On the Ising Phase Transition in the Infrared-Divergent Spin Boson Model

TL;DR

This paper proves a phase transition in the infrared-divergent spin boson model by linking the absence of ground states at large coupling to long-range order in an associated Ising model, using percolation theory.

Contribution

It establishes the existence of a critical coupling constant where the spin boson model transitions from having a ground state to none, connecting quantum field theory with statistical mechanics.

Findings

Ground states exist at small coupling

No ground states at large coupling

Phase transition occurs at a critical coupling

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.