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数理解析研究所講究録

On the Spin-Boson Model(Quantum Stochastic Analysis and Related Fields)

Asao Arai, Masao Hirokawa

Published Aug 1, 1996• 0 citations
OpenAlex

Abstract

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.

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We review recent results on the existence of ground states for the infrared-critical spin boson model, which describes the interaction of a massless bosonic field with a two-state quantum system. Explicitly, we derive a critical coupling λc>0λ_{\mathsf c}>0λc​>0 such that the spin boson model exhibits a ground state for coupling constants λλλ with ∣λ∣<λc|λ|<λ_{\mathsf c}∣λ∣<λc​. 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.