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

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 $\mathrm{O}\left(2\right)$ rotational invariance, has a $\mathrm{U}\left(1\right)\times \mathrm{U}\left(1\right)$ 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 $\mathrm{O}\left(2\right)\times \mathrm{U}\left(1\right)$ global symmetry associated to spatial isotropy and the internal $\mathrm{U}\left(1\right)$ 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.

Figure 1

Visualization of the ${\alpha }_1$ phase. (a) The vacuum manifold (thick blue line) of $\vec{s}$ defined in Eq. (4.4), which lies on the $xz$ great circle of the Bloch sphere such that $s^y=0$. The position of $\vec{s}$ depends on $arg(u/v)$ defined from Eq. (4.6). Not shown is the remaining Hopf fiber component $arg\left(uv\right)$ that takes values in $S^1$. (b) The nematic Fermi surfaces in the ${\alpha }_1$ phase with the isospin texture plotted. The $\langle {\psi }_{\mathbf{k}}^{†}{\sigma }^{x,y}{\psi }_{\mathbf{k}}\rangle$ texture does not wind around the Fermi surface and is shown here as being horizontal and vanishes along the high symmetry directions $\pm \pi /4,\pm 3\pi /4$.

Figure 2

Visualization of the ${\beta }_1$ phase. (a) The vacuum manifold (dark blue dots) of $\vec{s}$ defined in Eq. (4.4), which lies on the $\pm y$ poles of the Bloch sphere. Not shown is the remaining Hopf fiber component $arg\left(u\right)$ from Eq. (4.19) that takes values in $S^1$. (b) The Fermi surfaces in the ${\beta }_1$ phase with the isospin texture plotted. The $\langle {\psi }_{\mathbf{k}}^{†}{\sigma }^{x,y}{\psi }_{\mathbf{k}}\rangle$ texture winds around the Fermi surface twice in this case.

Figure 3

The first-order loop corrections to the quadratic and quartic interactions.

Figure 4

Integrated RG flow of the marginal couplings.

Figure 5

Schematic phase diagram implied by the RG equations at one-loop order. Not shown is the $\alpha$ axis where it is required that $\alpha >\left|\lambda \right|>0$ for stability.

Figure 6

Schematic of a layered structure comprised of two thin films of completely spin-polarized metals (“half-metals”) that are oppositely polarized. At the interface between the layers lies a two-dimensional electron liquid that remains ferromagnetically ordered due to the imbalance of spin-polarized 2D bands that may be controlled by gating. Here the bulk half-metals are assumed to be magnetically hard or pinned by hard magnets (not shown) above and below.

Figure 7

Diagrammatic lines and fermions for the Feynman rules of the Hubbard-Stratanovich decoupled action and with coupling to electromagnetism. The form of the EM coupling in Eqs. (2.13) and (2.14) yields two additional vertices that couple to $\mathbf{A}$ that are due to distortions and isospin textures on the Fermi surfaces due to $\mathrm{\Gamma }$.

Figure 8

The susceptibility bubble for the ${\mathrm{\Gamma }}_{\mu i}$ field, which also determines its temporal dependence under a gradient expansion of $(q_0,\mathbf{q})$.

Figure 9

The quartic interaction bubble $I_{\mu i,\nu j,\rho g,\lambda h}^{\left(4\right)}$ for the static-homogeneous mode $\mathrm{\Gamma }{(q=0)}_{\mu i}$.

Figure 10

The density $n$ with its perturbative correction $\delta n$ due to $\mathrm{\Gamma }$ at order $\mathrm{O}\left({\mathrm{\Gamma }}^2\right)$.

Figure 11

The conventional density-density response bubble.

Figure 12

The simplest quartic interaction terms at finite momentum $q$ that lead to damping from interband scattering in the broken symmetry mean-field phases. Here, $k^{\pm }=k\pm q/2$, where $k$ is the internal loop momentum. The diagrammatic notation follows that of Fig. 7 in Appendix pp1.