Many-body localization and delocalization in large quantum chains

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 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]

Time dynamics with matrix product states: Many-body localization transition of large systems revisited

“…On the other hand, the recent study [39] of larger systems (size L = 100) gives the estimate of the transition at a much larger value W c ≈ 5.5 on the basis of time dynamics obtained using TDVP. This obviously contradicts the above mentioned result based on finite size scaling of data for systems with L ≤ 22 and suggests that either the time-dynamics of observables gives different answer than analysis of level statistics and properties of eigenvectors or results obtained by TDVP [39] have to be reconsidered. We analyse a time-evolution obtained for the Hamiltonian (3) starting from the Néel state with every second spin pointing up and every second spin down |ψ = | ↑↓ .…”

“…This is a typical system size met in cold atoms experiments [31]. The previous TDVP based analysis [39] has concentrated on L = 100 showing that L = 100 and L = 50 do not differ substantially. Still L = 50 is computationally less expensive and, on the other hand, it seems sufficiently large to ensure that boundary effects are small.…”

“…Firstly, in the interesting interval of disorder values, χ = 64 or even χ = 128, as used e.g. in [39], lead to unconverged results affecting the shape and the decay of imbalance curves. Only for W > 4.5 such small values of χ seem to produce results one could rely on.…”

Subdiffusion in the Anderson model on the random regular graph

“…(3), confirming that the system is in a fully ergodic phase [69]. For larger disorder, W ∈ [0.4W AT , 0.7W AT ] [8,13], Π(0, t) decays as a stretched exponential ∼ e −Γt β(W ) , where the exponent is well approximated by (5) and goes to zero at the Anderson transition. As a consequence, the drastic change in the time evolution of Π(0, t) provides evidence of the existence of two dynamically distinct phases, one fully ergodic and the other NEE (see Appendices B and C).…”

“…The MBL phase itself also seems to hold more surprises than what was thought before. For instance, recent studies [13,17,23] show that there can be residual slow dynamics even for disorder values deep in the MBL phase. Whether this dynamics is transient or implies underestimation of the critical disorder strength remains to be explored.…”