Many-body (de)localization in long quantum chains
We study the quench dynamics of an isolated Heisenberg spin chain with a random on-site magnetic field. We employ a combination of the time-dependent variational principle as applied to matrix product states and machine learning, with a focus on long chains up to 100 spins in length. For the analysis of the data, three complementary approaches are used: (i) determination of the exponent characterizing the power-law decay of the antiferromagnetic imbalance with time; (ii) similar analysis of the exponent characterizing the decay of the Schmidt gap in the entanglement spectrum, (iii) machine learning (supervised and unsupervised) that uses as an input the full time dependence of spin densities in the whole chain. We find that considering larger system sizes substantially increases the estimate for the critical disorder that separates ergodic and many-body localized regimes. In the range of disorder strengths where previous studies on shorter chains have reported the existence of the many-body localization transition, we find instead slow but finite transport characterized by subdiffusive power-low exponents.