Bath-induced delocalization in interacting disordered spin chains

Letter

Bath-induced delocalization in interacting disordered spin chains

Dries Sels

Phys. Rev. B 106, L020202 – Published 27 July 2022

Abstract

Over a decade of work has culminated in the consensus that one-dimensional systems, subject to sufficiently large disorder, fail to thermalize and possess an extensive set of local integrals of motion. In this Letter, I will provide numerical evidence for the contrary. In particular, this work studies the dynamics of disordered spin chains which are weakly coupled to a Markovian bath. Within this approach, the critical disorder for stability to quantum avalanches exceeds $W^{*} ≳ 20$ in the random field Heisenberg chain. In stark contrast to the Anderson insulator, the avalanche threshold drifts considerably with system size, with no evidence of saturation in the studied regime.

Received 1 December 2021
Revised 19 May 2022
Accepted 21 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.L020202

Physics Subject Headings (PhySH)

Many-body localization

Markovian processes Spin chains

Statistical Physics & Thermodynamics

Authors & Affiliations

Dries Sels

Department of Physics, New York University, New York, New York 10003, USA and Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA

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Vol. 106, Iss. 2 — 1 July 2022

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Images

Figure 1
Rare thermal inclusion: No matter how large the disorder, in the thermodynamic limit the system will contain regions of arbitrary size $ℓ_{0}$ where the disorder is so weak that it can effectively be neglected. In interacting systems, these rare regions serve as a heat bath for the rest of the system. The fate of the system depends critically on how fast particles in the vicinity of the bath thermalize with it. In this work we consider ideal asymptotically large baths $ℓ_{0} \to \infty$ , such that all computational resources are used to describe the dynamics in the strongly disordered region $ℓ$ of interest.
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Figure 2
Relaxation rate: Curves show the 80th percentile of the distribution of the smallest (nonzero) eigenvalue of the Liouvillian superoperator in a Heisenberg chain over disorder realizations. By rescaling with $4^{L}$ the crossing point in the data becomes indicative of the avalanche stability threshold. Different curves show different system sizes, ranging from $L = 7$ to 14. The crossing point drifts substantially from around $W^{*} \approx 8$ for the smallest system to $W^{*} > 20$ for the largest available system size. The inset shows the numerically extracted minimal value of $Γ 4^{L}$ with $99 %$ bootstrap confidence intervals.
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Figure 3
Relaxation rate II: Curves show the behavior 80th percentile of the distribution of the smallest (nonzero) eigenvalue of the Liouvillian superoperator in a Heisenberg chain over disorder realizations. Different curves correspond to different value of the disorder strength $W = 8 - 20$ ; blue squares serve as a reference and correspond to weak disorder $W = 1$ . Black lines show quadratic fits, from which we extract the minimal value of the rescaled rate shown in the inset in Fig. 2.
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Figure 4
Relaxation rate III: The typical slowest relaxing operator in a disordered Heisenberg chain. Curves show the 80th percentile of the distribution of the smallest (nonzero) eigenvalue of the Liouvillian superoperator over disorder realizations. Different curves correspond to different systems sizes ranging from $L = 4$ to 10. (a) shows the bare rate $Γ$ and (b) shows the rate rescaled by its expected asymptotic $W^{- 2 (L - 1)}$ behavior. The dashed lines show the extracted asymptotes and dotted lines show the extrapolated function.
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Figure 5
Asymptotic scaling: At sufficiently large disorder $W$ the slowest relaxation rate is expected to behave as $C / W^{2 (L - 1)}$ . This figure shows the change in the numerically extracted prefactor with system size $L$ . Red circles and blue squares correspond to the Heisenberg chain and free fermions, respectively. Shaded areas indicate $99 %$ bootstrap confidence intervals. The inset shows the same data as a function of $L log L$ .
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Figure 6
Relaxation time Anderson: The typical slowest relaxing operator in a disordered chain of noninteracting fermions. Curves show the 80th percentile of the distribution of the smallest (nonzero) eigenvalue of the Liouvillian superoperator over disorder realizations. (a) Different curves correspond to different systems sizes ranging from $L = 4$ to 10. By rescaling with $4^{L}$ the crossing point in the data becomes indicative of the avalanche stability threshold. A stable crossing is observed around $W \approx 1.4$ . The dashed lines show the decay rate of the slowest single-particle operator. (b) The same data are shown as a function of $L$ for different values of the disorder strength $W$ .
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