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Progress of Theoretical Physics

Ground State of a Spin-Boson System: The Dynamic Compensation Theorem and the Orthogonality Theorem

T. Tsuzuki

Published Nov 1, 1989• 6 citations
Publisher

Abstract

We present a dynamic theory of the ground state of a spin boson system. We show the infrared divergence due to the zero-point fluctuation of displaced bosons is completely compensated by that due to the dynamical fluctuation of coherently excited bosons for arbitrary positive power sof the frequency dependence of the spectral density of interaction. We propose this as the theorem of dynamic compensation of the infrared divergence. In this sense our theory is a dynamic extension of the Anderson orthogonality theorem. Solving the eigenvalue equation in the lowest approximation we find that (1) there exists a critical value αc of the interaction strenght below which the ground state (GS) is the tunneling state for arbitrary positive s, (2) for s>1/2 the GS is the localized state above αc but (3) for 1/2>s>0 we find the localized GS above another critical value α'c>αc and we have found nothing definite, for the moment, for αc≪α≪α'c and (4) the tunneling gap tends to zero continuously when α approaches αc.

DOI
10.1143/ptp.82.917

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We present a dynamic theory of the ground state of a spin boson system. We show the infrared divergence due to the zero-point fluctuation of displaced bosons is completely compensated by that due to the dynamical fluctuation of coherently excited bosons for arbitrary positive power sof the frequency dependence of the spectral density of interaction. We propose this as the theorem of dynamic compensation of the infrared divergence. In this sense our theory is a dynamic extension of the Anderson orthogonality theorem. Solving the eigenvalue equation in the lowest approximation we find that (1) there exists a critical value αc of the interaction strenght below which the ground state (GS) is the tunneling state for arbitrary positive s, (2) for s>1/2 the GS is the localized state above αc but (3) for 1/2>s>0 we find the localized GS above another critical value α'c>αc and we have found nothing definite, for the moment, for αc≪α≪α'c and (4) the tunneling gap tends to zero continuously when α approaches αc.
The physics of an inversion-symmetric (IS) spin-boson system is first solved rigorously in the white limit of boson system on the basis of the concept of dynamic compensation which the author proposed previously. Two kinds of displaced bosons are introduced. They are defined freely out of the infrared divergence due to the dynamic compensation. It is shown that this IS system allows both IS-preserving and IS-breaking states when the bare zero-point contribution of bosons brings about the infrared divergence. The ground state and the excited state are determined explicitly in the white limit. Using them, after the brief study of thermodynamics, we construct the density matrix and study the problem of the persistence of oscillation of quantum coherence under the influence of environment. We describe the persistence of it up to a limited strength of interaction, taking a reasonable initial condition. An explicit formula is given.
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.
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.
An analytic ground state is proposed for the unbiased spin-boson Hamiltonian, which is non-Gaussian and beyond the Silbey-Harris ground state with lower ground state energy. The infrared catastrophe in Ohmic and sub-Ohmic bosonic bath plays an important role in determining the degeneracy of the ground state. We show that the infrared divergence associated with the displacement of the nonadiabatic modes in bath may be removed from the proposed ground state for the coupling α < αc. Then αc is the quantum critical point of a transition from non-degenerate to degenerate ground state and our calculated αc agrees with previous numerical results.
We study the ground state energy and the excitation spectra of a spin-boson system in the case of white bath limit as well as in the case of white bath on the basis of a recent theory proposed by the author. We confirm the previous numerical results in the case of non-white bath. Our analytical findings are as follows. (1) The ground state in the case of white bath is the tunneling state for s > 1 and the localized state for 1 ≥ s > 0, where s is the power of the frequency dependence of the spectral density of interaction. (2) The ground state in the white bath limit differs from that in the white bath for s = 1, that is, the former does not change into the latter continuously, while the latter is the limit of the former for 1 > s > 0. (3) The low-lying excitations form gapless continuum above the ground state energy. There appears another continuum shifted upward by the tunneling gap when the tunneling ground state exists.