Heisenberg spin chain affects many-body localization. By analyzing entanglement entropy and gap ratio in static and dynamical settings, the authors show that an avalanche destabilizing MBL occurs only when the thermal region scales with system size with $P/L>0.1$, and they identify an intermediate, partially thermalized phase at strong disorder that diminishes with increasing system size. The combination of finite-size scaling and dynamical growth patterns provides a coherent picture: MBL can persist against small inclusions, but sufficiently large thermal regions induce global thermalization. These results clarify the competition between disorder strength and thermal-region size in finite systems and offer insights into the thermodynamic fate of MBL under inclusions.

Abstract

Many-body localization is a profound phase of matter affecting the entire spectrum which emerges in the presence of disorder in interacting many-body systems. Recently, the stability of many-body localization has been challenged by the avalanche mechanism, in which a small thermal region can spread, destabilizing localization and leading to global thermalization of the system. A key unresolved question is the critical competition between the thermal region's influence and the disorder strength required to trigger such an avalanche. Here, we numerically investigate many-body localization stability in an isolated Heisenberg spin chain of size $L$ subjected to a disordered magnetic field. By embedding a tunable thermal region of size $P$, we analyze the system's behavior in both static and dynamical regimes using entanglement entropy and the gap ratio. Our study yields two main findings. Firstly, for strong disorder, the avalanche only occurs if the thermal region scales with system size, specifically when $P/L$ exceeds a threshold value. Secondly, at strong disorder, we identify an intermediate phase between many-body localization and ergodic behavior as $P$ increases. This intermediate phase leaves its finger print in both static and dynamic properties of the system and tends to vanish in the thermodynamic limit. Although our simulations are restricted to finite system sizes, the analysis suggests that these results hold in the thermodynamic limit for isolated many-body systems.

Figures (12)

• Figure 1: (a) Schematic of our model system. While the dark points are exposed to strong random disorder, the light points are subjected to a weak random disorder of strength $w{=}0.5$. (b) Graphical visualization of the phase diagram as a function of disorder strength $W$ and the size of the weak-disorder region $P{/}L$. The solid (dashed) line indicates the phase boundary controlled by the critical $W$ ($P{/}L$). The intermediate area gives a rough approximation of the region wherein the MBL is unstable in the thermodynamic limit. (c) Schematic of two different types of chain partitioning for calculation of the entanglement entropy.
• Figure 2: First row: the averaged EE as a function of disorder strength $W$ in (a) fully disordered chains, and in (b-d) disordered chains with thermal regions of different sizes $P$ (indicated in the insets). The system sizes $L$ used in numerical simulation are indicated in the insets. Second row: the averaged GR as a function of disorder strength $W$ for the same set of $P$ values. The results for EE and GR are obtained using $1000{-}2000$ and $10000{-}25000$ realizations, respectively.
• Figure 3: (a) The averaged EE $\overline{\langle S \rangle}$, computed for half chain slicing as a function of $P{/}L$. (b) the averaged EE $\overline{\langle S_{\mathrm{mbl}} \rangle}$, computed by separating the strongly disordered region on the left side from the rest of the chain as a function of $P{/}L$. In both panels, the size of the system varies and $W{=}9$. (c) The difference between the results presented in (a) and (b). (d) The averaged GR as a function of $P{/}L$ for $W{=}9$. The results for EE and GR are obtained using $1000{-}2000$ and $10000{-}25000$ realizations, respectively.
• Figure 4: (a) averaged EE $\overline{\langle S\rangle}{/}S_{P}$ and (b) averaged GR $\overline{\langle r \rangle}$ as a function of $W$ and $P{/}L$ in a system of size $L{=}16$. In both panels, the solid circles represent the estimated $W_c^{*}(L_{av})$ for fixed $P$'s. The other colorful markers represent the extracted $(P{/}L_{av})_{c}^{*}$ for fixed $W$'s. As graphically visualized in Fig. \ref{['fig:1']}(b), three distinct regions are classified as $1.$ ergodic, $2.$ intermediate, and $3.$ localized.
• Figure 5: (a) The dynamic of EE in a system of size $L{=}16$ for different values of $P$, when the system with $P{=}0$ is determined as localized in the presence of strong disorder $W{=}9$. (b) the saturated values of the EE, namely $\overline{S(t{\rightarrow\infty)}}{/}S_{P}$, as a function of $P{/}L$ when the size of the system varies and $W{=}9$. The results are obtained using $200{-}400$ realizations.