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Chinese Physics

Comparison of calculation methods for the tunnel splitting at excited states of biaxial spin models

Cui Xiao-Bo, Zhide Chen

Published Jun 30, 2004• 0 citations

Abstract

We present the calculation and comparison of tunnel splitting at excited levels of biaxial spin models by various methods, including the generalized instanton method, the generalized path integral method for coherent spin states, the perturbation method and the exact method by numerical diagonalization of the Hamiltonian. It is found that, for integer spin with spin number around 10, tunnel splitting predicted by the generalized path integral for coherent spin states is about 10 − n times of the exact numerical result for the n th excited level, while the ratio of the results of the perturbation method and the exact numerical method diverges in the large spin limit. We thus conclude that the generalized instanton method is the best approximate way for calculating tunnel splitting in spin models.

DOI: 10.1088/1009-1963/13/7/028

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We present the calculation and comparison of tunnel splitting at excited levels of biaxial spin models by various methods, including the generalized instanton method, the generalized path integral method for coherent spin states, the perturbation method and the exact method by numerical diagonalization of the Hamiltonian. It is found that, for integer spin with spin number around 10, tunnel splitting predicted by the generalized path integral for coherent spin states is about 10 − n times of the exact numerical result for the n th excited level, while the ratio of the results of the perturbation method and the exact numerical method diverges in the large spin limit. We thus conclude that the generalized instanton method is the best approximate way for calculating tunnel splitting in spin models.

A continuous path integral is used to obtain the semiclassical expressions of quantum propagators by means of coherent states in the holomorphic representation together with the appropriate boundary conditions. It allows us to express the total actions including boundary terms, classical trajectories, and extra phases in the stationary phase approximation. In particular, in the semiclassical expression for the quantum spin propagator, we pay attention to the extra quantum phase discovered by Solari and Kochetov. It is shown that this extra phase is related to an anomaly in the fluctuation determinant. It is verified that once this extra phase is included, the semiclassical propagator leads to the correct short time behaviour to O(T²), and shows its consistency under dissection of the path. In the application to the instanton methods, the extra quantum phase yields the correct formula for the tunnel splitting including both the exponential factor and the prefactor. The quantum phase leads to the correct spectrum with a shift of the spin quantum number by 1/2. We apply the semiclassical propagators and the tunnel splitting to a few models in magnetic molecular systems. It is shown that the formula gives the correct exponential factor and prefactors for quantum tunneling of a large spin if the extra quantum phase is taken into account.

A consistent theory of the ground-state energy and its splitting due to the process of tunnelling for the Lipkin - Meshkov - Glick (LMG) model is presented. We calculate accurately the trivial and the instanton saddle-point contributions to the functional integral for the partition function of the model, in terms of the spin coherent states. We show that such a calculation has to be performed very accurately taking into account the discrete nature of the functional integral. This accurate consideration leads to the replacement of the magnitude of the spin s by s + 1/2, in the formula for the ground-state splitting obtained by a naive continuous method. We compare the numerical calculation of the ground-state energy and the splitting due to tunnelling with the results obtained by the quasiclassical method and obtain excellent agreement.