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

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Abstract

We discuss the emergence of rings of zero-energy excitations in momentum space for superfluid phases of ultra-cold fermions when spin-orbit, Zeeman fields and interactions are varied. We show that phases containing rings of nodes possess non-trivial topological invariants, and that phase transitions between distinct topological phases belong to the Lifshitz class. Upon crossing phase boundaries, existing massless Dirac fermions in the gapless phase anihilate to produce bulk zero-mode Majorana fermions at phase boundaries and then become massive Dirac fermions in the gapped phase. We characterize these tunable 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 transitions in momentum space, we conclude that Lifshitz is the lord of the rings.

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\(Z_2\) Topological Order and the Quantum Spin Hall Effect

C. L. Kane, E. Mele (2005)

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. We show that this phase is associated with a novel \(Z_2\) topological invariant, which distinguishes it from an ordinary insulator. The \(Z_2\) classification, which is defined for time reversal invariant Hamiltonians, is analogous to the Chern number classification of the quantum Hall effect. We establish the \(Z_2\) order of the QSH phase in the two band model of graphene and propose a generalization of the formalism applicable to multi band and interacting systems.

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A spin-orbit coupled Bose-Einstein condensate

I. B. Spielman, K Jiménez-García, Y. Lin (2011)

Spin-orbit (SO) coupling -- the interaction between a quantum particle's spin and its momentum -- is ubiquitous in nature, from atoms to solids. In condensed matter systems, SO coupling is crucial for the spin-Hall effect and topological insulators, which are of extensive interest; it contributes to the electronic properties of materials such as GaAs, and is important for spintronic devices. Ultracold atoms, quantum many-body systems under precise experimental control, would seem to be an ideal platform to study these fascinating SO coupled systems. While an atom's intrinsic SO coupling affects its electronic structure, it does not lead to coupling between the spin and the center-of-mass motion of the atom. Here, we engineer SO coupling (with equal Rashba and Dresselhaus strengths) in a neutral atomic Bose-Einstein condensate by dressing two atomic spin states with a pair of lasers. Not only is this the first SO coupling realized in ultracold atomic gases, it is also the first ever for bosons. Furthermore, in the presence of the laser coupling, the interactions between the two dressed atomic spin states are modified, driving a quantum phase transition from a spatially spin-mixed state (lasers off) to a phase separated state (above a critical laser intensity). The location of this transition is in quantitative agreement with our theory. This SO coupling -- equally applicable for bosons and fermions -- sets the stage to realize topological insulators in fermionic neutral atom systems.