Phase-space equilibrium distributions and their applications to spin systems with nonaxially symmetric Hamiltonians

The Fourier series representation of the equilibrium quasiprobability density function $W_S(\vartheta ,\phi )$ or Wigner function of spin “orientations” for arbitrary spin Hamiltonians in a representation (phase) space of the polar angles $(\vartheta ,\phi )$ (analogous to the Wigner function for translational motion) arising from the generalized coherent state representation of the density operator is evaluated explicitly for some nonaxially symmetric problems including a uniaxial paramagnet in a transverse external field, a biaxial, and a cubic system. It is shown by generalizing transition state theory to spins [i.e., calculating the escape rate using the equilibrium density function $W_S(\vartheta ,\phi )$ only] that one may evaluate the reversal time of the magnetization. The quantum corrections to the transition state theory escape rate equation for classical magnetic dipoles appear both in the prefactor and in the exponential part of the escape rate and exhibit a marked dependence on the spin number. Furthermore, the phase-space representation allows us to estimate the switching field curves and/or surfaces for spin systems because quantum effects in these fields can be estimated via Thiaville’s geometrical method [Phys. Rev. B 61, 12221 (2000)] for the study of the magnetization reversal of single-domain ferromagnetic particles. The calculation is accomplished (just as the determination of the equilibrium quasiprobability distributions in the phase space of the polar angles) by calculating switching field curves and/or surfaces using the Weyl symbol ($c$-number representation) of the Hamiltonian operator for given magnetocrystalline-Zeeman energy terms. Examples of such calculations for various spin systems are presented. Moreover, the reversal time of the magnetization allows us to estimate thermal effects on the switching fields for spin systems.

Figure 1

(Color online) Three-dimensional (3D) plot of $\beta V(\vartheta ,\phi )$ for ${\sigma }^{\prime }=10$, $h=0.1$, and various values of $S=1$, 2, 4, and $S\to \infty$ (classical limit).

Figure 2

(Color online) 3D plot of $\beta V(\vartheta ,\phi )$ for $S=2$, ${\sigma }^{\prime }=5$, and ${\delta }^{\prime }=2$.

Figure 3

(Color online) 3D plot of $\beta V(\vartheta ,\phi )$ for positive (left) and negative (right) cubic anisotropies at $S=4$ and $\xi =0$.

Figure 4

Spin dependence of switching field curves for (a) uniaxial, (b) biaxial at ${\sigma }_1∕\delta =0.25$, (c) cubic, and (d) mixed (at ${\sigma }_1∕{\sigma }_2=0.5$ and $\chi =0$) anisotropies calculated at $S=2$ (solid lines), 5 (dashed lines), and $S\to \infty$ (dotted lines; classical limit).