The Spin-Boson model describes the interaction between a quantum-mechanical two-state system, a spin, and a Bosonic field. Its Hamiltonian is said to have a ground state if the infimum of its spectrum is an eigenvalue. Using the Feynman-Kac formula, one can express matrix elements of the semigroup generated by the Hamiltonian in terms of a self-attractive jump process. We will discuss how one can apply this representation in order to derive probabilistic criteria for the existence and non-existence of ground states. We will summarize how these criteria have been applied to show that, in the infrared-critical case, a phase transition occurs: as the coupling strength increases, the system transitions from having a ground state to having none. Based on joint work with Volker Betz, Benjamin Hinrichs and Mino Nicola Kraft.
