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Physical review. B, Condensed matter

Numerical solution of spin systems and the S=(1/2 Heisenberg antiferromagnet using guided random walks

Ted Barnes, Geoff J. Daniell

Published Mar 1, 1988• 26 citations
Publisher

Abstract

In this paper we describe a numerical method for measuring ground-state expectation values and energies of quantum spin systems. The technique exploits the similarity between the Euclidean-time Schr\"odinger equation and a random walk in state space to generate configurations with a weighted distribution proportional to \ensuremath{\Vert}${\ensuremath{\psi}}_{0}$${\ensuremath{\Vert}}^{2}$. A model application to a single spin is described first, followed by application of the method to the isotropic spin-(1/2 Heisenberg antiferromagnet. For the antiferromagnet we measure the singlet ground-state energy and the singlet-triplet energy gap on chains of up to 48 spins. These results are shown to be in good agreement with energies obtained using Lanczos techniques on chains of up to 20 spins for the energy gap and 24 spins for the ground-state energy.

DOI
10.1103/physrevb.37.3637

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In this paper we describe a numerical method for measuring ground-state expectation values and energies of quantum spin systems. The technique exploits the similarity between the Euclidean-time Schr"odinger equation and a random walk in state space to generate configurations with a weighted distribution proportional to \ensuremath{\Vert}{\ensuremath{\psi}}_{0}$${\ensuremath{\Vert}}^{2}. A model application to a single spin is described first, followed by application of the method to the isotropic spin-(1/2 Heisenberg antiferromagnet. For the antiferromagnet we measure the singlet ground-state energy and the singlet-triplet energy gap on chains of up to 48 spins. These results are shown to be in good agreement with energies obtained using Lanczos techniques on chains of up to 20 spins for the energy gap and 24 spins for the ground-state energy.
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 .
Abstract The quantum mechanical underpinnings of magnetism are explored via the Heisenberg model of antiferromagnetism. The Lanczos algorithm is developed and applied to obtain ground state properties of the anisotropic antiferromagnetic Heisenberg spin chain. In particular, the phase diagram for the system magnetization is determined. A quantum Monte Carlo method that is appropriate for discrete systems is also presented. The method leverages the similarity between the Schrödinger equation and the diffusion equation to compute energy levels. The formalism necessary to compute ground state matrix elements is also developed. Finally, the method is tested with an application to the spin chain.
We describe a numerical simulation of the isotropic spin-\textonehalf{} Heisenberg antiferromagnet on square lattices of size 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. A Hamiltonian Monte Carlo method is used to measure the energies of the lowest-lying singlet and triplet states, and we extrapolate these results to give estimates of the properties of large antiferromagnets. Our result for the ground-state energy per spin is $\frac{{E}_{0}}{{N}^{2}}=\ensuremath{-}0.672\ifmmode\pm\else\textpm\fi{}0.001$ for an infinite system. Within our statistical accuracy, the singlet-triplet gap behaves as ${E}_{1}\ensuremath{-}{E}_{0}\ensuremath{\approx}\frac{2.1}{N}$, which implies that the infinite system is gapless.
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.
Numerical Solution of the spin-32\frac {3}{2}23​ Heisenberg Antiferromagnetic Chains Using a Projector Method
An efficient projector method is used to calculate the energies of the ground state and the first excited state, and hence the energy gap between them, of the spin- Heisenberg antiferromagnetic chains with linear sizes up to . For the chains up to the exact results are obtained, and for longer chains up to they are estimated by imposing on Monte Carlo technique. It is found that the size dependence of the ground state energy per site has the form of , and the value of the ground state energy per site as is . Our results show that the energy gap trends to zero as , which is consistent with the rigorous proof.