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Slow decay in a spin model

Raf Dekeyser, M. Howard Lee

Published Jan 1, 1991• 0 citations
OpenAlex

Abstract

The time evolution of a spin in the spin-1/2 van der Waals model has been obtained by solving the Heisenberg equation of motion exactly. Using this solution, we have obtained the long time behavior of the autocorrelation function of a spin. The origin and mechanism of slow decay are described.

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Journal of the Korean Physical Society•Sep 1, 2012•0 citations
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 autocorrelation function of a single spin 〈s(t)\ensuremath{\cdot}s(0)〉 has been obtained from the time-evolution solutions given in the preceding paper by carrying out the ensemble averages explicitly. Slow decay is found in the transverse component and only at high temperatures (T>Tc{\mathit{T}}_{\mathit{c}}Tc​). The exponent \ensuremath{\kappa}, where 〈sx{\mathit{s}}^{\mathit{x}}sx(t)sx{\mathit{s}}^{\mathit{x}}sx(0)〉\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\kappa}}}$, as t\ensuremath{\rightarrow}\ensuremath{\infty}, is found to depend discontinuously on the spin-spin interaction strength R=Jz{\mathit{J}}_{\mathit{z}}Jz​/J: \ensuremath{\kappa}=2 if R=0, \ensuremath{\kappa}=3 if 0R\ensuremath{\le}2 (R\ensuremath{\ne}1), \ensuremath{\kappa}=\ensuremath{\infty} if R=1, and \ensuremath{\kappa}=\ensuremath{\infty} if R>2. Slow decay in this model is attributed to nondimensional effects, e.g., cooperativity. Physical and mathematical mechanisms of the slow decay are described.
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 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.