Many-Body Delocalization via Emergent Symmetry

Open Access

Many-Body Delocalization via Emergent Symmetry

N. S. Srivatsa, Roderich Moessner, and Anne E. B. Nielsen

Phys. Rev. Lett. 125, 240401 – Published 8 December 2020

Abstract

Many-body localization (MBL) provides a mechanism to avoid thermalization in many-body quantum systems. Here, we show that an emergent symmetry can protect a state from MBL. Specifically, we propose a $Z_{2}$ symmetric model with nonlocal interactions, which has an analytically known, SU(2) invariant, critical ground state. At large disorder strength, all states at finite energy density are in a glassy MBL phase, while the lowest energy states are not. These do, however, localize when a perturbation destroys the emergent SU(2) symmetry. The model also provides an example of MBL in the presence of nonlocal, disordered interactions that are more structured than a power law. Finally, we show how the protected state can be moved into the bulk of the spectrum.

Received 1 March 2020
Accepted 2 November 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.240401

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Localization Many-body localization

1-dimensional systems Disordered systems Quantum spin models Spin glasses

Exact diagonalization Finite-size scaling Monte Carlo methods

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

N. S. Srivatsa , Roderich Moessner, and Anne E. B. Nielsen^*

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany

^*On leave from: Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark.

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Issue

Vol. 125, Iss. 24 — 11 December 2020

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Images

Figure 1
(a) Probability that spin $i$ interacts most strongly with one of the spins $i \pm m$ modulus $N$ for different disorder strengths. (b) The adjacent gap ratio [averaged over $10^{4}$ disorder realizations ( $10^{3}$ for $N = 16$ ) and shown as a function of system size] is close to the Gaussian orthogonal ensemble (GOE) for weak disorder and close to the Poisson distribution, indicating MBL, for strong disorder. (c) The transition to the MBL phase is also seen in the entanglement entropy $S$ of half of the chain for the state closest to the middle of the spectrum averaged over $10^{5}$ disorder realizations as a function of the disorder strength $δ$ for different system sizes $N$ . The blue dashed lines indicate the thermal value $[N \ln (2) - 1] / 2$ of the entropy. The inset shows that the mean entropy follows a volume law for weak disorder and an area law for strong disorder. (d) The variance $σ^{2}$ of the entanglement entropy computed from the same set of data shows a peak at the transition point.
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Figure 2
(a) Disorder averaged spin glass order parameter $⟨ χ^{SG} ⟩$ for a state in the middle of the spectrum ( $E / E_{\max} = 0.5$ , circles), a low energy state ( $E / E_{\max} = 0.01$ , triangles), and the ground state ( $E = 0$ , dashed line) as a function of the disorder strength $δ$ . The excited states show glassiness for strong disorder, and the ground state is not glassy. (b) A finite size scaling collapse for the state in the middle of the spectrum. The inset shows the scaling collapse for the low energy state. For both cases, the predicted phase transition point is $δ_{c} \approx 1.05$ . (c) $⟨ χ^{SG} ⟩$ for all eigenstates for a system with 12 spins and weak ( $δ = 0.1$ ) or strong ( $δ = 12$ ) disorder. The states close to the ground state have different values of $⟨ χ^{SG} ⟩$ compared to the states in the middle of the spectrum. The inset shows that the disorder averaged energy spectrum (normalized) at $δ = 12$ has gaps, and these coincide with the jumps in $⟨ χ^{SG} ⟩$ . (d) When we destroy the emergent SU(2) symmetry by perturbing the Hamiltonian (we modify $F_{i j}^{A}$ to $- 1.9 w_{i j}^{2}$ ), the ground state becomes glassy for $δ ≳ 1$ . In all cases, the number of disorder realizations is $10^{4}$ .
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Figure 3
(a) The Renyi entropy of the ground state (plotted for $N = 600$ and $δ = 0.75$ in the inset) follows the logarithmic relation (6). The main plot shows the coefficient $C$ as a function of the system size for $δ = 0.75$ . The results might suggest a Luttinger liquid, but the method is not accurate enough to make clear conclusions. We use $10^{3}$ disorder realizations, and the error from the Monte Carlo simulations and disorder averaging is of order $10^{- 4}$ . (b) The coefficient $ξ$ in the second cumulant $C_{2}$ computed for different systems divided into two halves is close to the value for the Luttinger liquid both for the clean ( $δ = 0$ ) and the disordered ( $δ = 4$ ) Haldane-Shastry state. Each data point is averaged over $10^{5}$ disorder realizations. The inset shows the logarithmic scaling of the averaged Renyi entropy for $δ = 4$ and $N = 50$ . RSP stands for random singlet phase.
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