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Nonrelativistic axion electrodynamics in $p+is$ superconductors

In previous works, axion electrodynamics in three dimensional $p+is$ superconductors is discussed by borrowing the results from superconducting Dirac systems. However, in this work, based on a systematic path integral approach, we show that the axion electrodynamics in $p+is$ superconductors exhibits a nonrelativistic form, which is different from the superconducting Dirac and Weyl systems. More precisely, the induced electric field does not enter into the axion action, and gauge invariance is ensured by the combination of the electric potential and the Nambu-Goldstone phase mode. Furthermore, unlike the axion angle in the Dirac case which is equal to the difference between the superconducting phases on the two Fermi surfaces of different helicities, the axion angle in the present case contains an additional sinusoidal term. As applications of the derived nonrelativistic axion electrodynamics, physical effects related to vortex lines and Witten effect are discussed. Our work reveals the differences for axion electrodynamics between the relativistic and nonrelativistic systems.