Semiclassical Husimi functions for spin systems

Rigorous derivation of semiclassical approximations in phase space via path integrals [1] for systems with one degree of freedom have recently received considerable attention, both for continuous variables and spin systems. Baranger et al [2], for example, have discussed the canonical coherent state path integral and its semiclassical approximation in some detail, including an initial-value representation and the Green function. The study of semiclassical propagation of wave packets, using complex [3] or nearly real [4] trajectories, for regular and chaotic [5] systems, has developed considerably over the last few years. The spin path integral, and its semiclassical approximation, has found an important application in the study of spin tunnelling and topological effects [6]. Stone et al have derived the spin coherent state semiclassical propagator in detail [7], paying particular attention to the so-called Solari-Kochetov [8] correction. This correction is related to the difference between the average value of the Hamiltonian in coherent states and its Weyl symbol [9], and has a counterpart in the canonical case [2]. To obtain semiclassical approximations for the energy levels En and stationary states 〈x|n〉 = ψn(x) of onedimensional bound systems, on the other hand, one normally resorts to the usual Bohr-Sommerfeld (BS) and WKB theories [10]. A coherent state version of these theories, which works in phase space, is also available [2] and produces a BS formula and a semiclassical approximation to the Husimi functions Hn(z) = |〈z|n〉|2. Recently, Garg and Stone [11] have derived a semiclassical (BS-like) quantization condition for spin systems, including the first quantum corrections (see also [12]). By taking the trace of the semiclassical Green function, they obtained the energy levels as the location of its poles. In the present work we have obtained the semiclassical Husimi functions for spin systems. The non-normalized spin coherent states are defined by |z〉 = exp{zJ+}|j,−j〉, and the semiclassical approximation to the propagator K = 〈zf |e|zi〉 is [7]