Many-body localization in large systems: Matrix-product-state approach

Recent developments in matrix-product-state (MPS) investigations of many-body localization (MBL) are reviewed, with a discussion of benefits and limitations of the method. This approach allows one to explore the physics around the MBL transition in systems much larger than those accessible to exact diagonalization. System sizes and length scales that can be controllably accessed by the MPS approach are comparable to those studied in state-of-the-art experiments. Results for 1D, quasi-1D, and 2D random systems, as well as 1D quasi-periodic systems are presented. On time scales explored (up to $t \approx 300$ in units set by the hopping amplitude), a slow, subdiffusive transport in a rather broad disorder range on the ergodic side of the MBL transition is found. For 1D random spin chains, which serve as a "standard model" of the MBL transition, the MPS study demonstrates a substantial drift of the critical point $W_c(L)$ with the system size $L$: while for $L \approx 20$ we find $W_c \approx 4$, as also given by exact diagonalization, the MPS results for $L = 50$--100 provide evidence that the critical disorder saturates, in the large-$L$ limit, at $W_c \approx 5.5$. For quasi-periodic systems, these finite-size effects are much weaker, which suggests that they can be largely attributed to rare events. For quasi-1D ($d\times L$, with $d \ll L$) and 2D ($L\times L$) random systems, the MPS data demonstrate an unbounded growth of $W_c$ in the limit of large $d$ and $L$, in agreement with analytical predictions based on the rare-event avalanche theory.