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Progress of Theoretical Physics

Statistical Mechanics of a Spin-Boson System: The Dynamic Compensation Theorem and the Exact Partition Function

T. Tsuzuki

Published Jul 1, 1990• 1 citation
Publisher

Abstract

We first extend the dynamic compensation theorem of infrared divergence for the case of finite temperatures and then obtain the exact partition function of the spin-boson system, which is different from the previous ones. We find that there exists no thermodynamic equivalence between the spin-boson system and the Kondo system in general and that such equivalence is restricted to the case of high temperature limit.

DOI
10.1143/ptp/84.1.81

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Progress of Theoretical Physics•Feb 1, 1991•0 citations
We first extend the dynamic compensation theorem of infrared divergence for the case of finite temperatures and then obtain the exact partition function of the spin-boson system, which is different from the previous ones. We find that there exists no thermodynamic equivalence between the spin-boson system and the Kondo system in general and that such equivalence is restricted to the case of high temperature limit.
A closed integral equation for the partition function of a spin-boson system is obtained by the use of the coherent state representation of boson. This is accomplished by the use of new method of taking the thermal average by differential operations. The dynamic compensation theorem for the thermodynamic fluctuation is proved without employing a series expansion of the partition function as done in our previous works. The zeroth approximation to the equation is shown to be the decoupling approximation introduced previously, as naturally expected.
A closed integral equation for the partition function of a spin-boson system is obtained by the use of the coherent state representation of boson. This is accomplished by the use of new method of taking the thermal average by differential operations. The dynamic compensation theorem for the thermodynamic fluctuation is proved without employing a series expansion of the partition function as done in our previous works. The zeroth approximation to the equation is shown to be the decoupling approximation introduced previously, as naturally expected.
We study the non-adiabatic effect of the bosonic bath on the spin dynamics by the light of the dynamic compensation theorem. We examine the time-evolutions of the probability of finding the up-state of spin and the thermal average of σz(t). The dynamics is formulated in competition between the tunneling coherence of spin and the non-adiabatic effect of bath in the equitable treatment. We prove the complete compensation of the infrared divergences due to the zero-point and thermodynamic fluctuations by the dynamical fluctuation of the coherent boson (the dynamic compensation). We further present the exact expressions of the above two quantities in the series form.
The partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is shown to be exactly equal to that of a one-dimensional gas and also, in a delicate limit, to that of a one-dimensional Ising model. Both of these are peculiar in that they must provide a free energy which is an intensive quantity. For special values of the parameters in the Hamiltonian, the partition function is equal to that first derived by Anderson and Yuval, and by Hamann, for the Kondo problem.
We study the non-adiabatic effect of the bosonic bath on the spin dynamics by the light of the dynamic compensation theorem. We examine the time-evolutions of the probability of finding the up-state of spin and the thermal average of σz(t). The dynamics is formulated in competition between the tunneling coherence of spin and the non-adiabatic effect of bath in the equitable treatment. We prove the complete compensation of the infrared divergences due to the zero-point and thermodynamic fluctuations by the dynamical fluctuation of the coherent boson (the dynamic compensation). We further present the exact expressions of the above two quantities in the series form.