The density-matrix renormalization group

2004/09/11 by Ulrich Schollwöck, Ulrich Schollwoeck · 1 voice · 3,389 citations
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #Density matrix #Density matrix renormalization group #Hilbert space #Mathematics #Matrix multiplication #Matrix product state #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum algorithm #Quantum dynamics #Quantum mechanics #Quantum process #Quantum, superfluid, helium dynamics #Statistical physics #Statistics #Truncation (statistics) #cond-mat.stat-mech #cond-mat.str-el

paper · pdf · doi:10.1103/revmodphys.77.259

published in Reviews of Modern Physics 77(1), 259-315 (American Physical Society) · accepted by Rev. Mod. Phys. in July 2004; scheduled to appear in the January 2005 issue

arxiv created 2004/09/11 · arxiv published 2004/09/11 · arxiv updated 2004/09/11 · openalex publication_date 2005/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

Citations

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Discussions

• The state is $|\psi_n\rangle \in \mathcal{H}$ , where $\mathcal{H}$ is the Hilbert space. The wavefunction is the inner product of a position "eigenstate" with the state: $\langle \vec{r}|\psi_n\rangl [stackexchange, 2 points]