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arXiv:1807.05051 · Physical review. B./Physical review. B

Many-body localization and delocalization in large quantum chains

Elmer V. H. Doggen, Frank Schindler, Konstantin S. Tikhonov, A. D. Mirlin, Titus Neupert, Dmitry G. Polyakov, I. V. Gornyi

Published Nov 8, 2018• 186 citations
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Abstract

We theoretically study the quench dynamics for an isolated Heisenberg spin chain with a random on-site magnetic field, which is one of the paradigmatic models of a many-body localization transition. We use the time-dependent variational principle as applied to matrix product states, which allows us to controllably study chains of a length up to $L=100$ spins, i.e., much larger than $L\ensuremath{\simeq}20$ that can be treated via exact diagonalization. For the analysis of the data, three complementary approaches are used: (i) determination of the exponent $\ensuremath{\beta}$ which characterizes the power-law decay of the antiferromagnetic imbalance with time; (ii) similar determination of the exponent ${\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}$ which characterizes the decay of a Schmidt gap in the entanglement spectrum; and (iii) machine learning with the use, as an input, of the time dependence of the spin densities in the whole chain. We find that the consideration of the larger system sizes substantially increases the estimate for the critical disorder ${W}_{c}$ that separates the ergodic and many-body localized regimes, compared to the values of ${W}_{c}$ in the literature. On the ergodic side of the transition, there is a broad interval of the strength of disorder with slow subdiffusive transport. In this regime, the exponents $\ensuremath{\beta}$ and ${\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}$ increase, with increasing $L$, for relatively small $L$ but saturate for $L\ensuremath{\simeq}50$, indicating that these slow power laws survive in the thermodynamic limit. From a technical perspective, we develop an adaptation of the ``learning by confusion'' machine-learning approach that can determine ${W}_{c}$.

DOI
10.1103/physrevb.98.174202

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We theoretically study the quench dynamics for an isolated Heisenberg spin chain with a random on-site magnetic field, which is one of the paradigmatic models of a many-body localization transition. We use the time-dependent variational principle as applied to matrix product states, which allows us to controllably study chains of a length up to L=100L=100L=100 spins, i.e., much larger than L\ensuremath≃20L\ensuremath{\simeq}20L\ensuremath≃20 that can be treated via exact diagonalization. For the analysis of the data, three complementary approaches are used: (i) determination of the exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ which characterizes the power-law decay of the antiferromagnetic imbalance with time; (ii) similar determination of the exponent \ensuremathβ\ensuremathΛ{\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}\ensuremathβ\ensuremathΛ​ which characterizes the decay of a Schmidt gap in the entanglement spectrum; and (iii) machine learning with the use, as an input, of the time dependence of the spin densities in the whole chain. We find that the consideration of the larger system sizes substantially increases the estimate for the critical disorder Wc{W}_{c}Wc​ that separates the ergodic and many-body localized regimes, compared to the values of Wc{W}_{c}Wc​ in the literature. On the ergodic side of the transition, there is a broad interval of the strength of disorder with slow subdiffusive transport. In this regime, the exponents \ensuremathβ\ensuremath{\beta}\ensuremathβ and \ensuremathβ\ensuremathΛ{\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}\ensuremathβ\ensuremathΛ​ increase, with increasing LLL, for relatively small LLL but saturate for L\ensuremath≃50L\ensuremath{\simeq}50L\ensuremath≃50, indicating that these slow power laws survive in the thermodynamic limit. From a technical perspective, we develop an adaptation of the ``learning by confusion'' machine-learning approach that can determine Wc{W}_{c}Wc​.
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Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to L=28L=28L=28 sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\ensuremath{\propto}{t}^{1/z}$ with a disorder-dependent exponent $z\ensuremath{\ge}1$, in contrast with the pure ballistic growth z=1z=1z=1 of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance $\mathcal{I}(t)\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\zeta}}$ with a continuously varying disorder-dependent exponent $\ensuremath{\zeta}$, vanishing at the transition. This provides a clear experimental signature for detecting this nonconventional regime.
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