Matrix-product-state comparison of the numerical renormalization group and the variational formulation of the density-matrix renormalization group

2008/04/01 by Hamed Saberi, Andreas Weichselbaum, Jan von Delft · 5 citations
Mathematics · Physics and Astronomy · #Density matrix renormalization group #Eigenvalues and eigenvectors #Mathematical physics #Mathematics #Matrix (chemical analysis) #Matrix multiplication #Matrix product state #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum and electron transport phenomena #Quantum many-body systems #Quantum mechanics #Renormalization group #cond-mat.mes-hall #cond-mat.str-el

paper · pdf · doi:10.1103/physrevb.78.035124

published as Phys. Rev. B 78, 035124 (2008) · 19 pages, 14 figures

arxiv created 2008/04/01 · openalex publication_date 2008/07/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Wilson's numerical renormalization group (NRG) method for solving quantum impurity models yields a set of energy eigenstates that have the form of matrix product states (MPS). White's density-matrix renormalization group (DMRG) for treating quantum lattice problems can likewise be reformulated in terms of MPS. Thus, the latter constitute a common algebraic structure for both approaches. We exploit this fact to compare the NRG approach for the single-impurity Anderson model with a variational matrix product state approach (VMPS), equivalent to single-site DMRG. For the latter, we use an ``unfolded'' Wilson chain, which brings about a significant reduction in numerical costs compared to those of NRG. We show that all NRG eigenstates (kept and discarded) can be reproduced using VMPS, and compare the difference in truncation criteria, sharp vs smooth in energy space, of the two approaches. Finally, we demonstrate that NRG results can be improved upon systematically by performing a variational optimization in the space of variational matrix product states, using the states produced by NRG as input.

Citations

• Kondo effect in quantum dots
• Numerical renormalization group method for quantum impurity systems
• Density Matrix Renormalization Group and Periodic Boundary Conditions: A Quantum Information Perspective
• Density-matrix algorithms for quantum renormalization groups
• High-energy dynamics of the single-impurity Anderson model
• The renormalization group: Critical phenomena and the Kondo problem
• Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains
• Real-Time Dynamics in Quantum-Impurity Systems: A Time-Dependent Numerical Renormalization-Group Approach

Cited by

• Two-bath spin-boson model: Phase diagram and critical properties
• Ancilla-assisted sequential approximation of nonlocal unitary operations
• Chebyshev expansion for impurity models using matrix product states
• Quantum tricritical point emerging in the spin-boson model with two dissipative spins in staggered biases
• Uncontrolled disorder effects in fabricating photonic quantum simulators on a kagome geometry: A projected-entangled-pair-state versus exact-diagonalization analysis

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• The density-matrix renormalization group in the age of matrix product states
• Accurate variational electronic structure calculations with the density matrix renormalization group
• Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems
• An Impurity Solver Using the Time-Dependent Variational Matrix Product State Approach
• Matrix Product Renormalization Group: Potential Universal Quantum Many-Body Solver