Skip to content
Tutorials
Density Matrix Renormalization Group (DMRG)

Density Matrix Renormalization Group (DMRG)

The density-matrix renormalization group (DMRG) finds accurate approximations to the ground state (and a few low-lying excited states) of one-dimensional quantum lattice models by iteratively truncating the Hilbert space to its most relevant DD-dimensional subspace. These tutorials work through the ALPS dmrg application on the spin-1/2 and spin-1 antiferromagnetic Heisenberg chains, a pair of models that look superficially similar but differ fundamentally in their low-energy physics, making them an ideal testbed for the method.

DMRG was introduced by Steven White in two seminal papers, Density matrix formulation for quantum renormalization groups (Phys. Rev. Lett. 69, 2863, 1992) and Density-matrix algorithms for quantum renormalization groups (Phys. Rev. B 48, 10345, 1993), which respectively laid out the method and its finite-system refinement into the algorithm used by ALPS today. See the DMRG reference page for further background and references.

Introduction

  • DMRG-01 Introduction — introduces the dmrg executable and the DMRG algorithm (infinite- and finite-system sweeps, truncation error) and its control parameters.

Model Physics and Ground State Energies

  • DMRG-02 Heisenberg Spin Chains — surveys the physics of the two models in depth: the critical, gapless spin-1/2 chain solvable by the Bethe ansatz, and the gapped, non-critical spin-1 (Haldane) chain, with the benchmark values used throughout the rest of the series.
  • DMRG-03 Ground State Energies — runs the first dmrg calculations, computing ground state energies of the spin-1/2 and spin-1 chains at fixed length and extrapolating to the energy per site (or bond) in the thermodynamic limit.

Excitations and Correlations

  • DMRG-04 Gaps — computes the singlet-triplet gap of the spin-1/2 chain and the Haldane gap of the spin-1 chain at finite length, and extrapolates both to the thermodynamic limit.
  • DMRG-05 Local Observables — uses the local magnetization profile to distinguish boundary from bulk excitations in the spin-1 chain, a subtlety arising from DMRG’s preference for open boundary conditions.
  • DMRG-06 Correlations — computes spin-spin correlation functions, extracting the critical power-law exponent of the spin-1/2 chain and the correlation length of the spin-1 chain.