Physical review. B, Condensed matter

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

Minoru Takahashi

Published Sep 1, 1988• 71 citations

Abstract

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

DOI: 10.1103/physrevb.38.5188

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

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Physical Review LettersMar 26, 1990101 citations

Spin correlation function of theS=1antiferromagnetic Heisenberg chain by the large-cluster-decomposition Monte Carlo method

K. Nomura

We calculate correlation functions of a one-dimensional spin S=1 antiferromagnetic Heisenberg model by the large-cluster-decomposition Monte Carlo method. We find that the correlation functions are well approximated by modified Bessel functions. This result supports Haldane's conjecture, and the correlation length agrees with that obtained from spin-wave theory using elementary excitation data.

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Quantum Monte Carlo Approach to Elementary Excitations of Antiferromagnetic Heisenberg Chains

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Physical Review LettersOct 30, 199551 citations

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

Physical review. B, Condensed matterDec 1, 198832 citations

Effects of edges inS=1 Heisenberg antiferromagnetic chains

Seiji Miyashita, Shoji Yamamoto

Physical review. B, Condensed matterJul 1, 1993123 citations

Ground state of the random-bond spin-1 Heisenberg chain

Sara Bergkvist, Patrik Henelius, Anders Rosengren

arXiv:cond-mat/0206092Physical review. B, Condensed matterOct 9, 200225 citations

Physical review. B, Condensed matter

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

Minoru Takahashi

Published Sep 1, 1988• 71 citations

Publisher

Abstract

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}

DOI: 10.1103/physrevb.38.5188

Related papers

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

Shoudan Liang

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.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.

Physical Review LettersMar 26, 1990101 citations

Spin correlation function of theS=1antiferromagnetic Heisenberg chain by the large-cluster-decomposition Monte Carlo method

K. Nomura

Physical review. B, Condensed matterAug 1, 198997 citations

Quantum Monte Carlo Approach to Elementary Excitations of Antiferromagnetic Heisenberg Chains

Shoji Yamamoto

A world-line Monte Carlo approach is proposed to calculate the elementary excitations of quantum systems. The lowest energies as a function of momentum are successfully extracted from an imaginary-time correlation function evaluated at a sufficiently low temperature. The method is applied to the antiferromagnetic Heisenberg chains of $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2},1,\frac{3}{2},$ and $2$ with length up to 64. The spectra for $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ and 1 coincide well with the result by des Cloizeaux and Pearson and with the result by Takahashi, respectively. The spectra for $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{3}{2}$ and 2 turn out to be massless and massive, respectively. The $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ Haldane gap is estimated to be $0.049\ifmmode\pm\else\textpm\fi{}0.018$.A world-line Monte Carlo approach is proposed to calculate the elementary excitations of quantum systems. The lowest energies as a function of momentum are successfully extracted from an imaginary-time correlation function evaluated at a sufficiently low temperature. The method is applied to the antiferromagnetic Heisenberg chains of $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2},1,\frac{3}{2},$ and $2$ with length up to 64. The spectra for $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ and 1 coincide well with the result by des Cloizeaux and Pearson and with the result by Takahashi, respectively. The spectra for $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{3}{2}$ and 2 turn out to be massless and massive, respectively. The $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ Haldane gap is estimated to be $0.049\ifmmode\pm\else\textpm\fi{}0.018$.

Physical Review LettersOct 30, 199551 citations

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

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.

Physical review. B, Condensed matterDec 1, 198832 citations

Effects of edges inS=1 Heisenberg antiferromagnetic chains

Seiji Miyashita, Shoji Yamamoto

Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈 ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ 〉 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.Properties of the ground state of open chains of the S=1 Heisenberg antiferromagnet are studied by a quantum Monte Carlo method. Size (length) dependences of the ground-state energy, the energy gap, the staggered magnetization 〈${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$〉 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.

Physical review. B, Condensed matterJul 1, 1993123 citations

Ground state of the random-bond spin-1 Heisenberg chain

Sara Bergkvist, Patrik Henelius, Anders Rosengren

Stochastic series expansion quantum Monte Carlo is used to study the ground state of the antiferromagnetic spin-1 Heisenberg chain with bond disorder. Typical spin- and string-correlation functions behave in accordance with real-space renormalization group predictions for the random-singlet phase. The average string-correlation function decays algebraically with an exponent of $\ensuremath{-}0.378(6),$ in very good agreement with the prediction of $\ensuremath{-}(3\ensuremath{-}\sqrt{5})/2\ensuremath{\simeq}\ensuremath{-}0.382,$ while the average spin-correlation function is found to decay with an exponent of about $\ensuremath{-}1,$ quite different from the expected value of $\ensuremath{-}2.$ By implementing the concept of directed loops for the spin-1 chain we show that autocorrelation times can be reduced by up to two orders of magnitude.Stochastic series expansion quantum Monte Carlo is used to study the ground state of the antiferromagnetic spin-1 Heisenberg chain with bond disorder. Typical spin- and string-correlation functions behave in accordance with real-space renormalization group predictions for the random-singlet phase. The average string-correlation function decays algebraically with an exponent of $\ensuremath{-}0.378(6),$ in very good agreement with the prediction of $\ensuremath{-}(3\ensuremath{-}\sqrt{5})/2\ensuremath{\simeq}\ensuremath{-}0.382,$ while the average spin-correlation function is found to decay with an exponent of about $\ensuremath{-}1,$ quite different from the expected value of $\ensuremath{-}2.$ By implementing the concept of directed loops for the spin-1 chain we show that autocorrelation times can be reduced by up to two orders of magnitude.

arXiv:cond-mat/0206092Physical review. B, Condensed matterOct 9, 200225 citations

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}

^{4}\mathrm{He}

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2},1,\frac{3}{2},

and

2

with length up to 64. The spectra for

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

and 1 coincide well with the result by des Cloizeaux and Pearson and with the result by Takahashi, respectively. The spectra for

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{3}{2}

and 2 turn out to be massless and massive, respectively. The

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2

Haldane gap is estimated to be $0.049\ifmmode\pm\else\textpm\fi{}0.018$.

{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.