Sharpness of Ising-based criteria for ground-state phase transition

Determine whether, for the continuum Ising model associated to the infrared-divergent spin boson Hamiltonian via g(t)=∫_{R^d}|v(k)|^2 e^{-|t| ω(k)} dk, absence of long-range order in the one-sided model on [0,∞) (i.e., inf_{t≥0} τ_{α,1}(t)=0) immediately implies that the magnetic susceptibility in the corresponding two-sided model on R is finite. Establishing this implication would prove the sharpness of the sufficient criteria that link ground-state existence and nonexistence in the spin boson model to Ising correlation functions.

Background

The paper connects ground-state existence in the spin boson model to correlation functions of a one-sided, long-range continuum Ising model obtained via a Feynman–Kac representation. Prior work provided sufficient criteria (in terms of Ising correlations and susceptibility bounds) for both existence and absence of ground states.

The authors note that in discrete Ising models, sharpness across the phase transition (e.g., between long-range order and finite susceptibility) is established, but extending such results to the continuum and one-sided setting relevant here is nontrivial.