Theory of the low- and high-field superconducting phases of UTe$_2$

Recent nuclear magnetic resonance (NMR) and calorimetric experiments have observed that UTe$_2$ exhibits a transition between two distinct superconducting phases as a function of magnetic field strength for a field applied along the crystalline $b$-axis. To determine the nature of these phases, we employ a microscopic two-band minimal Hamiltonian with the essential crystal symmetries and structural details. We also adopt anisotropic ferromagnetic exchange terms. We study the resulting pairing symmetries and properties of these low- and high-field phases in mean field theory.