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Canadian Journal of Physics

The relaxation function of the spin van der Waals model in a weak magnetic field

Y. K. Lee, I. M. Kim

Published Jan 1, 1990• 1 citation
Publisher

Abstract

We consider the spin van der Waals model in a weak magnetic field. For this model we obtain the relaxation function by applying the perturbative method that was developed for the evaluation of the relaxation function represented as an infinite continued fraction. To show the applicability of the perturbative method, we make comparisons between an exact result and various degrees of approximation.

DOI
10.1139/p90-007

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Physical review. B, Condensed matter•Jan 1, 1979•54 citations
We consider the spin van der Waals model in a weak magnetic field. For this model we obtain the relaxation function by applying the perturbative method that was developed for the evaluation of the relaxation function represented as an infinite continued fraction. To show the applicability of the perturbative method, we make comparisons between an exact result and various degrees of approximation.
The exact time evolution expressions for the total spin components are obtained by solving the equations of motion for the classical spin van der Waals model in transverse magnetic fields. They are given in terms of Jacobi elliptic functions and are examined in detail. They are also used to obtain the time-dependent total spin correlation functions by adopting equilibrium configurations acquired via a heat bath algorithm. The time-dependent total spin correlation functions are interpreted by utilizing the motion of the tip of the total spin vector on the surface of a Bloch sphere.
The authors calculate the classical partition function for a mean-field effective Hamiltonian with a finite number of spins. They find a single-valued magnetisation as a function of the external magnetic field, with a sharp transition at zero magnetic field, below the Curie point. The authors then calculate the thermodynamic limit for the same system (equivalent to the infinite-dimensional Ising model) in a straightforward fashion, and find the corresponding discontinuous behaviour at the phase transition point, with no van der Waals loop.
The spin van der Waals model has received some attention in recent years as a soluble model, useful for demonstrating collective behavior analytically. By solving the Heisenberg equation of motion, we obtain the time evolution of a single spin in this model when N\ensuremath{\rightarrow}\ensuremath{\infty}, where N is the number of spins in the model. Our solution is characterized by rotations in spin space having the appearance of a Larmor precession. In the accompanying paper, we use this time-evolution solution to obtain the spin autocorrelation function and determine its long-time behavior.
The time-dependent autocorrelation function has been derived for the xxx component of the total spin for the S=12S=\frac{1}{2}S=21​ constant-coupling anisotropic Heisenberg model, i.e., the Van der Waals system. For T>TcT>{T}_{c}T>Tc​, the time correlation is shown to be Gaussian for both XY\mathrm{XY}XY-like and Ising-like regimes of the model. For T<TcT<{T}_{c}T<Tc​, the correlation function is still Gaussian if the model is XY\mathrm{XY}XY-like; but it is oscillatory if the model is Ising-like. Critical slow down appears only with the Ising-like system for this time-correlation function.