I review recent studies that predict quantum liquid-crystalline orders in resonant atomic gases. As examples of such putative systems I will discuss an s-wave resonant imbalanced Fermi gas and a p-wave resonant Bose gas. In the former, the liquid-cry
stalline smectic, nematic and rich variety of other descendant states emerge from strongly quantum- and thermally- fluctuating Fulde-Ferrell and Larkin-Ovchinnikov states, driven by a competition between resonant pairing and Fermi-surface mismatch. In the latter, at intermediate detuning the p-wave resonant interaction generically drives Bose-condensation at a finite momentum, set by a competition between atomic kinetic energy and atom-molecule hybridization. Because of the underlying rotationally-invariant environment of the atomic gas trapped isotropically, the putative striped superfluid is a realization of a quantum superfluid smectic, that can melt into a variety of interesting phases, such as a quantum nematic. I will discuss the corresponding rich phase diagrams and transitions, as well the low-energy properties of the phases and fractional topological defects generic to striped superfluids and their fluctuation-driven descendants.
Motivated by a realization of imbalanced Feshbach-resonant atomic Fermi gases, we formulate a low-energy theory of the Fulde-Ferrell and the Larkin-Ovchinnikov (LO) states and use it to analyze fluctuations, stability, and phase transitions in these
enigmatic finite momentum-paired superfluids. Focusing on the unidirectional LO pair-density wave state, that spontaneously breaks the continuous rotational and translational symmetries, we show that it is characterized by two Goldstone modes, corresponding to a superfluid phase and a smectic phonon. Because of the liquid-crystalline softness of the latter, at finite temperature the 3d state is characterized by a vanishing LO order parameter, quasi-Bragg peaks in the structure and momentum distribution functions, and a charge-4, paired Cooper-pairs, off-diagonal-long-range order, with a superfluid-stiffness anisotropy that diverges near a transition into a nonsuperfluid state. In addition to conventional integer vortices and dislocations the LO superfluid smectic exhibits composite half-integer vortex-dislocation defects. A proliferation of defects leads to a rich variety of descendant states, such as the charge-4 superfluid and Fermi-liquid nematics and topologically ordered nonsuperfluid states, that generically intervene between the LO state and the conventional superfluid and the polarized Fermi-liquid at low and high imbalance, respectively. The fermionic sector of the LO gapless superconductor is also quite unique, exhibiting a Fermi surface of Bogoliubov quasiparticles associated with the Andreev band of states, localized on the array of the LO domain-walls.
General symmetry arguments, dating back to de Gennes dictate that at scales longer than the pitch, the low-energy elasticity of a chiral nematic liquid crystal (cholesteric) and of a Dzyaloshinskii-Morya (DM) spiral state in a helimagnet with negligi
ble crystal symmetry fields (e.g., MnSi, FeGe) is identical to that of a smectic liquid crystal, thereby inheriting its rich phenomenology. Starting with a chiral Frank free-energy (exchange and DM interactions of a helimagnet) we present a transparent derivation of the fully nonlinear Goldstone mode elasticity, which involves an analog of the Anderson-Higgs mechanism that locks the spiral orthonormal (director/magnetic moment) frame to the cholesteric (helical) layers. This shows explicitly the reduction of three orientational modes of a cholesteric down to a single phonon Goldstone mode that emerges on scales longer than the pitch. At a harmonic level our result reduces to that derived many years ago by Lubensky and collaborators.
We develop a low-energy model of a unidirectional Larkin-Ovchinnikov (LO) state. Because the underlying rotational and translational symmetries are broken spontaneously, this gapless superfluid is a smectic liquid crystal, that exhibits fluctuations
that are qualitatively stronger than in a conventional superfluid, thus requiring a fully nonlinear description of its Goldstone modes. Consequently, at nonzero temperature the LO superfluid is an algebraic phase even in 3d. It exhibits half-integer vortex-dislocation defects, whose unbinding leads to transitions to a superfluid nematic and other phases. In 2d at nonzero temperature, the LO state is always unstable to a charge-4 nematic superfluid. We expect this superfluid liquid-crystal phenomenology to be realizable in imbalanced resonant Fermi gases trapped isotropically.
We present an exact three-dimensional solitonic solution to a sine-Gordon-type Euler-Lagrange equation, that describes a configuration of a three-dimensional vector field n constrained to a surface p-vortex, with a prescribed polar tilt angle on a pl
anar substrate and escaping into the third dimension in the bulk. The solution is relevant to characterization of a schlieren texture in nematic liquid-crystal films with tangential (in-plane) substrate alignment. The solution is identical to a section of a point defect discovered many years ago by Saupe [Mol. Cryst. Liq. Cryst. 21, 211 (1973)], when latter is restricted to a surface.
The atomic Bose gas is studied across a Feshbach resonance, mapping out its phase diagram, and computing its thermodynamics and excitation spectra. It is shown that such a degenerate gas admits two distinct atomic and molecular superfluid phases, wit
h the latter distinguished by the absence of atomic off-diagonal long-range order, gapped atomic excitations, and deconfined atomic pi-vortices. The properties of the molecular superfluid are explored, and it is shown that across a Feshbach resonance it undergoes a quantum Ising transition to the atomic superfluid, where both atoms and molecules are condensed. In addition to its distinct thermodynamic signatures and deconfined half-vortices, in a trap a molecular superfluid should be identifiable by the absence of an atomic condensate peak and the presence of a molecular one.
