On the Ising Phase Transition in the Infrared-Divergent Spin Boson Model
Volker Betz, Benjamin Hinrichs, Mino Nicola Kraft, Steffen Polzer
TL;DR
The paper addresses the existence of ground states in the infrared-divergent spin boson model and proves a coupling-driven phase transition by mapping the problem to a one-dimensional continuum Ising model via a Feynman-Kac representation. It expresses the ground-state overlap $\rho(\lambda)$ in terms of Ising correlation functions and analyzes long-range order through a continuum FK-percolation framework, establishing that long-range order at large coupling forces $\rho(\lambda)=0$. The results combine a vacuum-overlap expansion with percolation-based arguments to demonstrate the uniqueness of the phase transition and connect non-relativistic QFT infrared behavior to continuum Ising physics. This provides a rigorous link between ground-state existence in quantum field models and long-range order phenomena in associated continuum Ising systems, with potential implications for bounding critical couplings in related models.
Abstract
We prove absence of ground states in the infrared-divergent spin boson model at large coupling. Our key argument reduces the proof to verifying long range order in the dual one-dimensional continuum Ising model, i.e., to showing that the respective two point function is lower bounded by a strictly positive constant. We can then use known results from percolation theory to establish long range order at large coupling. Combined with the known existence of ground states at small coupling, our result proves that the spin boson model undergoes a phase transition with respect to the coupling strength. We also present an expansion for the vacuum overlap of the spin boson ground state in terms of the Ising $n$-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.