Hokkaido Mathematical Journal

On generalized spin-boson models with singular perturbations

Toshimitsu Takaesu

Published Oct 1, 2010• 14 citations

Abstract

In this paper we consider generalized spin-boson models with singular perturbations. It is proven that under the infrared regularity condition Hamiltonians have the unique ground state for sufficiently small values of coupling constants. In addition it is shown that the asymptotic creation and annihilation operators of massless boson field exist.

DOI: 10.14492/hokmj/1288357972

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Hokkaido Mathematical Journal

On generalized spin-boson models with singular perturbations

Toshimitsu Takaesu

Published Oct 1, 2010• 14 citations

Publisher

Abstract

DOI: 10.14492/hokmj/1288357972

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arXiv:2102.13373Communications in Mathematical PhysicsAug 9, 202118 citations

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The existence and uniqueness of ground states of the spin-boson Hamiltonian

H_{\mathrm{S}\mathrm{B}}

are con- sidered.The main results in the case of massive bosons include:

(\mathrm{i})(\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e})

there exists a ground state without restriction for the strength of the coupling constant;

(\mathrm{i}\mathrm{i})(\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s})

under a mild (nonperturbative) condition for the parameters contained in

H_{\mathrm{S}\mathrm{B}},

H_{\mathrm{S}\mathrm{B}}

has only one ground state ; (iii) (degeneracy) under a certain condition for the parameters of

H_{\mathrm{S}\mathrm{B}}

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

H_{\mathrm{S}\mathrm{B}}

is shown as a limit of ground states of the massive case.The methods used are nonperturbative.A generalization of the model is proposed.

λ_{\mathsf c}>0

such that the spin boson model exhibits a ground state for coupling constants

λ

with

|λ|<λ_{\mathsf c}

. The proof combines a Feynman-Kac-Nelson formula for the spin boson model with external magnetic field, a 1D-Ising model correlation bound and a compactness argument in Fock space. Elaborating on the connection to a long-range 1D-Ising model, we briefly discuss the conjecture that the spin boson model does not have a ground state at large coupling. This note is based on joint work with David Hasler and Oliver Siebert.

n

-point functions, which implies that the phase transition is unique, i.e., that there is a critical coupling constant below which a ground state exists and above which none can exist.