Quantum phase transition in the spin boson model

TL;DR

This paper explores quantum critical phenomena in the spin boson model, analyzing how different bath spectra influence the quantum phase transition and employing renormalization group techniques for various bath densities.

Contribution

It provides a detailed RG analysis of the spin boson model with different bath spectra and discusses the derivation of Shiba's relation and quantum-classical mapping.

Findings

Quantum critical point behavior depends on bath density of states.

Renormalization group treatment varies with bath spectrum.

Explicit derivation of Shiba's relation for sub-ohmic baths.

Abstract

In this paper we give a general introduction to quantum critical phenomena, which we practically illustrate by a detailed study of the low energy properties of the spin boson model (SBM), describing the dynamics of a spin 1/2 impurity (or more generically a two-level system) coupled to a bath of independent harmonic oscillators. We show that the behavior of the model is very sensitive to the bath spectrum, in particular how the properties of the quantum critical point in the SBM are affected by the functional form of the bath Density of States (DoS). To this effect, we review the renormalization group (RG) treatment of the SBM for various bath DoS, based on an unconventional Majorana representation of the spin 1/2 degree of freedom. We also discuss the derivation of Shiba's relation for the sub-ohmic SBM, and explicitely derive an effective action vindicating the quantum to classical mapping.