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Physical Review Letters

Critical Properties of the XY Model

Donald D. Betts, M. Howard Lee

Published Jun 24, 1968• 35 citations
Publisher

Abstract

The $\mathrm{XY}$ model of a quantum lattice fluid or a ferromagnet is studied by the method of exact high-temperature series expansion. Nine terms are obtained in the free-energy series and seven in the series for the square of the fluctuation in the long-range order. Analysis of these series yields the critical values $\frac{k{T}_{c}}{J}=4.84\ifmmode\pm\else\textpm\fi{}0.06$, $\ensuremath{\gamma}=1.00\ifmmode\pm\else\textpm\fi{}0.07$, and $\ensuremath{\alpha}=\ensuremath{-}0.20\ifmmode\pm\else\textpm\fi{}0.20$ for the fcc lattice.

DOI
10.1103/physrevlett.20.1507

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The XY\mathrm{XY}XY model of a quantum lattice fluid or a ferromagnet is studied by the method of exact high-temperature series expansion. Nine terms are obtained in the free-energy series and seven in the series for the square of the fluctuation in the long-range order. Analysis of these series yields the critical values kTcJ=4.84\ifmmode±\else\textpm\fi0.06\frac{k{T}_{c}}{J}=4.84\ifmmode\pm\else\textpm\fi{}0.06JkTc​​=4.84\ifmmode±\else\textpm\fi0.06, \ensuremathγ=1.00\ifmmode±\else\textpm\fi0.07\ensuremath{\gamma}=1.00\ifmmode\pm\else\textpm\fi{}0.07\ensuremathγ=1.00\ifmmode±\else\textpm\fi0.07, and \ensuremathα=\ensuremath−0.20\ifmmode±\else\textpm\fi0.20\ensuremath{\alpha}=\ensuremath{-}0.20\ifmmode\pm\else\textpm\fi{}0.20\ensuremathα=\ensuremath−0.20\ifmmode±\else\textpm\fi0.20 for the fcc lattice.
A method is developed for the exact high temperature series expansion of thermodynamic properties of the spin 1/2 XY model of ferromagnetism or of a quantum lattice gas. Eleven coefficients in the specific heat series for the f.c.c. and b.c.c. lattices and nine coefficients in the series for the fluctuation in the long range order for the f.c.c., b.c.c., and s.c. lattices are calculated. From the fluctuation series the critical temperatures Kc−1 ≡ kTc/J are estimated to be 4.520, 2.902, and 2.02 respectively, while the critical index for the square of the fluctuation in the long range order is γ = 1.35.
The universality hypothesis for critical phenomena considers spin to be an irrelevant quantity. The spin-SSS XY\mathrm{XY}XY model through its fluctuation series provides a convenient way of directly testing this hypothesis. For $S=\frac{1}{2} \mathrm{and} \ensuremath{\infty}$, it is known from high-temperature series expansions for the fluctuation that $\ensuremath{\gamma}\ensuremath{\approx}\frac{4}{3}$ at both limits. We provide here information on the fluctuation series expansions for intermediate values of spin ranging from S=1S=1S=1 to S=106S={10}^{6}S=106. Although our series are short, our results seem to be consistent with the idea of spin independence.
The spin one half XY model of ferromagnetism is defined by the interaction Hamiltonian [MATH] where the σ's are Pauli matrices defined at the sites i and j of a regular lattice. This model is appropriate for magnetic insulators for which g⊥ >> g|| temperature series expansions for the fluctuation in the long range order yield estimates of the critical temperatures for the f.c.c., b.c.c. and s.c. lattices and a susceptibility index γ ≈ 4/3. The fourth order fluctuation series yields the estimate Δ ≈ 5/3 from which by the scaling hypothesis all other indices are obtained. A further linearity hypothesis leads to predictions for the Heisenberg model indices from the Ising and XY indices.
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.
Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.