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Progress of Theoretical Physics

Ground State of a Spin-Boson System. II: Wavefunction and Variational Study

T. Tsuzuki

Published Mar 1, 1992• 2 citations
Publisher

Abstract

We first present a clarifying explanation of the wavefunction of the author's dynamic compensation theory of a spin-boson system. The ground state energy is examined by the variational method to justify the diagonal approximation introduced in the first part of the present series of works. We present a unified description of the ground state and the resulting phase diagram of the tunneling-localization transition with varying the strength of spin-boson interaction.

DOI
10.1143/ptp/87.3.569

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We first present a clarifying explanation of the wavefunction of the author's dynamic compensation theory of a spin-boson system. The ground state energy is examined by the variational method to justify the diagonal approximation introduced in the first part of the present series of works. We present a unified description of the ground state and the resulting phase diagram of the tunneling-localization transition with varying the strength of spin-boson interaction.
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.
Ground states of a spin-boson Hamiltonian, describing one two-level system (a spin) coupled to infinitely many harmonic oscillators (bosons), are studied. This spin-boson Hamiltonian is a prototype for the description of a ‘‘small’’ system (e.g., a molecule) coupled to its environment. The respective ground-state vector(s) are approximated by a linear combination of two coherent-state vectors corresponding to the two levels of the spin. Interest concentrates mainly on phase-transition phenomena (generation of superselection rules) in case the parameters of the Hamiltonian (level splitting of the spin, frequencies of the field modes, and coupling constants) exhibit an infrared singularity. The resulting phase diagrams are shown to satisfy reasonably well the rigorous bounds derived by Spohn, and in particular distinguish between the superohmic, ohmic, and subohmic regime in the sense of Leggett. Nevertheless, the approximation method used is simple enough so that everything can be explicitly calculated. Former results by Pfeifer as well as results by Emery and Luther, Zwerger and Harris and Silbey are extended and discussed.
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.
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.