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Australian Journal of Physics

Density Matrix Renormalisation Group Method and Symmetries of the Hamiltonian

Ian P. McCulloch, M. Gulácsi

Published Feb 23, 2001• 38 citations
Publisher

Abstract

Substantial improvements in the computational effort in a density matrix renormalisation program can be made by utilising symmetries of the Hamiltonian. Extra quantum numbers are always desirable to include in the calculation, since it allows the Hilbert space of the superblock to be refined. Since the speed of the calculation is approximately O ( n 3 ) in superblock states, the speed increase in targeting a specific total spin state can be considerable. In this paper a new density matrix renormalisation algorithm is presented which conserves the total spin. The general procedure obtained works for any operator, even operators that do not commute with the Hamiltonian.

DOI
10.1071/ph00023

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Substantial improvements in the computational effort in a density matrix renormalisation program can be made by utilising symmetries of the Hamiltonian. Extra quantum numbers are always desirable to include in the calculation, since it allows the Hilbert space of the superblock to be refined. Since the speed of the calculation is approximately O ( n 3 ) in superblock states, the speed increase in targeting a specific total spin state can be considerable. In this paper a new density matrix renormalisation algorithm is presented which conserves the total spin. The general procedure obtained works for any operator, even operators that do not commute with the Hamiltonian.
The infinite Density Matrix Renormalisation Group (iDMRG) algorithm is a highly successful numerical algorithm for the study of low-dimensional quantum systems. However, it may fail at high filling fractions when exploiting global symmetries such as particle number conservation to increase simulation speed. This paper introduces a modified iDMRG algorithm capable of exploiting these symmetries while still producing accurate results at high filling fractions, and at a substantially reduced computational cost.
We have applied the momentum space version of the density-matrix renormalization-group method (k-DMRG) in quantum chemistry in order to study the accuracy of the algorithm in this new context. We have shown numerically that it is possible to determine the desired accuracy of the method in advance of the calculations by dynamically controlling the truncation error and the number of block states using a novel protocol that we dubbed dynamical block state selection protocol. The relationship between the real error and truncation error has been studied as a function of the number of orbitals and the fraction of filled orbitals. We have calculated the ground state of the molecules CH2,{\mathrm{CH}}_{2},CH2​, H2O,{\mathrm{H}}_{2}\mathrm{O},H2​O, and F2{\mathrm{F}}_{2}F2​ as well as the first excited state of CH2.{\mathrm{CH}}_{2}.CH2​. Our largest calculations were carried out with 57 orbitals, the largest number of block states was 1500--2000, and the largest dimensions of the Hilbert space of the superblock configuration was 800 000--1 200 000.
We adapt White's density matrix renormalisation group (DMRG) to the direct study of critical phenomena. We use the DMRG to generate transformations in the space of coupling constants. We postulate that a study of density matrix eigenvalues leads to a natural identification of renormalised blocks, operators and Hamiltonians. We apply the scheme to the phase transition in the anisotropic spin-1/2 Heisenberg chain. In the simplest case where the two most probable states in odd sized blocks are used to construct approximate renormalisation group transformations, we find qualitative improvement upon the standard real space renormalisation group method for the thermal exponent ννν.
We parallelize density-matrix renormalization group to directly extend it to 2-dimensional (nnn-leg) quantum lattice models. The parallelization is made mainly on the exact diagonalization for the superblock Hamiltonian since the part requires an enormous memory space as the leg number nnn increases. The superblock Hamiltonian is divided into three parts, and the correspondent superblock vector is transformed into a matrix, whose elements are uniformly distributed into processors. The parallel efficiency shows a high rate as the number of the states kept mmm increases, and the eigenvalue converges within only a few sweeps in contrast to the multichain algorithm.
Abstract. – We describe here the extension of the density matrix renormalization group algorithm to the case where the Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the the one-dimensional Hubbard model as the symmetry group is enlarged from U(1) × U(1), up to SU(2) × SU(2). In past years, the density matrix renormalization group (DMRG) method [1] has been extensively used to study one and two dimensional strongly correlated electron systems [2]. This method became very popular when it was realized that it enabled a level of numerical accuracy for one dimensional systems that was not possible using other methods [3]. One major drawback of DMRG is that calculations are performed in a subspace of purely Abelian symmetries, such as the U(1) symmetries of total particle number and the z component of the total spin. Thus one can only obtain a few states in different total particle