  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.
