The density-matrix renormalization group
Ulrich Schollwoeck
TL;DR
DMRG targets efficient truncation of the Hilbert space for strongly correlated quantum systems, achieving unprecedented accuracy in one-dimensional models. It uses density-matrix–based truncation to build an optimal renormalization flow, enabling high-precision results for large 1D systems while avoiding the sign problem. The paper surveys the algorithmic foundations, the relationship to matrix-product states and quantum information theory, and a broad range of applications—including static, dynamic, and thermodynamic properties, with extensions to two-dimensional systems, quantum chemistry, small grains, nuclei, and nonequilibrium/time-dependent phenomena. It highlights both the method's power and its limitations, pointing toward further algorithmic and application developments in higher dimensions and broader fields.
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.