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Anapole superconductivity from $\mathcal{PT}$-symmetric mixed-parity interband pairing

Recently, superconductivity with spontaneous time-reversal or parity symmetry breaking is attracting much attention owing to its exotic properties, such as nontrivial topology and nonreciprocal transport. Particularly fascinating phenomena are expected when the time-reversal and parity symmetry are simultaneously broken. This work shows that time-reversal symmetry-breaking mixed-parity superconducting states generally exhibit an unusual asymmetric Bogoliubov spectrum due to nonunitary interband pairing. For generic two-band models, we derive the necessary conditions for the asymmetric Bogoliubov spectrum. We also demonstrate that the asymmetric Bogoliubov quasiparticles lead to the effective anapole moment of the superconducting state, which stabilizes a nonuniform Fulde-Ferrell-Larkin-Ovchinnikov state at zero magnetic fields. The concept of anapole order employed in nuclear physics, magnetic materials science, strongly correlated electron systems, and optoelectronics is extended to superconductors by this work. Our conclusions are relevant for any multiband superconductors with competing even- and odd-parity pairing channels. Especially, we discuss the superconductivity in UTe$_2$.