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Canadian Journal of Physics

On the critical behaviour of the two dimensional spin XY model

Jos Rogiers

Published Mar 1, 1982• 2 citations
Publisher

Abstract

Transformation methods are used to analyse the series for the second order fluctuation of the transverse magnetization for the triangular and square lattices. For the triangular lattice some evidence is found for an exponential behaviour of this quantity near the critical point with a tentative estimate for the exponent [Formula: see text].

DOI
10.1139/p82-051

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In this paper we report analyses of high temperature series expansions for the spin [Formula: see text] XY model on the triangular and square lattices. Quantities for which series are analyzed include the fluctuation in the transverse magnetization, fourth order fluctuations in the same quantity, second and fourth moments of the transverse spin–spin correlations, specific heat, and entropy. The evidence favours a phase transition at a finite temperature with conventional power law critical singularities. Scaling seems to hold but hyperscaling seems to be violated. Estimates for critical exponents include γ = 2.50 ± 0.3. Δ = 2.38 ± 0.2, and ν = 143 ± 0.10. The specific heat exhibits no singular behaviour at Tc.

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Canadian Journal of Physics•Apr 1, 1978•15 citations
Transformation methods are used to analyse the series for the second order fluctuation of the transverse magnetization for the triangular and square lattices. For the triangular lattice some evidence is found for an exponential behaviour of this quantity near the critical point with a tentative estimate for the exponent [Formula: see text].
In this paper we report analyses of high temperature series expansions for the spin [Formula: see text] XY model on the triangular and square lattices. Quantities for which series are analyzed include the fluctuation in the transverse magnetization, fourth order fluctuations in the same quantity, second and fourth moments of the transverse spin–spin correlations, specific heat, and entropy. The evidence favours a phase transition at a finite temperature with conventional power law critical singularities. Scaling seems to hold but hyperscaling seems to be violated. Estimates for critical exponents include γ = 2.50 ± 0.3. Δ = 2.38 ± 0.2, and ν = 143 ± 0.10. The specific heat exhibits no singular behaviour at Tc.
The first nine coefficients have been derived in the exact high temperature expansion of the fluctuation in the long range order in the XY model on the triangular and square lattices. For the triangular lattice seven coefficients have also been obtained in the expansion of the fourth-order fluctuation. The method of conformal transformations has been developed in a form suitable for the analysis of series of this type. Clear evidence is obtained in favor of a phase transition for which the critical index of the fluctuations is γ ≈ 3/2. The critical temperatures kTC/J for the triangular and square lattices are 1.500 ± 0.006 and 0.900 ± 0.006 respectively.
The series for the fluctuation of the order parameter and the fourth-order fluctuation of the order parameter of the S = [Formula: see text] XY model on four-dimensional lattices are analysed. Assuming a simple power law we find γ = 1.125 ± 0.010 for the simple hypercubic and γ = 1.113 ± 0.005 for the face-centred hypercubic. An alternative method of analysis which includes a logarithmic correction factor and assumes γ = 1 gives as power for the logarithmic correction p = 0.36 ± 0.05 for the simple hypercubic lattice.
Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .
A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-12\frac{1}{2}21​ $X\ensuremath{-}Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.