EPJ B Colloquium - Density-matrix renormalization group: a pedagogical introduction

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Published on 17 August 2023

The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm have been put forth, that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions.

In a new Colloquium published in EPJB, G. Catarina and Bruno Murta provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. This pedagogical review is expected to find wide readership amongst students and researchers who are taking their first steps in numerical simulations via DMRG.