Majorana representation for dissipative spin systems

TL;DR

This paper introduces a Majorana fermion-based method for analyzing dissipative spin systems, simplifying the calculation of certain spin correlation functions and applying it to the Bose-Kondo model to derive relaxation rates.

Contribution

It develops a path-integral approach using Majorana representation to efficiently compute spin correlation functions in dissipative systems, including a saddle point analysis for the Bose-Kondo model.

Findings

Simplified calculation of N-point spin correlations as N-point Majorana correlators.

Derived spin relaxation rate from saddle point solution.

Fluctuations around the saddle point do not affect single-spin correlation functions.

Abstract

The Majorana representation of spin operators allows for efficient field-theoretical description of spin-spin correlation functions. Any N-point spin correlation function is equivalent to a 2N-point correlator of Majorana fermions. For a certain class of N-point spin correlation functions (including "auto" and "pair-wise" correlations) a further simplification is possible, as they can be reduced to N-point Majorana correlators. As a specific example we study the Bose-Kondo model. We develop a path-integral technique and obtain the spin relaxation rate from a saddle point solution of the theory. Furthermore, we show that fluctuations around the saddle point do not affect the correlation functions as long as the latter involve only a single spin projection. For illustration we calculate the 4-point spin correlation function corresponding to the noise of susceptibility.