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Physical Review Letters

Time-Dependent Behavior of the Spin-½ Anisotropic Heisenberg Model in Infinite Lattice Dimensions

M. Howard Lee, I. M. Kim, Raf Dekeyser

Published Apr 30, 1984• 51 citations
Publisher

Abstract

The spin van der Waals model may be regarded as an infinite-lattice-dimensional limit of the spin-\textonehalf{} anisotropic Heisenberg model. By solution of the generalized Langevin equation, time-dependent behavior is obtained. The geometry of realized Hilbert spaces lends a simple interpretation for critical dynamics.
DOI
10.1103/physrevlett.52.1579

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Physical review. B, Condensed matter•May 1, 1996•1 citation
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.
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.
A phenomenological dynamic scaling method is proposed and utilized to interpret finite-size effects in the time-dependent correlation function for the classical spin van der Waals model. We claim that the equations of motion for the classical spin van der Waals model are well defined only within a certain characteristic time. The order of magnitude for the size-dependent maximum time interval of the time-dependent correlation function, analytically obtained in the large N limit via the method of Laplace by Dekeyser and Lee, is found to be related to our characteristic time. As a by-product of this work, time scaling variables are found with which we can completely collapse the time-dependent correlation functions of the classical spin van der Waals model into one curve. \textcopyright{} 1996 The American Physical Society.