Many-Body Resonances in the Avalanche Instability of Many-Body Localization

Hyunsoo Ha, Alan Morningstar, and David A. Huse

Phys. Rev. Lett. 130, 250405 – Published 23 June 2023

Abstract

Many-body localized (MBL) systems fail to reach thermal equilibrium under their own dynamics, even though they are interacting, nonintegrable, and in an extensively excited state. One instability toward thermalization of MBL systems is the so-called “avalanche,” where a locally thermalizing rare region is able to spread thermalization through the full system. The spreading of the avalanche may be modeled and numerically studied in finite one-dimensional MBL systems by weakly coupling an infinite-temperature bath to one end of the system. We find that the avalanche spreads primarily via strong many-body resonances between rare near-resonant eigenstates of the closed system. Thus we find and explore a detailed connection between many-body resonances and avalanches in MBL systems.

Received 24 January 2023
Accepted 3 June 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.250405

Physics Subject Headings (PhySH)

Many-body localization Quantum statistical mechanics

Quantum many-body systems

Statistical Physics & ThermodynamicsQuantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Hyunsoo Ha ¹, Alan Morningstar ^1,2, and David A. Huse ¹

¹Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
²Department of Physics, Stanford University, Stanford, California 94305, USA

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Vol. 130, Iss. 25 — 23 June 2023

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Images

Figure 1
(a) The avalanche model. We model the large avalanche (seeded by a bare thermal rare region) spreading, but still far away ( $d ≫ 1$ ) by connecting the Markovian infinite-temperature bath in the weak-coupling limit with the one-dimensional MBL system of length $L$ spins. Specifically, we analyze the decay of the slowest mode ( ${\hat{τ}}_{S}$ ), which is localized near the end of the system farthest from the bath; ${\hat{τ}}_{S}$ is a “localized integral of motion” in the MBL phase. (b) Schematic decay of ${\hat{τ}}_{S}$ . A large fraction of the probability current in the decay of ${\hat{τ}}_{S}$ passes through four eigenstates associated with a rare near-resonance.
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Figure 2
Probability distribution over samples of the ratio $R = D_{μ ν} / (2^{L} Γ)$ (see text). In the thermal regime (dark red to yellow), the ratio exponentially decays with increasing $L$ . In comparison, $R$ barely drifts with $L$ for the MBL case (blue). We observe that $R$ never exceeds the value 0.5 (gray dashed line), which is explained with the minimal model in the main text. In this figure, we used $α = 30$ and $α = 1$ for MBL and thermal regimes, respectively.
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Figure 3
Two distinct cases for the set of four most important eigenstates of ${\hat{U}}_{F}$ for the avalanche to propagate. The near-resonant pair ${A, B}$ are coupled with ${\hat{X}}_{L}$ and ${\hat{Y}}_{L}$ jump operators to $B^{'}$ and $A^{'}$ , respectively, i.e., they have anomalously large matrix elements. In the $Z$ case (b), the near-resonant pair involves a spin flip next to the bath (site $L$ ), and the bath can additionally couple the resonant pair with the ${\hat{Z}}_{L}$ jump operator with matrix element $S_{A B}$ twice as large as $S_{A^{'} B}$ and $S_{A B^{'}}$ . The two states with the largest $D_{μ ν}$ are colored red. In the $X Y$ case (a), this pair is not the near-resonant pair, and ${μ, ν} = {A, B^{'}}$ , while they are the same in the $Z$ case, so ${μ, ν} = {A, B}$ . In some samples, ${A^{'}, B^{'}}$ are also near-resonant (not shown).
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Figure 4
Consistency with the near-resonant eigenstate set. All data here are for $α = 30$ . (a) The near-resonance between ${A, B}$ causes the $A B^{'}$ and $A^{'} B$ matrix elements to be roughly equal, as described in the minimal model. The distribution of $S_{A B^{'}} / S_{A^{'} B}$ is indeed peaked near 1. The inset shows $Z_{jump}$ of $A B^{'}$ and $A^{'} B$ are peaked at zero (coupled with ${\hat{X}}_{L}$ and ${\hat{Y}}_{L}$ ). (b) When the near-resonant pair involves the spin flip of the last site ( $Z$ case), the bath can directly couple ${A, B}$ . The distribution of the shown ratio of matrix elements demonstrates that $S_{A B} ≃ 2 S_{A B^{'}} ≃ 2 S_{A^{'} B}$ . The inset shows $Z_{jump}$ of $A B$ is peaked at 1, implying ${A, B}$ are coupled with ${\hat{Z}}_{L}$ . The bottom two panels show the demixing angles for (c) the $X Y$ case and (d) the $Z$ case. The points correspond to $A B$ (red), $A^{'} B$ (gray $\times$ ), $A B^{'}$ (gray dot), and $A^{'} B^{'}$ (blue). The near-resonant pair identified as in the main text ( $A B$ ) has the largest demixing angles for all cases. The calculations in (a),(b) are done with $L = 11$ and $10^{4}$ disorder realizations.
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