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Parity violating superfluidity in ultra-cold fermions under the influence of artificial non-Abelian gauge fields

We discuss the creation of parity violating Fermi superfluids in the presence of non-Abelian gauge fields involving spin-orbit coupling and crossed Zeeman fields. We focus on spin-orbit coupling with equal Rashba and Dresselhaus (ERD) strengths which has been realized experimentally in ultra-cold atoms, but we also discuss the case of arbitrary mixing of Rashba and Dresselhaus (RD) and of Rashba-only (RO) spin-orbit coupling. To illustrate the emergence of parity violation in the superfluid, we analyze first the excitation spectrum in the normal state and show that the generalized helicity bands do not have inversion symmetry in momentum space when crossed Zeeman fields are present. This is also reflected in the superfluid phase, where the order parameter tensor in the generalized helicity basis violates parity. However, the pairing fields in singlet and triplet channels of the generalized helicity basis are still parity even and odd, respectively. Parity violation is further reflected on ground state properties such as the spin-resolved momentum distribution, and in excitation properties such as the spin-dependent spectral function and density of states.