Equality of thermal and spectral critical couplings (Spohn’s αc vs spectral λ0)

Determine whether the critical parameter αc governing the phase transition of KMS (thermal) states in the periodic-boundary continuum Ising representation of the spin boson model coincides with the spectral critical coupling λ0 (under the mapping α(λ)=λ^2/8) at which the spin boson Hamiltonian on Fock space ceases to have a ground state, thereby establishing equality of the thermal and spectral phase transitions.

Background

Spohn (1989) proved a phase transition for KMS states of the Ohmic spin boson model via a continuum Ising model with periodic boundary conditions, identifying a critical coupling αc at which magnetization becomes nonzero under + boundary conditions and the KMS state changes character.

The present paper proves a spectral phase transition in the spin boson model (existence vs absence of ground states) and formulates a Feynman–Kac mapping to a one-sided continuum Ising model. The authors conjecture that the thermal critical parameter αc equals the spectral critical coupling λ0 (modulo α=λ^2/8), but note boundary-condition and domain differences that complicate a proof.