(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

P. Butera, M. Pernici

arXiv:0806.2100Physical Review BAug 6, 20089 citations

arXiv:1202.2464 · Nuclear Physics B

Lifshitz-point correlation length exponents from the large-n expansion

Mykola A. Shpot, Yu. M. Pis’mak

Published Apr 16, 2012• 10 citations

arXiv HTML PDF

DOI: 10.1016/j.nuclphysb.2012.04.011

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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

P. Butera, M. Pernici

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.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.

arXiv:0806.2100Physical Review BAug 6, 20089 citations

Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8

ndash

d)2

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.