Physical Review Letters

Monte Carlo calculations of the correlation functions for Heisenberg spin chains atT=0

Shoudan Liang

Published Mar 26, 1990• 101 citations

Abstract

We compute the ground-state spin-spin correlation function for the one-dimensional antiferromagnetic Heisenberg model of spin 1/2, 1, and 3/2. The spin-1 chain has a disordered ground state with a correlation length of 6.2 lattice spacings whereas the spin-1/2 and the spin-3/2 chains both have (ln(r)${)}^{\mathrm{\ensuremath{\sigma}}}$/r decaying correlation functions. The logarithmic corrections are different for the spin-1/2 and the spin-3/2 chains. Finite-size scaling is used to analyze the data for chains of up to 128 spins.

DOI: 10.1103/physrevlett.64.1597

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Physical Review Letters

Monte Carlo calculations of the correlation functions for Heisenberg spin chains atT=0

Shoudan Liang

Published Mar 26, 1990• 101 citations

Publisher

Abstract

{)}^{\mathrm{\ensuremath{\sigma}}}

/r decaying correlation functions. The logarithmic corrections are different for the spin-1/2 and the spin-3/2 chains. Finite-size scaling is used to analyze the data for chains of up to 128 spins.

DOI: 10.1103/physrevlett.64.1597

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

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.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.

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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 $^{4}\mathrm{He}$.

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{)}^{\mathrm{\ensuremath{\sigma}}}

/r decaying correlation functions. The logarithmic corrections are different for the spin-1/2 and the spin-3/2 chains. Finite-size scaling is used to analyze the data for chains of up to 128 spins.

{r}_{0}

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

{\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}