The density-matrix renormalization group

TL;DR

The density-matrix renormalization group (DMRG) is a highly precise numerical method for studying low-dimensional strongly correlated quantum systems, with broad applications across physics and chemistry, based on efficient Hilbert space truncation.

Contribution

This paper reviews the DMRG algorithm, its theoretical foundations, and its diverse applications, highlighting its status as the preferred method for one-dimensional quantum systems.

Findings

Achieved unprecedented precision in 1D quantum systems

Extended DMRG applications to 2D systems, quantum chemistry, and more

Linked DMRG to matrix-product states and quantum information theory

Abstract

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly acquired the status of method of choice for numerical studies of one-dimensional quantum systems. Its applications to the calculation of static, dynamic and thermodynamic quantities in such systems are reviewed. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and non-equilibrium statistical physics, and time-dependent phenomena is discussed. This review also considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by DMRG.