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

TL;DR

This article presents a thorough synthesis of density-matrix renormalization group methods through the lens of matrix product states, clarifying how DMRG can be implemented and extended entirely within MPS/MPO formalisms. It lays out canonical MPS representations, efficient contractions, and compression schemes, then builds up to ground-state and time-evolution algorithms, including real and imaginary time dynamics and finite-temperature approaches. The work also connects DMRG with NRG, infinite-size methods, and advanced strategies to push time scales and thermal simulations, highlighting entanglement as the central bottleneck and proposing concrete methods to mitigate it. Overall, the paper maps a cohesive, modular toolkit for one-dimensional quantum many-body problems and points toward promising directions for further algorithmic development and practical implementations.

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.

Figures (63)

• Figure 1: Our toy model: a chain of length $L$ with open ends, where a spin-$\frac{1}{2}$ sits on each site and interacts with its nearest neighbours.
• Figure 2: The left and right half of the figure present the iterations taken in the infinite-system and finite-system DMRG procedures respectively. In both cases, new blocks are formed from integrating a site into a block, with a state space truncation according to the density-matrix prescription of DMRG. Whereas in the infinite-system version this growth happens on both sides of the chain, leading to chain growth, in the finite-system algorithm it happens only for one side at the expense of the other, leading to constant chain length.
• Figure 3: Plaquette current on a $t$-$J$-$V$-$V'$-ladder, $J=0.4t$, $V=3t$, $V'=t$ ($V$ nearest, $V'$ next-nearest neighbour interaction) at hole doping $\delta=0.1$, system size $2\times 60$, as induced by a finite boundary current on rung $1$. The absolute current strength is shown; whereas infinite-system DMRG and the first sweep indicate the generation of a long-ranged pattern, a fully converged calculation (here after 6 to 7 sweeps) reveals an exponential decay into the bulk. Taken from Ref. Schollwoeck03.
• Figure 4: Resulting matrix shapes from a singular value decomposition (SVD), corresponding to the two rectangular shapes that can occur. The singular value diagonal serves as a reminder that in $M=USV^{\ \dagger}$$S$ is purely non-negative diagonal.
• Figure 5: Graphical representation of an iterative construction of an exact MPS representation of an arbitrary quantum state by a sequence of singular value decompositions.