Nambu-Eliashberg theory for multi-scale quantum criticality : Application to ferromagnetic quantum criticality in the surface of three dimensional topological insulators

We develop an Eliashberg theory for multi-scale quantum criticality, considering ferromagnetic quantum criticality in the surface of three dimensional topological insulators. Although an analysis based on the random phase approximation has been performed for multi-scale quantum criticality, an extension to an Eliashberg framework was claimed to be far from triviality in respect that the self-energy correction beyond the random phase approximation, which originates from scattering with $z = 3$ longitudinal fluctuations, changes the dynamical exponent $z = 2$ in the transverse mode, explicitly demonstrated in nematic quantum criticality. A novel ingredient of the present study is to introduce an anomalous self-energy associated with the spin-flip channel. Such an anomalous self-energy turns out to be essential for self-consistency of the Eliashberg framework in the multi-scale quantum critical point because this off diagonal self-energy cancels the normal self-energy exactly in the low energy limit, preserving the dynamics of both $z = 3$ longitudinal and $z = 2$ transverse modes. This multi-scale quantum criticality is consistent with that in a perturbative analysis for the nematic quantum critical point, where a vertex correction in the fermion bubble diagram cancels a singular contribution due to the self-energy correction, maintaining the $z = 2$ transverse mode. We also claim that this off diagonal self-energy gives rise to an artificial electric field in the energy-momentum space in addition to the Berry curvature. We discuss the role of such an anomalous self-energy in the anomalous Hall conductivity.