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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Interaction-round-a-face density-matrix renormalization-group method applied to rotational-invariant quantum spin chains

Tatsuaki Wada

Published Mar 1, 2000• 34 citations
Publisher

Abstract

An interaction-round-a-face density-matrix renormalization-group (IRF-DMRG) method is developed for higher integer spin chain models which are rotational invariant. The expressions of the IRF weights associated with the nearest-neighbor spin-$S$ interaction ${\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}$ are explicitly derived. Using the IRF-DMRG with these IRF weights, the Haldane gaps $\ensuremath{\Delta}$ and the ground state energy densities ${e}_{0}$ for both $S=1$ and $S=2$ isotropic antiferromagnetic Heisenberg quantum spin chains are calculated by keeping up to only $m=90$ states.

DOI
10.1103/physreve.61.3199

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An interaction-round-a-face density-matrix renormalization-group (IRF-DMRG) method is developed for higher integer spin chain models which are rotational invariant. The expressions of the IRF weights associated with the nearest-neighbor spin-SSS interaction Si\ensuremath⋅Si+1{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}Si​\ensuremath⋅Si+1​ are explicitly derived. Using the IRF-DMRG with these IRF weights, the Haldane gaps \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ and the ground state energy densities e0{e}_{0}e0​ for both S=1S=1S=1 and S=2S=2S=2 isotropic antiferromagnetic Heisenberg quantum spin chains are calculated by keeping up to only m=90m=90m=90 states.
An interaction-round-a-face density-matrix renormalization-group (IRF-DMRG) method is developed for higher integer spin chain models which are rotational invariant. The expressions of the IRF weights associated with the nearest-neighbor spin-S interaction Si⋅Si+1{\bf S}_{i} \cdot {\bf S}_{i+1}Si​⋅Si+1​ are derived explicitly. Using the IRF-DMRG with these IRF weights, the Haldane gaps ΔΔΔ and the ground state energy densities e0e_{0}e0​ for both S=1 and S=2 isotropic antiferromagnetic Heisenberg quantum spin chains are calculated by keeping up to only m=90 states.
Given a Hamiltonian with a continuous symmetry one can generally factorize that symmetry and consider the dynamics on invariant Hilbert Spaces. In Statistical Mechanics this procedure is known as the vertex-IRF map, and in certain cases, like rotational invariant Hamiltonians, can be implemented via group theoretical techniques. Using this map we translate the DMRG method, which applies to 1d vertex Hamiltonians, into a formulation adequate to study IRF Hamiltonians. The advantage of the IRF formulation of the DMRG method ( we name it IRF-DMRG), is that the dimensions of the Hilbert Spaces involved in numerical computations are smaller than in the vertex-DMRG, since the degeneracy due to the symmetry has been eliminated. The IRF-DMRG admits a natural and geometric formulation in terms of the paths or string algebras used in Exactly Integrable Systems and Conformal Field Theory. We illustrate the IRF-DMRG method with the study of the SOS model which corresponds to the spin 1/2 Heisenberg chain and the RSOS models with Coxeter diagram of type A, which correspond to the quantum group invariant XXZ chain.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
An efficient density matrix renormalization group (DMRG) algorithm is presented and applied to Y junctions, systems with three arms of nnn sites that meet at a central site. The accuracy is comparable to DMRG of chains. As in chains, new sites are always bonded to the most recently added sites and the superblock Hamiltonian contains only new or once renormalized operators. Junctions of up to $N=3n+1\ensuremath{\approx}500$ sites are studied with antiferromagnetic (AF) Heisenberg exchange JJJ between nearest-neighbor spins SSS or electron transfer ttt between nearest neighbors in half-filled Hubbard models. Exchange or electron transfer is exclusively between sites in two sublattices with ${N}_{A}\ensuremath{\ne}{N}_{B}$. The ground state (GS) and spin densities ${\ensuremath{\rho}}_{r}=\ensuremath{\langle}{S}_{r}^{z}\ensuremath{\rangle}$ at site rrr are quite different for junctions with S=1/2S=1/2S=1/2, 1, 3/2, and 2. The GS has finite total spin SG=2S(S){S}_{G}=2S(S)SG​=2S(S) for even (odd) NNN and for MG=SG{M}_{G}={S}_{G}MG​=SG​ in the SG{S}_{G}SG​ spin manifold, ${\ensuremath{\rho}}_{r}>0(<0)$ at sites of the larger (smaller) sublattice. S=1/2S=1/2S=1/2 junctions have delocalized states and decreasing spin densities with increasing NNN. S=1S=1S=1 junctions have four localized Sz=1/2{S}_{z}=1/2Sz​=1/2 states at the end of each arm and centered on the junction, consistent with localized states in S=1S=1S=1 chains with finite Haldane gap. The GS of S=3/2S=3/2S=3/2 or 2 junctions of up to 500 spins is a spin density wave with increased amplitude at the ends of arms or near the junction. Quantum fluctuations completely suppress AF order in S=1/2S=1/2S=1/2 or 1 junctions, as well as in half-filled Hubbard junctions, but reduce rather than suppress AF order in S=3/2S=3/2S=3/2 or 2 junctions.