Comment on "Equivalence of the variational matrix product method and the density matrix renormalization group applied to spin chains"

TL;DR

This paper critiques prior comparisons between the matrix product method and the density matrix renormalization group, clarifying that previous DMRG results were inaccurate and demonstrating that both methods yield equivalent energies with proper superblock configurations.

Contribution

The authors correct previous DMRG results and explicitly show that with proper superblock construction, DMRG and MPM produce identical energies, confirming their theoretical equivalence.

Findings

DMRG results in prior studies were in error.

Proper superblock configuration yields identical energies for DMRG and MPM.

Artificial bound states in DMRG are very small with standard configurations.

Abstract

Dukelsky, Mart\'in-Delgado, Nishino and Sierra (Europhys. Lett., 43, 457 (1998) - hereafter referred to as DMNS) investigated the matrix product method (MPM), comparing it with the infinite-size density matrix renormalization group (DMRG). For equivalent basis size, the MPM produces an improved variational energy over that produced by DMRG and, unlike DMRG, produces a translationally-invariant wavefunction. The DMRG results presented were significantly worse than the MPM, caused by a shallow bound state appearing at the join of the two DMRG blocks. They also suggested that the DMRG results can be improved by using an alternate superblock construction $[B] \bullet [B]$ for the last few steps of the calculation. In this comment, we show that the DMRG results presented by DMNS are in error and the artificial bound state produced by the standard superblock configuration is very small even for $m=2$ states kept. In addition, we calculate explicitly the energy and wavefunction for the $[B] \bullet [B]$ superblock structure and verify that the energy coincides with that of the MPM, as conjectured by S. Ostlund and S. Rommer (Phys. Rev. Lett., 75, 3537 (1995)).