FKN Formula and Ground State Energy for the Spin Boson Model with External Magnetic Field

TL;DR

This paper derives a path integral formula for the spin boson model with an external magnetic field, enabling the analysis of ground state energy and its derivatives, and establishing ground state existence in challenging conditions.

Contribution

It introduces a Feynman-Kac-Nelson formula for the model and links ground state energy expansion to Ising model correlations, advancing understanding of magnetic field effects.

Findings

Path integral formula for the heat kernel established.

Ground state energy expansion coefficients expressed via Ising correlations.

Proved finiteness of the second derivative and existence of ground states in infrared-singular cases.

Abstract

We consider the spin boson model with external magnetic field. We prove a path integral formula for the heat kernel, known as Feynman-Kac-Nelson (FKN) formula. We use this path integral representation to express the ground state energy as a stochastic integral. Based on this connection, we determine the expansion coefficients of the ground state energy with respect to the magnetic field strength and express them in terms of correlation functions of a continuous Ising model. From a recently proven correlation inequality, we can then deduce that the second order derivative is finite. As an application, we show existence of ground states in infrared-singular situations.