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

摘要

We develop an Eliashberg theory for multi-scale quantum criticality,considering ferromagnetic quantum criticality in the surface of threedimensional topological insulators. Although an analysis based on the randomphase approximation has been performed for multi-scale quantum criticality, anextension to an Eliashberg framework was claimed to be far from triviality inrespect 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, explicitlydemonstrated in nematic quantum criticality. A novel ingredient of the presentstudy is to introduce an anomalous self-energy associated with the spin-flipchannel. Such an anomalous self-energy turns out to be essential forself-consistency of the Eliashberg framework in the multi-scale quantumcritical point because this off diagonal self-energy cancels the normalself-energy exactly in the low energy limit, preserving the dynamics of both $z= 3$ longitudinal and $z = 2$ transverse modes. This multi-scale quantumcriticality is consistent with that in a perturbative analysis for the nematicquantum critical point, where a vertex correction in the fermion bubble diagramcancels a singular contribution due to the self-energy correction, maintainingthe $z = 2$ transverse mode. We also claim that this off diagonal self-energygives rise to an artificial electric field in the energy-momentum space inaddition to the Berry curvature. We discuss the role of such an anomalousself-energy in the anomalous Hall conductivity.