Topological feedback for superconducting states

The p-wave superfluid 3He has two distinct phases in zero fields. The pairing order parameter of A-phase has point nodes and, in the weak coupling approximation, B-phase with a fully gapped order parameter gains more condensation energy than A-phase. The reason for the overcoming stability of A-phase is the spin fluctuation feedback [1]. The pairing interaction due to the paramagnetic spin fluctuation becomes anisotropic in spin space due to the A-phase pair condensation, and the anisotropy gives the feedback and supports the A-phase pairing. More recently, it was found that the chiral p-wave pairing, which is a two-dimensional and superconducting analog of 3He-A phase, would have another type of feedback effect [2]. The chiral pairing leads to a spontaneous induction of a Chern-Simons(CS)-like term in the low energy effective action for the electromagnetic gauge field [3]. The coefficient of the induced term is closely related to the Chern integer, i.e., the chirality of the Cooper pair. Due to the charge-flux attachment by the CS-like term, each electron in a chiral Cooper pair feels the Lorentz force that strengthens the attractive interaction (see Figure 1). The topological nature of pairing thus causes this chiral feedback effect and supports the pairing itself.
We then present our attempt to find topological feedback mechanisms for the other types of superconducting states. The helical p-wave state, which is the time-reversal-symmetric (TRS) extension of the chiral p-wave state, would have a feedback effect caused by the induced BF term (TRS extension of the CS term) leading to the spin-dependent Lorentz force. We also investigate the three-dimensional extension of the chiral feedback effect for Weyl superconductors, which possess the axion term in the effective action.