The combination of spin-orbit coupling and magnetic field produces an interesting bandstructure with a co-existing Dirac point and Fermi points. In the first part of my talk I will review how proximity-induced superconductivity is expected to realize Majorana zero modes. I will discuss theoretical understanding of the results from the experimental search for Majorana zero modes in spin-orbit coupled semiconductor-based topological superconducting systems. I will conclude this part discussing the possibility of interferometric detection of topological superconducivity.
In the second part we discuss strongly interacting Bosons subject to the same Hamiltonian, which would be the Tonks gas version of the spin-orbit coupled gases realized in experiments. We use the effective Lagrangian formalism for classifying non-relativistic Nambu-Goldstone modes due to derive the existance of an Ising transition between two gapless phases that can be thought of as a superfluid minimally coupled to an Ising gauge field. This coupling between the Ising model and the superfluid turns out to modify the dynamical critical exponent from Lorentz-like (z~1) to Lifshitz-like (z~2). We find these predictions to be consistent with DMRG simulations.