Physical review. B, Condensed matter

Monte Carlo simulation of the S=(1/2 antiferromagnetic Heisenberg chain and the long-distance behavior of the spin-correlation function

Kenn Kubo, T. A. Kaplan, J. Borysowicz

Published Dec 1, 1988• 32 citations

Abstract

We present a new Monte Carlo simulation of the correlation function C(r) between two spins separated by a distance r in the ground state of the spin-1/2 Heisenberg chain. The calculation is based on the Sutherland mapping between the spin-chain ground state and the six-vertex model at its critical temperature. New results are obtained for rings with the number of spins N=32 and 40. We give evidence that the asymptotic decay of C(r) at large r is slower than the 1/r behavior of Luther and Peschel, and that this might be attributed to a logarithmic factor, i.e., C(r)\ensuremath{\sim}(A/r)(lnr/${r}_{0}$${)}^{\ensuremath{\sigma}}$, with \ensuremath{\sigma}\ensuremath{\approxeq}0.2 to 0.3.

DOI: 10.1103/physrevb.38.11550

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Physical review. B, Condensed matter

Monte Carlo simulation of the S=(1/2 antiferromagnetic Heisenberg chain and the long-distance behavior of the spin-correlation function

Kenn Kubo, T. A. Kaplan, J. Borysowicz

Published Dec 1, 1988• 32 citations

Publisher

Abstract

{r}_{0}$${)}^{\ensuremath{\sigma}}

, with \ensuremath{\sigma}\ensuremath{\approxeq}0.2 to 0.3.

DOI: 10.1103/physrevb.38.11550

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Spin-correlation function of theS=1antiferromagnetic Heisenberg chain atT=0

Minoru Takahashi

The correlation function \ensuremath{\rho}(l)=〈 ${S}_{i}^{z}$ ${S}_{i+l}^{z}$ 〉 is calculated for the spin-1 Heisenberg antiferromagnetic chain (H=J${\ensuremath{\Sigma}}_{i}$ ${\mathrm{scrS}}_{i}$ ${\mathrm{scrS}}_{i+1}$ , ${\mathrm{scrS}}_{N+1}$ \ensuremath{\equiv} ${\mathrm{scrS}}_{1}$ ) at the ground state. Using the Monte Carlo method of Hirsch, Sugar, Scalopino, and Blankenbecler we find that \ensuremath{\rho}(l) decays exponentially in contrast to the S=half case where \ensuremath{\rho}(l) decays algebraically. This fact coincides with Haldane's prediction and recent numerical calculations. We calculate the upper bound of elementary excitation from the structure factor using a variational method which resembles the Feynman theory for elementary excitation of liquid $^{4}\mathrm{He}$ .The correlation function \ensuremath{\rho}(l)=〈${S}_{i}^{z}$${S}_{i+l}^{z}$〉 is calculated for the spin-1 Heisenberg antiferromagnetic chain (H=J${\ensuremath{\Sigma}}_{i}$${\mathrm{scrS}}_{i}$${\mathrm{scrS}}_{i+1}$, ${\mathrm{scrS}}_{N+1}$\ensuremath{\equiv}${\mathrm{scrS}}_{1}$) at the ground state. Using the Monte Carlo method of Hirsch, Sugar, Scalopino, and Blankenbecler we find that \ensuremath{\rho}(l) decays exponentially in contrast to the S=half case where \ensuremath{\rho}(l) decays algebraically. This fact coincides with Haldane's prediction and recent numerical calculations. We calculate the upper bound of elementary excitation from the structure factor using a variational method which resembles the Feynman theory for elementary excitation of liquid $^{4}\mathrm{He}$.

Physical review. B, Condensed matterSep 1, 198871 citations

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 and the spin-correlation function 〈

{\mathit{S}}_{1}^{\mathit{z}}

{\mathit{S}}_{\mathit{i}}^{\mathit{z}}

〉 are investigated under free, fixed, and periodic boundary conditions. The size dependences are significant in chains with less than 25 spins, where the magnitude of the staggered magnetization is rather large. However, the fourfold degeneracy of the ground state and the exponential decay of correlations that are inherent to the Haldane system are recovered when the chain length becomes long enough. Both the staggered magnetization and the spin-correlation function decay exponentially. The correlation length is about six in lattice spacing, which agrees with the value obtained in the bulk.

The correlation function \ensuremath{\rho}(l)=〈

{S}_{i}^{z}

{S}_{i+l}^{z}

〉 is calculated for the spin-1 Heisenberg antiferromagnetic chain (H=J${\ensuremath{\Sigma}}_{i}$

{\mathrm{scrS}}_{i}

{\mathrm{scrS}}_{i+1}

{\mathrm{scrS}}_{N+1}

\ensuremath{\equiv}

{\mathrm{scrS}}_{1}

) at the ground state. Using the Monte Carlo method of Hirsch, Sugar, Scalopino, and Blankenbecler we find that \ensuremath{\rho}(l) decays exponentially in contrast to the S=half case where \ensuremath{\rho}(l) decays algebraically. This fact coincides with Haldane's prediction and recent numerical calculations. We calculate the upper bound of elementary excitation from the structure factor using a variational method which resembles the Feynman theory for elementary excitation of liquid

^{4}\mathrm{He}