The recent critical amplitude calculation of Barouch, McCoy, and Wu for the susceptibility of the Ising model on a square lattice is extended to yield amplitudes for the triangular, honeycomb, and kagom\'e lattices. The extension assumes the validity of the generalized law of corresponding states---as first postulated by Betts, Guttmann, and Joyce---for the two-dimensional Ising model.

Physical review. B, Solid stateJun 1, 197414 citations

Physics Letters A

The spin-spin correlation functions and susceptibility amplitudes for the two-dimensional Ising model; triangular lattice

H. G. Vaidya

Published May 1, 1976• 19 citations

Publisher

DOI: 10.1016/0375-9601(76)90432-1

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The high-temperature susceptibility of the spin- $S$ Ising model on the triangular net is analyzed as a function of $S$ using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.The high-temperature susceptibility of the spin-$S$ Ising model on the triangular net is analyzed as a function of $S$ using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.

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Exact Universal Amplitude Ratios for Two-Dimensional Ising Models and a Quantum Spin Chain

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Let ${f}_{N}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for ${a}_{k}$ and ${b}_{k}$ in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio ${b}_{k}/{a}_{k}$ is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.Let ${f}_{N}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for ${a}_{k}$ and ${b}_{k}$ in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio ${b}_{k}/{a}_{k}$ is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.

arXiv:cond-mat/0009102Physical Review LettersMay 28, 200162 citations

Spin-spin correlation function of the plane Ising lattice

R. Z. Bariev, M.P. Zhelifonov

Theoretical and Mathematical PhysicsOct 1, 19752 citations

Susceptibility amplitudes for the two-dimensional Ising model

Anthony John Guttmann

Physical review. B, Solid stateJun 1, 197414 citations

The high-temperature susceptibility of the spin-

S

Ising model on the triangular net is analyzed as a function of

S

using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.

Let

{f}_{N}

and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for

{a}_{k}

and

{b}_{k}

in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio

{b}_{k}/{a}_{k}

is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.