Effective field theory of an anomalous Hall metal from interband quantum fluctuations

Abstract

We construct an effective field theory, a two-dimensional two-component metallic system described by a model with two Fermi surfaces ("pockets"). This model describes a translationally invariant metallic system with two types of fermions, each with its own Fermi surface, with forward scattering interactions. This model, in addition to the $O(2)$ rotational invariance, has a $U(1) \times U(1)$ symmetry of separate charge conservation for each Fermi surface. For sufficiently attractive interactions in the $d$-wave (quadrupolar) channel this model has an interesting phase diagram that includes a spontaneously generated anomalous Hall metal phase. We derive the Landau-Ginzburg effective action of quadrupolar order parameter fields which enjoys an $O(2)\times U(1)$ global symmetry associated to spatial isotropy and the internal $U(1)$ relative phase symmetries respectively. We show that the order parameter theory is dynamically local with a dynamical scaling of $z=2$ and perform a one-loop renormalization group analysis of the Landau-Ginzburg theory. The electronic liquid crystal phases that result from spontaneous symmetry breaking are studied and we show the presence of Landau damped Nambu-Goldstone modes at low momenta that is a signature of non-Fermi liquid behavior. Electromagnetic linear response is also analyzed in both the normal and symmetry broken phases from the point of view of the order parameter theory. The nature of the coupling of electromagnetism to the order parameter fields in the normal phase is non-minimal and decidedly contains a precursor to the anomalous Hall response in the form of a order-parameter-dependent Chern-Simons term in the effective action.