The density-matrix renormalization group in the age of matrix product states

TL;DR

This paper reviews the density-matrix renormalization group (DMRG) method, emphasizing its formulation with matrix product states (MPS), and discusses potential future algorithmic improvements for simulating one-dimensional quantum systems.

Contribution

It provides a detailed exposition of DMRG in the MPS framework and explores new directions for algorithmic development and enhancement.

Findings

Deep understanding of DMRG's operation on MPS.

Clarification of implementation strategies for DMRG algorithms.

Identification of promising future algorithmic improvements.

Abstract

The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms.