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 20 that can be treated via exact diagonalization. For the analysis of the data, three complementary approaches are used: (i) determination of the exponent β which characterizes the power-law decay of the antiferromagnetic imbalance with time; (ii) similar determination of the exponent βΛ which characterizes the decay of a Schmidt gap in the entanglement spectrum, (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 that separates the ergodic and many-body localized regimes, compared to the values of 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 β and βΛ increase, with increasing L, for relatively small L but saturate for L 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. arXiv:1807.05051v4 [cond-mat.dis-nn]

We study quench dynamics in an interacting spin chain with a quasi-periodic on-site field, known as the interacting Aubry-André model of many-body localization. Using the time-dependent variational principle, we assess the late-time behavior for chains up to L = 50. We find that the choice of periodicity Φ of the quasi-periodic field influences the dynamics. For Φ = ( √ 5 − 1)/2 (the inverse golden ratio) and interaction ∆ = 1, the model most frequently considered in the literature, we obtain the critical disorder Wc = 4.8 ± 0.5 in units where the non-interacting transition is at W = 2. At the same time, for periodicity Φ = √ 2/2 we obtain a considerably higher critical value, Wc = 7.8 ± 0.5. Finite-size effects on the critical disorder Wc are much weaker than in the purely random case. This supports the enhancement of Wc in the case of a purely random potential by rare "ergodic spots," which do not occur in the quasi-periodic case. Further, the data suggest that the decay of the antiferromagnetic order in the delocalized phase is faster than a power law. arXiv:1901.06971v5 [cond-mat.dis-nn]

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