arXiv:2102.13373 · Communications in Mathematical Physics

On Existence of Ground States in the Spin Boson Model

David Hasler, Benjamin Hinrichs, Oliver Siebert

Published Aug 9, 2021• 18 citations

Abstract

Abstract We show the existence of ground states in the massless spin boson model without any infrared regularization. Our proof is non-perturbative and relies on a compactness argument. It works for arbitrary values of the coupling constant under the hypothesis that the second derivative of the ground state energy as a function of a constant external magnetic field is bounded.

DOI: 10.1007/s00220-021-04185-w

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arXiv:2102.13373 · Communications in Mathematical Physics

On Existence of Ground States in the Spin Boson Model

David Hasler, Benjamin Hinrichs, Oliver Siebert

Published Aug 9, 2021• 18 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/s00220-021-04185-w

Related papers

Existence of Ground States in the Infrared-Critial Spin Boson Model

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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 $λ_{\mathsf c}>0$ 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.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 $λ_{\mathsf c}>0$ 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.

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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.

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.

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.