The T=0 dynamics of the one-dimensional s= 1 / 2 planar antiferromagnet is studied by an approach which consists of exact analytic calculations in the Bethe formalism and numerical finite-chain calculations on rings up to 10 spins. The method makes use of well known critical exponents for the correlation functions and of exact sum rules. The authors obtain approximate analytic expressions for both the out-of-plane and the in-plane dynamic structure factors, and for related quantities such as integrated intensities, susceptibilities and autocorrelation functions. The results are discussed in relation to possible experiments on quasi-1D magnetic compounds at low T. The calculations make clear that the T=0 dynamic structure factors are dominated by two-parameter continua of excitations rather than by single branches of spin-waves as predicted by classical spin-wave theory. By varying the planar anisotropy the autocorrelation functions display interesting features in their longtime asymptotic behaviour, such as a crossover from a uniform power-law decay to an oscillatory decay and a crossover between oscillatory decays with different frequencies. They conjecture a possibility of approaching the classical limit s= infinity starting from the quantum limit s= 1 / 2 . This provides a qualitative, and in some aspects even quantitative, understanding of the dynamical behaviour of s> 1 / 2 systems in terms of a quantum approach.
The low-temperature critical properties and crossovers of a spin-
21 planar ferromagnet in a longitudinal magnetic field are explored in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions in terms of an anisotropic bosonic action, suitable to describe the spin model in the low-temperature regime. This is performed adopting a procedure, which combines a preliminary averaging over dynamic degrees of freedom and a next static Wilson renormalization group transformation. We work to one-loop level to obtain results which are exact to first order in $\ensuremath{\epsilon}$ and believed to be qualitatively correct for
2<d<4. Within this static framework, we get the phase boundary, ending in a quantum critical point, and the general expressions for the correlation length and susceptibility as functions of the temperature and the applied magnetic field within the disordered phase. With these basic ingredients, the critical properties and a rich crossover scenario in the influence domain of the quantum critical point are derived in a simple and transparent way. Remarkably, one can speculate that the crossovers, found to occur decreasing the temperature with the magnetic field fixed at its quantum critical point value, might be actually observable in complex magnetic compounds, as suggested by recent experiments.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to
1922, for several temperatures below, at, and above
TKT. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function
Sxx also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in
Sxx, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}
TKT, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
Scaling relations are used to study cross-overs, due to anisotropic spin interactions or single ion anisotropy, and due to disorder, in the thermodynamics and correlation functions near quantum-critical transitions. The principal results are simple with a wide range of applications. The region of attraction to the stable anisotropic fixed point in the quantum-critical region is exponentially enhanced by the dynamical critical exponent
z compared to the region of attraction to the fixed point in the quantum disordered region. The result implies that, even for small anisotropy, the region of attraction to the stable incommensurate Ising or planar metallic anti-ferromagnetic critical points, which belong to the universality class of the XY model with
z→∞, covers the entire quantum-critical region. In crossovers due to disorder, the instability of the pure fixed point in the quantum disordered region is exponentially enhanced by
z compared to that in the quantum-critical region. This result suggests that for some classes of disorder and for large enough
z, one may find singularities in the correlations as a function of frequency and temperature down to very low temperatures even though the correlation length in space remains short range.