Spin Dynamics from Majorana Fermions

Using the Majorana fermion representation of spin-$1/2$ local moments, we show how the dynamic spin correlation and susceptibility are obtained directly from the one-particle Majorana propagator. We illustrate our method by applying it to the spin dynamics of a nonequilibrium quantum dot, computing the voltage-dependent spin relaxation rate and showing that, at weak coupling, the fluctuation-dissipation relation for the spin of a quantum dot is voltage dependent. We confirm the voltage-dependent Curie susceptibility recently found by Parcollet and Hooley [Phys. Rev. B 66, 085315 (2002)].

Figure 1

Leading order Feynman diagram for the Majorana self-energy ${\Sigma }_{ab}(\omega )$, which is isotropic in spin indices in zero applied field. The self-energy of the $f$-fermion $f^{†}=(1/\sqrt{2})({\eta }^1+i{\eta }^2)$ is given by $\Sigma (\omega )$.

Figure 2

Voltage-dependent spin-relaxation rate, computed to quadratic order in the Kondo coupling constant, showing the zero field, Korringa component, and the voltage-dependent inter-lead component. For the case chosen, ${\overline{J}}_{\lambda {\lambda }^{\prime }}=\overline{J}$ ($\lambda ,{\lambda }^{\prime }\in \{R,L\}$).

Figure 3

Voltage-dependent spin correlator $C(\omega )$ calculated for a sequence of voltages $V=0,0.3,1$ at a temperature $T=0.05$. Here $\alpha =1$ and $\overline{J}=0.2$. Inset, fluctuation dissipation function $h(\omega )$ for the same sequence of voltages.