The longitudinal susceptibility for the
S=21XY model on cubic lattices is discussed on the basis of its relationship to the free energy. A series expansion for the susceptibility is developed and its behavior is studied at an asymptotic limit. The asymptotic behavior of the series expansion shows that the longitudinal susceptibility is nondivergent at the critical point
Tc, its "singular" part behaving as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\alpha}+1}$, where $\ensuremath{\alpha}$ is the specific-heat exponent. A related quantity, the partial longitudinal susceptibility, which is important in the study of spin dynamics, is also shown to have a similar nondivergent critical behavior.
A previously unsolved problem has been solved numerically using transfer matrix renormalization group analysis, based on a density matrix renormalization group idea. This calculation leads to a high-precision prediction of the value of susceptibility in the x - or y - directions of an XY spin-chain, because in the calculation the transfer matrix idea is incorporated and a thermodynamic limit is ensured. From this calculation the susceptibility at zero temperature is predicted for a spin-½ antiferromagnetic XY model in one dimension.
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 $\ensuremath{\gamma}=1.3177(5)$ 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.
Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc.