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CRC Press eBooks

On an Expression of the Ground State Energy of the Spin-Boson Hamiltonian

Masao Hirokawa

Published Nov 27, 2020• 0 citations
Publisher

Abstract

An expression of the ground state energy E SB of the spin-boson Hamiltonian H SB is considered. The expression in the cases of both massive and massless bosons is given by a nonperturbative method.
DOI
10.1201/9781003078296-11

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The existence and uniqueness of ground states of the spin-boson Hamiltonian HSBH_{\mathrm{S}\mathrm{B}}HSB​ are con- sidered.The main results in the case of massive bosons include: (i)(existence)(\mathrm{i})(\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e})(i)(existence) there exists a ground state without restriction for the strength of the coupling constant; (ii)(uniqueness)(\mathrm{i}\mathrm{i})(\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s})(ii)(uniqueness) under a mild (nonperturbative) condition for the parameters contained in HSB,H_{\mathrm{S}\mathrm{B}},HSB​, HSBH_{\mathrm{S}\mathrm{B}}HSB​ has only one ground state ; (iii) (degeneracy) under a certain condition for the parameters of HSBH_{\mathrm{S}\mathrm{B}}HSB​which is weaker than that of (ii), the number of the ground states is at most two.In the case of massless bosons, the existence of a ground state of HSBH_{\mathrm{S}\mathrm{B}}HSB​ is shown as a limit of ground states of the massive case.The methods used are nonperturbative.A generalization of the model is proposed.
The method of “auxiliary variables” (collective coordinates) used by Bogolyubov and Zubarev is formulated in a somewhat different manner. Improvements are made in two points: 1) The Hamiltonian is given in an Hermitian form and 2) the number of the auxiliary variables can be limited, if necessary. We calculate the ground state energy and the excitation energy spectrum of the spinless bosons taking account of phonon-phonon interactions. The expression for the ground state energy is shown to be equivalent to the one derived by Bogolyubov and Zubarev.