I discuss the simultaneous effects of Zeeman and spin-orbit fields during the evolution from Bardeen-Coopper-Schrieffer (BCS) to Bose-Einstein Condensation (BEC) superfluidity for ultra- cold fermions. I focus on spin-orbit couplings with equal Rashba and Dresselhaus (ERD) strengths which are realized experimentally, and show that topological phase transitions of the Lifshitz class occur through the emergence of Majorana and/or Dirac fermions as Zeeman and spin-orbit fields are varied. Topological quantum phase transitions in superfluids with non-s- wave order parameters have been conjectured theoretically for p-wave and d-wave systems for many years, but not yet observed experimentally due to the absence of tunable parameters. However, Zeeman or spin-orbit fields and interactions can be tuned in the context of ultra-cold atoms and allow for the visitation of several different phases. For systems with finite ERD spin- orbit coupling, but zero Zeeman fields, the evol ution from BCS to BEC superfluidity is only a crossover [1] as the system remains fully gapped. In contrast, for finite Zeeman fields, ERD spin- orbit coupling induces a triplet component in the order parameter that produces nodes in the quasiparticle excitation spectrum leading to bulk topological phase transitions of the Lifshitz type. Additionally, a fully gapped phase exists, where a crossover from indirect to direct gap occurs. For spin-orbit couplings with equal Rashba and Dresselhaus strengths the nodal quasi- particles are Dirac fermions that live at and in the vicinity of rings of nodes. Transitions from and to nodal phases can occur via the emergence of zero-mode Majorana fermions at phase boundaries, where rings of nodes of Dirac fermions annihilate [2]. Furthermore, I characterize different superfluid phases at low temperatures via spectroscopic and thermodynamic properties to conclude that Lifshitz is the Lord of the Rings [3], and that crossing the critic al temperature of the superfluid as a function of interaction, Zeeman field and spin-orbit coupling can reveal manifestations of such phases at finite temperatures [4].

- Li Han, C. A. R. S de Melo, Evolution from BCS to BEC superfluidity in the presence of spin-orbit coupling, Physical Review A 85, 011606(R) (2012), see also arXiv:1106.3613v1.
- Kangjun Seo, Li Han and C. A. R. S de Melo, Topological phase transitions in ultra-cold Fermi superfluids: the evolution from BCS to BEC under artificial spin-orbit fields, Physical Review A 85, 033601 (2012), see also arXiv:1108.4068v2.
- Kangjun Seo, Li Han, and C. A. R. S de Melo, Emergence of Majorana and Dirac particles in ultracold fermions via tunable interactions, spin-orbit effects, and Zeeman fields, Physical Review Letters 109, 105303 (2012), see also arXiv:1201.0177v1.
- Philip D. Powell, Gordon Baym, C. A. R. S de Melo, Superfluid transition temperature of spin-orbit and Rabi coupled fermions with tunable interactions, arXiv:1709.07042v1.

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