Matrix multiplication

Quantum mechanics

Approx

Algorithm

Product (mathematics)

Matrix product state

Pure mathematics

Statistics

Materials science

Operating system

Eigenvalues and eigenvectors

Range (aeronautics)

Computer science

Limit (mathematics)

Classical mechanics

Statistical physics

Random matrix

Central limit theorem

Matrix (chemical analysis)

Physics

Amplitude

Mathematics

Condensed matter physics

Mathematical analysis

Geometry

Quantum

Composite material

Ergodic theory

Thermodynamic limit

State (computer science)

Annals of Physics

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≈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 Wc(L) with the system size L: while for L≈20 we find Wc≈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 Wc≈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×L, with d≪L) and 2D (L×L) random systems, the MPS data demonstrate an unbounded growth of Wc in the limit of large d and L, in agreement with analytical predictions based on the rare-event avalanche theory.

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