Kl-th 96/6 Application of Instantons: Quenching of Macroscopic Quantum Coherence and Macroscopic Fermi–particle Configurations

Starting from the coherent state representation of the evolution operator with the help of the path–integral, we derive a formula for the low–lying levels E = ǫ0 − 2△ǫ cos(s+ ξ)π of a quantum spin system. The quenching of macroscopic quantum coherence is understood as the vanishing of cos(s+ ξ)π in disagreement with the suppression of tunneling (i.e. △ǫ = 0) as claimed in the literature. A new configuration called the macroscopic Fermi–particle is suggested by the character of its wave function. The tunneling rate (2△ǫ π ) does not vanish, not for integer spin s nor for a half–integer value of s, and is calculated explicitly (for the position dependent mass) up to the one–loop approximation. The question whether or not superpositions of macroscopically distinct quantum states exist in nature has attracted considerable attention in recent years owing mainly to the development of technology in mesoscopic physics. Thus Leggett et al.[1] predicted that the most intriguing quantum effect which could take place on the macroscopic scale is quantum tunneling[1]. Macroscopic magnetisation tunneling is a subject which is being investigated extensively and is of growing interest [2, 3]. In discussing this, it is essential to distinguish between macroscopic quantum tunneling (MQT) and macroscopic quantum coherence (MQC). In the case of MQC, tunneling between neighboring degenerate (perturbation theory) vacua is dominated by the instanton configuration with nonzero topological charge and leads to a level splitting which (apart from a factor of h̄ ) is seen to be the frequency of oscillation between, for example, the easy directions (defined by the vacua) of the ferromagnetic particle. The tunneling removes the degeneracy of the ground states, and the true ground state is a superposition of the degenerate ground states, for instance, the two easy directions of the ferromagnetic particle which are macroscopically distinguishable. For the case of MQT the tunneling is dominated by the socalled bounce configuration [4] with zero topological charge and leads to the decay of metastable states. The quantum tunneling of magnetization in small ferromagnetic particles has been well established within the leading order approximation of the instanton method for various models [5], using the classical Hamiltonian. Recently an interesting phenomenon known as topological quenching of MQC has been studied extensively [6, 7, 8, 9]. The quenching of MQC there was analysed as the suppression of tunneling (i.e. the vanishing of the level splitting △ǫ as given by our eqs.(14,15) below which gives its relation to the transition amplitude ) by destructive interference of Euclidean paths in the barrier [6]. In the case of MQC the Hamiltonian has a periodic nature, that is, there is an infinite number of degenerate vacua by successive 2π extension. However, tunneling in the ferromagnetic particle has only been considered in the literature so far as that for the double–well potential, like the φ potential in field theory, where the side barriers are infinitely high. We argue here that the method of calculating the tunneling effect for a periodic potential [10] ought to be used in dealing with MQC, regarding the periodic potential as a one–dimensional super–lattice. The tunneling through one barrier leads to the level splitting which extends formally to an energy “band” by translation symmetry (the