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

TL;DR

This paper compares the matrix product state formulations of Wilson's NRG and DMRG, showing their algebraic equivalence, reproducing NRG states with VMPS, and improving NRG results via variational optimization.

Contribution

It demonstrates the algebraic equivalence of NRG and DMRG in MPS form, and introduces a variational approach to enhance NRG results using VMPS.

Findings

All NRG eigenstates can be reproduced with VMPS.

Unfolded Wilson chain reduces computational costs.

NRG results can be systematically improved with variational optimization.

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 to 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.