Majorana representation for dissipative spin systems

The Majorana representation of spin operators allows for efficient field-theoretical description of spin–spin correlation functions. Any NN-point spin correlation function is equivalent to a 2NN-point correlator of Majorana fermions. For a certain class of NN-point spin correlation functions (including “auto” and “pair-wise” correlations) a further simplification is possible, as they can be reduced to NN-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 for the Ohmic case 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.