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Physical review. B, Condensed matter

Numerical solution of larges=1/2 ands=1 Heisenberg antiferromagnetic spin chains using a truncated basis expansion

Mitchel D. Kovarik

Published Apr 1, 1990• 15 citations
Publisher

Abstract

A new method of determining the spectra of simple quantum-mechanical spin Hamiltonians by direct Hamiltonian-matrix diagonalization is presented. The method is illustrated by applying it to the isotropic one-dimensional Heisenberg antiferromagnet, for which calculations were performed for longer spin chains than by previous direct Hamiltonian-matrix diagonalizations. For spin s=1/2 it was possible to accurately determine the ground- and first-excited-state energies for chains of length N=32 and 64. For s=1 the longest chain for which a similar computation was performed was N=32; the singlet-triplet Haldane gap is found to be 0.421\ifmmode\pm\else\textpm\fi{}0.005 which is consistent with earlier Monte Carlo results. Long-spin-chain calculations are performed iteratively by solving smaller-chain problems, and constructing from their eigenvalues, and eigenstates, the longer-spin-chain basis and Hamiltonian matrices. The computations were performed on a small computer, which suggests that much larger problems could be handled with a parallel-processing supercomputer.

DOI
10.1103/physrevb.41.6889

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Numerical Solution of the spin-12\frac {1}{2}21​ Heisenberg Antiferromagnetic Chains Using a Projector Method
An efficient projector method is proposed to calculate the energy gap between the ground, state and the first excited state of the spin- Heisenberg antiferromagnetic chains. We exactly computed the two lowest-lying energies for the magnetization and the lowest-lying energy for on chains up to 16 spins. The lowest-lying energy stat6 for is a singlet ground state. The second lowest-lying energy state for and the lowest-lying energy states for are triplet degenerate, which are the first excited energy states. We also imposed the Monte Carlo technique to estimate the lowest-lying energies for both and 1 on longer chains up to 32 spins. We found that the singlet-triplet energy gap trends to zero as .
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An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-2 Heisenberg antiferromagnetic chains with linear sizes up to N=32. By direct extrapolation, we found that a Haldane gap exists, \ensuremath{\Delta}E=0.05, and the value of the ground-state energy per site is e0{\mathit{e}}_{0}e0​=-4.7545 as N\ensuremath{\rightarrow}\ensuremath{\infty}.
The properties of the spin-1/2 double chain Heisenberg antiferromagnet are studied by means of the numerical diagonalization and the projector Monte Carlo simulation. The model consists of the two spin 1/2 antiferromagnetic quantum Heisenberg chains coupled with each other via the ferromagnetic interchain coupling J ′ . It is shown that the first excited state has a gap for large enough J ′ . This gap continues to the Haldane gap in the limit J ′ →∞, where spins on the different chains form triplets on each site. On the other hand, the gap decreases rapidly between J ′ =1.0 and 0.6 and is undetectable for J ′ <0.6 within the present accuracy.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.