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We discuss the emergence of rings of zero-energy excitations in momentumspace for superfluid phases of ultra-cold fermions when spin-orbit, Zeemanfields and interactions are varied. We show that phases containing rings ofnodes possess non-trivial topological invariants, and that phase transitionsbetween distinct topological phases belong to the Lifshitz class. Upon crossingphase boundaries, existing massless Dirac fermions in the gapless phaseanihilate to produce bulk zero-mode Majorana fermions at phase boundaries andthen become massive Dirac fermions in the gapped phase. We characterize thesetunable topological phase transitions via several spectroscopic properties,including excitation spectrum, spectral function and momentum distribution.Since the emergence or disappearance of rings leads to topological transitionsin momentum space, we conclude that Lifshitz is the lord of the rings.

Who is the Lord of the Rings: Majorana, Dirac or Lifshitz? The Spin-Orbit-Zeeman Saga in Ultra-cold Fermions