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# High-temperature expansions through order 24 for the two-dimensional classical XY model on the square lattice

• ##### M. Pernici
Physical review. B, Condensed matter (Impact Factor: 3.77). 11/2007; DOI: 10.1103/PhysRevB.76.092406
Source: arXiv

ABSTRACT The high-temperature expansion of the spin-spin correlation function of the two-dimensional classical XY (planar rotator) model on the square lattice is extended by three terms, from order 21 through order 24, and analyzed to improve the estimates of the critical parameters.

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ABSTRACT: Using the tensor renormalization group method based on the higher-order singular value decom- position, 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 \pm 0.0019, consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent \delta is estimated as 14.5, close to the analytic value by Kosterlitz.
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##### Article: Further extensions of the high-temperature expansions for the two-dimensional classical XY model on the triangular and the square lattices
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ABSTRACT: The high-temperature expansions for the spin-spin correlation function of the two-dimensional classical XY (planar rotator) model are extended by two terms, from order 24 through order 26, in the case of the square lattice, and by five terms, from order 15 through order 20, in the case of the triangular lattice. The data are analyzed to improve the current estimates of the critical parameters of the models.
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##### Article: (m, d, N)=(1, 3, 2) Lifshitz point and the three-dimensional XY universality class studied by high-temperature bivariate series for XY models with anisotropic competing …
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ABSTRACT: High-temperature bivariate expansions have been derived for the two-spin-correlation function in a variety of classical lattice XY (planar rotator) models in which spatially isotropic interactions among first-neighbor spins compete with spatially isotropic or anisotropic (in particular uniaxial) interactions among next-to-nearest-neighbor spins. The expansions, calculated for cubic lattices of dimensions d=1, 2, and 3, are expressed in terms of the two variables K1=J1/kT and K2=J2/kT, where J1 and J2 are the nearest-neighbor and the next-to-nearest-neighbor exchange couplings, respectively. This paper deals in particular with the properties of the d=3 uniaxial XY model (ANNNXY model) for which the bivariate expansions have been computed through the 18th order, thus extending by 12 orders the results so far available and making a study of this model possible over a wide range of values of the competition parameter R=J2/J1. Universality with respect to R on the critical line separating the paramagnetic and the ferromagnetic phases can be verified, and at the same time the very accurate determination γ=1.3177(5) and ν=0.6726(8) of the critical exponents of the susceptibility and of the correlation length, in the three-dimensional XY universality class, can be achieved. For the exponents at the multicritical (m,d,N)=(1,3,2) Lifshitz point the estimates γl=1.535(25), ν⊥=0.805(15), and ν∥=0.40(3) are obtained. Finally, the susceptibility exponent is estimated along the boundary between the disordered and the modulated phases.
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