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Wilson's numerical renormalization group (NRG) method for solving quantumimpurity models yields a set of energy eigenstates that have the form of matrixproduct states (MPS). White's density matrix renormalization group (DMRG) fortreating 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-impurityAnderson model to a variational matrix product state approach (VMPS),equivalent to single-site DMRG. For the latter, we use an ``unfolded'' Wilsonchain, which brings about a significant reduction in numerical costs comparedto those of NRG. We show that all NRG eigenstates (kept and discarded) can bereproduced using VMPS, and compare the difference in truncation criteria, sharpvs. smooth in energy space, of the two approaches. Finally, we demonstrate thatNRG results can be improved upon systematically by performing a variationaloptimization in the space of variational matrix product states, using thestates produced by NRG as input.

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