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Time reversal invariant (TRI) topological superfluids (TSFs) and topologicalsuperconductors (TSCs) are robust symmetry protected gapped topological states.In this article, we study the evolution of these topological states in thepresence of time reversal symmetry breaking (TRB) fields and/or sufficientlylarge TRI fields. Physically, one of the realizations of TRB fields can beinternal spin exchange fields due to background magnetic ordering. We find thatthe fully gapped TSFs and TSCs are generically separated from other nodalstates by various zero temperature quantum critical points that arecharacterized by generalized quantum Lifshitz fields with distinct scalingproperties. These emergent Lifshitz fields also define finite temperatureproperties in quantum critical regimes. Moreover, for a certain subset of TRBfields, there exist a precursor to bulk transitions, where surface states canalso exhibit quantum critical behavior near zero fields. The possibility for afully gapped TSF/TSC to smoothly crossover into a topologically trivial statewithout a phase transition is also examined.

Topological Quantum Criticality in Superfluids and Superconductors: Surface criticality, Thermal properties, and Lifshitz Majorana fields