Quantum Liquid Crystals in an Imbalanced Fermi Gas: Fluctuations and Fractional Vortices in Larkin-Ovchinnikov States

Abstract

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 nematic superfluid. We expect this superfluid liquid-crystal phenomenology to be realizable in imbalanced resonant Fermi gases trapped isotropically.The tunability of interactions through Feshbach resonances has led to a realization of an s-wave paired superfluidity and BCS-BEC crossover [1,2,3], as well as promises of more exotic states such as gapless pwave[4] and periodic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) [5,6] superfluidity in strongly correlated degenerate alkali gases. The latter enigmatic state has been thoroughly explored within a BCS mean-field studies [7,8,9] and is expected to be realizable in a population-imbalanced (polarized) Feshbach resonant Fermi gas [10,11]. While recent experiments [12,13], have confirmed much of the predicted phenomenology of phase separation [11,14,15] in such systems, the FFLO states have so far eluded definitive observation.The simplest mean-field treatments [5,6,11] find that the FFLO type states are quite fragile, confined to a narrow range of polarization on the BCS side. However, motivated by earlier studies [7,8] and based on the finding of a negative domain-wall energy in an otherwise uniform singlet BCS superfluid [9,16], it has recently been argued that a more general periodic superfluid state that includes a larger set of collinear wavevectors may be significantly more stable. Much like a type-II superconductor undergoes a continuous transition into a vortex state at a lower-critical field H c1 , which is significantly below the thermodynamic field, here too, a Zeeman-field driven domain-wall nucleation (with the density increasing above the lower critical h c1 field) allows a continuous mechanism for a transition from a singlet paired superfluid to a LO-like periodic state [7,8,9,16].In this scenario the SF-LO transition is of a commensurate-incommensurate type as can be explicitly shown in one dimension (1d) [7,17]. The imposed species imbalance (excess of the majority fermionic atoms) can be continuously accommodated by the subgap states localized on the self-consistently induced domain-walls, with this picture resembling the doping of polyacetylene [18]. Such LO state can also be thought of as a periodically ordered micro-phase separation between the normal and BCS states, that thus naturally replaces the macro-phase separation ubiquitously found in the BEC-BCS detuning-polarization phase diag...

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

“…Consequently, as was originally anticipated by Shimahara [110], and was demonstrated in our recent work [112], to be explored in greater detail below, their Goldstone modes are qualitatively "softer", and therefore exhibit far stronger fluctuations. These can either completely destabilize the (otherwise energetically stable) FFLO state, or can qualitatively modify its meanfield form and properties.…”

“…However, given the extensive 45-year history of the topic, it is astounding that the equally basic complementary question of the nature of Goldstone modes description and their fluctuations within the FFLO states received so little attention, [110,111] until our study of the problem, reported in a recent Letter [112]. From the general symmetry principles the FFLO states' lowenergy phenomenology is expected to be significantly richer than that of a homogeneous fully gapped superconductor, whose low-energy phenomenological (GinzburgLandau and xy-model) description long predated the microscopic theory by BCS [99][100][101].…”

“…From the general symmetry principles the FFLO states' lowenergy phenomenology is expected to be significantly richer than that of a homogeneous fully gapped superconductor, whose low-energy phenomenological (GinzburgLandau and xy-model) description long predated the microscopic theory by BCS [99][100][101]. Namely, in addition to a local superfluid phase degree of freedom, the low-energy modes include the phonons (a single scalar one in the case of a uniaxial LO state) of the periodic superconducting structure, as well as gapless polarized (single-species) fermionic atoms confined to a fluctuating periodic array of two-dimensional domain walls [111,112].…”

Pair-density-wave superconducting order in two-leg ladders

“…FFLO states have been proposed over the years for different types of superconducting materials, most recently in the phase diagram of the heavy fermion superconductor CeCoIn 5 at finite magnetic fields, 14 although the experimental evidence for them is weak (at best). There are also recent theoretical proposals for FFLO-type states in cold atomic fermionic systems with unbalanced populations, 15 and in quantum wires of multi-valley semiconductors. 16 In the conventional theory of FFLO states one assumes a BCS-type system (a Fermi liquid) in which the spin up and down Fermi surfaces are split by the Zeeman interaction with an external magnetic field.…”

Pairing correlations in a trapped one-dimensional Fermi gas

“…Recent experiments [6] have explored two-species, attractively-interacting fermionic atomic gases in a quasi one-dimensional (1D) geometry, of interest since the regime of stability of the FFLO state is theoretically predicted [7,8] to be much wider than in the 3D case [9] (at least within the simplest mean-field approximation [10]). These experiments showed a remarkable quantitative agreement between experiment and theory for the local densities n σ (z) of the two species (σ =↑, ↓) of atoms within a theoretical approach that combined exact BetheAnsatz analysis of an infinite 1D gas with the local density approximation (LDA) to handle the spatial variation of the trap.…”