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arXiv:1102.4903 · Physical Review A

Fluctuations and phase transitions in Larkin-Ovchinnikov liquid-crystal states of a population-imbalanced resonant Fermi gas

Leo Radzihovsky

Published Aug 8, 2011• 127 citations
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Abstract

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, which 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 three-dimensional 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.

DOI
10.1103/physreva.84.023611

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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, which 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 three-dimensional 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.
We develop a low-energy model of an 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 nematic superfluid. We expect this superfluid liquid-crystal phenomenology to be realizable in imbalanced resonant Fermi gases trapped isotropically.
Motivated by the recent experimental realization of a candidate to the Fulde-Ferrell (FF) and the Larkin-Ovchinnikov (LO) states in one dimensional (1D) atomic Fermi gases, we study the quantum phase transitions in these enigmatic, finite momentum-paired superfluids. We focus on the FF state and investigate the effects of the induced interaction on the stability of the FFLO phase in homogeneous spin-imbalanced quasi-1D Fermi gases at zero temperature. When this is taken into account we find a direct transition from the fully polarized to the FFLO state. Also, we consider the effect of a finite lifetime of the quasi-particles states in the normal-superfluid instability. In the limit of long lifetimes, the lifetime effect is irrelevant and the transition is directly from the fully polarized to the FFLO state. We show, however, that for sufficiently short lifetimes there is a quantum critical point (QCP), at a finite value of the mismatch of the Fermi wave-vectors of the different quasi-particles, that we fully characterize. In this case the transition is from the FFLO phase to a normal partially polarized state with increasing mismatch.
We present an extensive study of two-dimensional Larkin-Ovchinnikov (LO) superfluidity in a spin-imbalanced two-component atomic Fermi gas. In the context of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, we explore a wide and generic class of pairing gap functions with explicit spatial dependency. The mean-field theory of such phases is applied through the Bogoliubov-de Gennes equations in which the pairing gap can be determined self-consistently. To systematically explore the configuration space we consider both the canonical and grand canonical ensembles where we control the polarization or chemical potentials of the system, respectively. The mean-field calculations enable us to understand the nature of the phase transitions in the fully paired Bardeen-Cooper-Schrieffer (BCS) state, exotic LO phase, and partially polarized free Fermi gas (NPP{N}_{\text{PP}}NPP​). The order of the phase transitions has been examined and, in particular, we find a weak first-order phase transition between the exotic inhomogeneous LO phase and the BCS phase. In comparison to the three-dimensional case, where the phase diagram is dominated by a generic separation phase, we predict a broader parameter space for the spatially inhomogeneous LO phase. By computing the superfluid density of the LO phase at different polarization, we show how the superfluidity of the system is suppressed with increasing spin polarization.
Fulde-Ferrell-Larkin-Ovchinnikov Superfluids with PPP-wave Symmetry in Spinless Fermionic Optical Lattices
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, whose Cooper pairing hosts a non-zero center-of-mass momentum, is predicted and well studied in spin-singlet superfluids in past decades. Different from previous works that engineer FFLO superfluids in spinful cold atoms, we show that the FFLO superfluids, with ppp-wave symmetry, can emerge in a spinless environment associated with nearest-neighbor-site interactions. The novel mechanism of the FFLO superfluids lies on the split Fermi surfaces at low filling factor, which naturally gives rise to Cooper pairings with finite momentum. The phase transition is accompanied by changed Chern numbers, in which, different from the conventional picture, the band gap does not close. The exotic topological transition can be explained by the spontaneously-broken translational invariance of order parameters. Finally, we investigate the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature of the FFLO state, and show its possibility to be observed at finite temperature.
In ultracold Fermi gases, the effect of spin imbalance on superfluidity has been the subject of intense study. One of the reasons for this is that spin imbalance frustrates the Bardeen-Cooper-Schrieffer (BCS) superfluid pairing mechanism, in which fermions in different spin states combine into Cooper pairs with zero momentum. In 1964, it was proposed that an exotic superfluid state called the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in which the Cooper pairs have nonzero momentum, could exist in a spin-imbalanced Fermi gas. At the saddle-point (mean-field) level, it has been shown that the FFLO state only occupies a very small sliver in the ground-state phase diagram of a three-dimensional (3D) Fermi gas. However, a question that remains to be investigated is as follows: What is the influence of phase fluctuations on the FFLO state? In this work, we show that phase fluctuations only lead to relatively small quantitative corrections to the presence of the FFLO state in the saddle-point phase diagram of a 3D spin-imbalanced Fermi gas. Starting from the partition function of the system, we calculate the effective action within the path-integral adiabatic approximation. The action is then expanded up to second order in the fluctuation field around the saddle point, leading to the fluctuation free energy. Using this free energy, we calculate corrections due to phase fluctuations to the BCS-FFLO transition in the saddle-point phase diagram. At temperatures at which the FFLO state exists, we find only small corrections to the size of the FFLO area. Our results suggest that fluctuations of the phase of the FFLO order parameter, which can be interpreted as an oscillation of its momentum vector, do not cause an instability of the FFLO state with respect to the BCS state.
The possibility of a Fulde-Ferrell-Larkin-Ovchinnikov-like (FFLO-like) state in a population-imbalanced Fermi gas with a vortex is proposed. Employing the Bogoliubov-de Gennes formalism, we determine self-consistently the superfluid order parameter and the particle number density in the presence of a vortex. We find that, upon increasing the population imbalance, the superfluid order parameter spatially oscillates around the vortex core in the radial direction, indicating that the FFLO-like state becomes stable. We find that the radial FFLO-like states cover a wide region of the phase diagram in the weak-coupling regime at T=0T=0T=0, in contrast with the conventional case without a vortex. We show that this inhomogeneous superfluidity can be detected as peak structures of the local polarization rate associated with the node structure of the superfluid order parameter. Since the vortex in the three-dimensional Fermi gas with population imbalance has been already realized in experiments, our proposal is a promising candidate of a FFLO-like state in cold atom physics.