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Effective field theory, chiral anomaly and vortex zero modes for odd parity topological superconducting state of three dimensional Dirac materials

The low energy quasiparticle dispersion of various narrow gap and gapless semiconductors are respectively described by three dimensional massive and massless Dirac fermions. The three dimensional Dirac spinor structure admits a time-reversal invariant, odd parity and Lorentz pseudoscalar topological superconducting state. Here we derive the effective field theory of this topological paired state for massless Dirac fermions in the presence of a fluctuating Zeeman term, which appears as a chiral gauge field. The effective theory consists of a mixed electromagnetic and chiral anomaly term in the bulk, and a combination of pure and mixed anomalies for the surface. In this paper we demonstrate the existence of fermion zero modes in the dilute vortex phase under generic conditions. Guided by the existence of the zero modes and its intimate connection with the anomaly, we propose an effective topological field theory in the presence of Dirac mass. We briefly discuss the experimental consequences of the effective field theory and the zero modes for the low temperature unconventional superconducting states of $\mathrm{Cu}_x\mathrm{Bi}_2\mathrm{Se}_3$ and $\mathrm{Sn}_{1-x}\mathrm{In}_x\mathrm{Te}$.