Equivalence of KMS and spectral critical couplings for the spin–boson model

Ascertain whether the critical parameter α_c governing the KMS (thermal) phase transition for the spin boson model coincides with the spectral critical coupling λ_0 (equivalently α(λ_0)=λ_0^2/8) at which the spin boson Hamiltonian H_λ loses a ground state; that is, prove or refute that α_c equals α(λ_0).

Background

Spohn (1989) established a phase transition for KMS states of the spin–boson model using a continuum Ising representation with periodic boundary conditions, yielding a critical α_c at which spontaneous magnetization emerges.

The present work establishes a spectral phase transition for ground states of H_λ via a different Ising representation (one-sided and with free boundary conditions). The authors suspect that the KMS critical α_c matches the spectral threshold α(λ_0), but boundary conditions and domain differences in the long-range Ising models obstruct a direct identification.