arXiv:0909.2202 · Physical Review B

Extended scaling behavior of the spatially anisotropic classical XY model in the crossover from three to two dimensions

P. Butera, M. Pernici

Published Aug 14, 2009• 6 citations

Abstract

The bivariate high-temperature expansion of the spin-spin correlation function of the three-dimensional classical $XY$ (planar rotator) model, with spatially anisotropic nearest-neighbor couplings, is extended from the tenth through the 21st order. The computation is carried out for the simple cubic lattice, in the absence of magnetic field, in the case in which the coupling strength along the $z$ axis of the lattice is different from those along the $x$ and the $y$ axes. It is then possible to determine accurately the critical temperature as function of the parameter $R$ which characterizes the coupling anisotropy and to check numerically the universality, with respect to $R$, of the critical exponents of the three-dimensional anisotropic system. The analysis of our data also shows that the main predictions of the generalized scaling theory for the crossover from the three-dimensional to the two-dimensional critical behavior are compatible with the series extrapolations.

DOI: 10.1103/physrevb.80.054408

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arXiv:0909.2202 · Physical Review B

Extended scaling behavior of the spatially anisotropic classical XY model in the crossover from three to two dimensions

P. Butera, M. Pernici

Published Aug 14, 2009• 6 citations

arXiv HTML PDF

Abstract

The bivariate high-temperature expansion of the spin-spin correlation function of the three-dimensional classical

XY

(planar rotator) model, with spatially anisotropic nearest-neighbor couplings, is extended from the tenth through the 21st order. The computation is carried out for the simple cubic lattice, in the absence of magnetic field, in the case in which the coupling strength along the

z

axis of the lattice is different from those along the

x

and the

y

axes. It is then possible to determine accurately the critical temperature as function of the parameter

R

which characterizes the coupling anisotropy and to check numerically the universality, with respect to

R

, of the critical exponents of the three-dimensional anisotropic system. The analysis of our data also shows that the main predictions of the generalized scaling theory for the crossover from the three-dimensional to the two-dimensional critical behavior are compatible with the series extrapolations.

DOI: 10.1103/physrevb.80.054408

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(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 interactions

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The bivariate high-temperature expansion of the spin-spin correlation function of the three-dimensional classical

XY

z

axis of the lattice is different from those along the

x

and the

y

axes. It is then possible to determine accurately the critical temperature as function of the parameter

R

which characterizes the coupling anisotropy and to check numerically the universality, with respect to

R

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

{K}_{1}={J}_{1}/kT

and

{K}_{2}={J}_{2}/kT

, where

{J}_{1}

and

{J}_{2}

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={J}_{2}/{J}_{1}

. 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 $\ensuremath{\gamma}=1.3177(5)$ and $\ensuremath{\nu}=0.6726(8)$ 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 ${\ensuremath{\gamma}}_{l}=1.535(25)$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=0.805(15)$, and ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=0.40(3)$ are obtained. Finally, the susceptibility exponent is estimated along the boundary between the disordered and the modulated phases.

s=1,3/2,\dots{}

and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.