Axions and superfluidity in Weyl semimetals

An effective field theory (EFT) for dynamical axions in Weyl semimetals (WSMs) is presented. A pseudoscalar axion excitation is predicted in WSMs at sufficiently low temperatures, independently of the strength of the Weyl fermion self-coupling. For strong fermion self-coupling the axion is the gapless Goldstone boson of chiral $\mathrm{U}{\left(1\right)}^{\text{ch}}$ spontaneous symmetry breaking. For weak fermion self-coupling an axion is also generated at nonzero chiral density for Weyl nodes displaced in energy, as a gapless collective mode of correlated fermion pair excitations of the Fermi surface. This is an explicit example of the extension of Goldstone's theorem to symmetry breaking by the axial anomaly itself. In both cases, the axion is a chiral density wave or phason mode of the superfluid state of the WSM, and the Weyl fermions form a chiral condensate $\langle \overline{\psi }\psi \rangle$ at low temperatures. In the presence of an applied magnetic field, the axion mode becomes gapped, in analogy to the Anderson–Higgs mechanism in a superconductor. 't Hooft anomaly matching from ultraviolet to infrared scales is directly verified in the EFT approach. WSMs thus provide an interesting quantum system in which superfluid, non-Fermi liquid behavior, and a dynamical axion are predicted to follow directly from the axial anomaly in a consistent EFT that may be tested experimentally.

Figure 1

The one-loop axial anomaly $\langle J_5^{\mu }{J}^{\alpha }{J}^{\beta }\rangle$ triangle diagram, which is ${\mathrm{\Gamma }}_5^{\mu \alpha \beta }(p,q)$ of (A4) in momentum space. The solid lines represent the propagators of Dirac fermions and the wavy lines external photon legs, which carry off four-momenta $p$ and $q$ from the electromagnetic current vertices $J^{\alpha }$ and $J^{\beta }$. The incoming four-momentum at the axial current vertex $J_5^{\mu }$ is $k=p+q$ by momentum conservation.

Figure 2

The $e^{+}{e}^{-}$ co-linear massless fermion pair state of opposite helicities from the triangle diagram of Fig. 1 represented as a single massless effective boson.

Figure 3

Two Weyl cones separated by $(2b^0,2\mathit{b})$ in momentum space. Here we assume zero charge chemical potential, so the Weyl cones are filled up to the Fermi energy, as indicated by the fill pattern. We also neglect the effects of nonlinear interactions at energies ${\mathrm{\Lambda }}_W$ far from the Weyl nodes, so that the EFT description applies for energy excitations and $|2b^0|\ll {\mathrm{\Lambda }}_W$.

Figure 4

The triangle diagram that contributes to the bosonic part of the axial current vertex $J_5^{\mu }\left[\mathrm{\Phi }\right]$ of (66) to two-photon amplitude ${\mathrm{\Gamma }}_5^{\mu \alpha \beta }(p,q)$, given by (92) in position space or (94) in momentum space. Its contraction with $k_{\mu }$ gives a contribution to the axial anomaly which is equal and opposite to the mass-dependent contribution of the fermion vertex $J_5^{\mu }\left[\psi \right]$ of (65) given by (A10).

Figure 5

The attractive channel of magnetic interactions of fermions leading to Cooper pairing. The photon line must be dressed by interactions in the chiral plasma if Debye screening is present at nonzero ${\mu }_5$.

Figure 6

The one-loop fermion self-energy $\mathrm{\Sigma }$ to be calculated for nonzero chiral potential ${\mu }_5$ needed for the gap equation in the ASB case. The photon propagator is the same as in Fig. 5.

Figure 7

The one-loop axial anomaly in two dimensions.