Density-matrix renormalization group: a pedagogical introduction

Abstract

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 that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. 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. Here, we 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. We expect this pedagogical review to find wide readership amongst students and researchers who are taking their first steps in numerical simulations via DMRG.

30 pages, 17 figures

References in corpus (15)

• The density-matrix renormalization group in the age of matrix product states
• The numerical renormalization group method for quantum impurity systems
• Real time evolution using the density matrix renormalization group
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• DMRG and periodic boundary conditions: a quantum information perspective
• From density-matrix renormalization group to matrix product states
• Entanglement spectrum in one-dimensional systems
• New Trends in Density Matrix Renormalization
• Minimally Entangled Typical Thermal State Algorithms
• Applying matrix product operators to model systems with long-range interactions
• A Strictly Single-Site DMRG Algorithm with Subspace Expansion
• Generic Construction of Efficient Matrix Product Operators
• Finite automata for caching in matrix product algorithms
• Tensor operators: constructions and applications for long-range interaction systems
• Density matrix renormalization group study of a three-orbital Hubbard model with spin-orbit coupling in one dimension

Cited by in corpus (9)

• Conformational tuning of magnetic interactions in coupled nanographenes
• A tensor network approach to sensing quantum light-matter interactions
• Charge and spin properties of a generalized Wigner crystal realized in the moiré WSe${}_2$/WS${}_2$ heterobilayer
• Modified large-$N$ approach to gapless spin liquids, magnetic orders, and dynamics: Application to triangular lattice antiferromagnets
• Interplay between local and non-local frustration in the 1D ANNNI chain I -- The even case
• Quantum Coherence of Topologically Frustrated Spin Chains
• Simulating Quantum Turbulence with Matrix Product States
• The Software Landscape for the Density Matrix Renormalization Group
• Numerically efficient unitary evolution for Hamiltonians beyond nearest-neighbors