Sharpness of Ising–correlation criteria for spin–boson ground states

Establish the sharpness of the sufficient criteria that relate the existence and absence of ground states for the spin boson Hamiltonian H_λ to correlation functions of the associated one-sided continuum Ising model; specifically, determine whether the absence of long-range order in the one-sided continuum Ising model implies that the magnetic susceptibility of the corresponding two-sided continuum Ising model is finite, thereby making these criteria sharp.

Background

The paper expresses ground-state properties of the infrared-divergent spin boson model in terms of correlation functions of a one-sided continuum Ising model. Using this duality, the authors obtain sufficient conditions for both existence and absence of ground states.

They point out that, unlike in discrete Ising models where sharpness of phase transitions is known, the analogous sharpness question for the continuum, long-range, one-sided versus two-sided Ising models remains unresolved. Establishing sharpness would align the thresholds governing long-range order and susceptibility, and would clarify whether the spectral phase transition in the spin boson model coincides with a sharp Ising transition in the continuum setting.