Quantum liquid-crystal order in resonant atomic gases

Quantum liquid-crystal order in resonant atomic gases


I review recent studies that predict a realization of 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-crystalline 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.


I Introduction

i.1 Resonant atomic gases

Experimental progress in trapping, cooling and coherently manipulating Feshbach-resonant atomic gases opened unprecedented opportunities to study degenerate strongly interacting quantum many-body systems in a broad range of previously unexplored regimes (1); (2); (3); (4); (5). These include paired fermionic superfluids (SF) (6); (7); (8); (9); (10); (11); (12); (13); (14); (15)), the associated Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover(16); (17); (18); (19); (20); (21); (22); (23); (24); (3), Bose-Fermi mixtures(25), bosonic molecular superfluids(26); (27); (28); (29), and many other states and regimes(30) under both equilibrium and nonequilibrium conditions(31); (23); (32).

Because degenerate atomic gases are free of the underlying crystalline matrix of the solid-state materials (though one can be imposed through a highly tunable optical lattice potential(1)), among this rich variety of states, they admit phases that spontaneously partially break continuous spatial symmetries and thereby exhibit concomitant strongly fluctuating Goldstone modes with corresponding rich phenomenology. Resonant atomic gases are thus uniquely suited for a realization of quantum liquid-crystalline states of matter, that have been somewhat of a holy-grail dating back to their studies in solid state materials such as the striped states in high-T superconductors, nickelates and other strongly correlated doped Mott insulators(33); (34); (35); (36); (37); (38), heavy-fermion and organic superconductors(39); (40), spiral states in helimagnets(41); (42); (43), and a two-dimensional electron gas with a partially-filled Landau level(44); (46); (47). In isotropic traps the putative quantum liquid-crystal order is expected to exhibit all the complexity of fluctuations and topological defects of conventional (mesogenic) liquid crystals(48); (49), but with the added enrichment of the accompanying quantum (off-diagonal) order of a superfluid. Another, not insignificant virtue is that (in contrast to other e.g., solid-state or nuclear matter systems) these dilute gases are extremely well-characterized at the two-body level, and are therefore described by microscopic (as opposed to effective) Hamiltonians with well-known couplings.

i.2 Candidate systems

Recent theoretical studies have predicted a number of such quantum liquid crystal realizations in degenerate atomic systems, that in addition to internal symmetries partially break spatial symmetries(50); (51); (52); (53); (54); (55); (56). These are typically driven by strong resonant and competing interaction that frustrates a spatially homogeneous and isotropic superfluidity. Known examples include bosonic and paired fermionic superfluids, where spatial order is driven by (i) dipolar interaction(60), (ii) pseudo-spin-orbit interaction(57) (realized through hyperfine states coupled by Raman transitions(58)), (iii) p-wave resonant interaction(53); (59), and (iv) a Fermi surface mismatch (realized through species number and/or mass imbalance), that leads to the Larkin-Ovchinnikov-Fulde-Ferrell finite-momentum pairing(50); (51), as well as their strongly fluctuating descendent states. In this brief review, I will focus on the last two realizations, and will discuss the associated microscopic models that I believe can realize a quantum superfluid liquid-crystal order, their phase behavior, fluctuations, topological defects, and a variety of experimental predictions and signatures. Much of the discussed low-energy phenomenology is shared more generally by systems exhibiting quantum liquid crystal orders in isotropic trap(54). For a more complete account, I refer the reader to the original literature and the more extensive reviews(2); (5).

I will not discuss the finite momentum states that depend on the lattice for their realization and stability, such as the p-band and FFLO superfluids in optical lattices(61); (62); (63); (64); (65). These are fascinating states, but are less relevant from the liquid-crystal perspective of this review.

Ii Imbalanced resonant Fermi gases

ii.1 Background

The most widely explored candidate for a realization of quantum liquid crystal order is a species-imbalanced Feshbach-resonant Fermi gas, (2); (3); (5), though it is only very recently that it was formulated and explored in these liquid-crystal terms(66); (67). These studies build on well-explored two-species Feshbach-resonant Fermi gas, that exhibits paired superfluidity, that can be tuned between a weakly-attractive Fermi-surface-driven BCS and a strongly-attractive molecular BEC superfluids(4); (3). While at a balanced gas exhibits no qualitative change of state, a quantitatively accurate description of this crossover, particularly around the strongly interacting and universal unitary regime (where in a vacuum a two-particle bound state first forms and the s-wave scattering length diverges) has presented a considerable challenge with much recent progress.

A species-number (and mass) imbalance in the two atomic hyperfine-states mixture offered a new extremely fruitful experimental knob(68); (69); (70); (71). The imbalance frustrates pairing(72); (73); (74); (75); (76); (77), driving quantum phase transitions out of the paired superfluid to a variety of possible ground states and thermodynamic phases(78); (52); (79); (80); (81); (82); (83). This rekindled considerable theoretical activity in the context of species-imbalanced resonant Fermi gases(84); (85); (86); (87); (88); (89); (90); (91); (92); (93); (94); (95); (96); (97); (98); (99); (100); (101); (102); (103); (104); (105); (106). The corresponding imbalance versus detuning BCS-BEC phase diagram, illustrated in Fig.1 (and its extension to finite temperature) is now well-established(79); (100); (103), showing qualitative agreement with experiments(68); (69); (70); (71). More recently, considerable progress has been made toward establishing quantitative details of this phase diagram through analytical(107); (108); (109); (110); (111), numerical(112); (113) and experimental approaches(70); (71); (114).

The identification of the number species imbalance with the magnetization of an electronic system, and the chemical potential difference with an effective Zeeman energy, connects these atomic gases studies with a large body of research on solid state electronic superconductors under a Zeeman field(115); (116); (117); (50); (51), as well as extensively studied realizations in nuclear and particle physics(118); (119); (120); (121). The obvious advantage of the newly-realized atomic system is the aforementioned tunability, disorder-free “samples”, and absence of the orbital part of the magnetic field, that always accompanies a solid-state charged superconductor in a magnetic field. In these neutral paired superfluids the orbital field effects can be independently controlled by a rotation of the atomic cloud(122).

As illustrated in Fig.1, among many interesting features, such as the gapless imbalanced superfluid ()(79); (101); (80); (81), ubiquitous phase separation(74); (79); (101); (100), tricritical point(100); (103); (106), etc., observed experimentally(68); (69); (70); (71) and studied extensively theoretically(5), the interaction–imbalance BEC-BCS phase diagram is also predicted(84); (79); (100); (81); (104); (105) to exhibit the enigmatic Fulde-Ferrell-Larkin-Ovchinnikov state (FFLO)(50); (51). First predicted in the context of solid-state superconductors over 45 years ago, the FFLO states has so far eluded a definitive observation, though some promising solid state(123) and quasi-1d atomic(124) candidate systems have recently been realized.

At its most generic level the FFLO state is a fermionic superfluid, paired at a finite center of mass momentum. It spontaneously “breaks” gauge and translational symmetry, a periodically-paired superfluid (superconductor), akin to a supersolid(125); (126); (127); (128), and thus can appropriately be called a pair-density wave (PDW)(129); (130). This state can be equivalently thought of as a periodically ordered micro-phase separation between the normal and paired states, that naturally replaces the macro-phase separation(121); (74) ubiquitously found in the BCS-BEC detuning-imbalance phase diagram(79); (101); (100); (103).

Microscopically, it is driven by Fermi surface mismatch(50); (51) due to an imposed pairing species number (and/or mass(132)) imbalance. As a compromise between the superfluid pairing and an imposed imbalance, at intermediate values of the latter, the superconducting order parameter condenses at a set of finite center-of-mass momenta determined by the details of the Fermi surface mismatch and interactions. At sufficiently large imbalance, no compromise is possible, and the resonant gas transitions to the normal state.

Figure 1: A mean-field zero-temperature phase diagram from Refs.(79); (100) of an imbalanced resonant Fermi gas, as a function of the inverse scattering length and normalized species imbalance , showing the magnetized (imbalanced) superfluid (), the FFLO state (approximated as the simplest FF state) confined to a narrow red sliver bounded by and , and the imbalanced normal Fermi liquid.

As illustrated in Fig.1, the key observation is that, despite strong interactions, within simplest mean-field treatments the conventional FFLO state(50); (51) remains quite fragile, confined to a narrow sliver of polarization in the BCS regime(79); (100); (131); (5). I emphasize that in fact above conclusion is only rigorously valid for the FF and not other forms (e.g., LO) of the FFLO class of states. Although the upper boundary, , just below the normal state is trustworthy, as it is shared by all FFLO states, the lower one, can strongly depend on the form of the FFLO state, but was determined by Sheehy and Radzihovsky only for the FF state(79); (100); (133). Furthermore, motivated by earlier studies of the Bogoluibov-de Gennes (BdG) equation for the LO state(134); (135); (136), combined with finding of a negative domain-wall energy in an otherwise fully-paired singlet BCS superfluid in Zeeman field(136); (137), these studies have quite convincingly argued, that a more generic pair-density wave state (that includes a larger set of collinear wavevectors(118); (119)) may be significantly more stable.

Figure 2: An illustration of a continuous commensurate-incommensurate (CI) transition at from a fully-gapped (balanced) paired-superfluid to an imbalanced Larkin-Ovchinnikov superfluid. The excess majority atoms are localized on the domain walls in (zeros of) the LO order parameter, whose number is then proportional to the imbalance and grows continuously with the chemical potential difference (Zeeman energy), .

The quantitative extent of the energetic stability of a PDW states in the imbalance-detuning phase diagram, in my view remains a widely open and urgent question. Consistent with above arguments but in absence of controlled quantitative analysis, I take the optimistic point of view that the LO states can extend over significantly wider region of the phase diagram, as schematically illustrated in Fig.3. Assuming it is indeed energetically stable, its phenomenology has been explored beyond its mean-field cartoon(50); (51) (latter only appropriate in the solid state and optical lattices, but not in the isotropically-trapped resonant atomic gases, where fluctuations are large).

Figure 3: A proposed vs. phase diagram for an imbalanced resonant Fermi gas, showing the more stable LO liquid crystal phases (discussed in the text and illustrated in detail Fig.5) replacing a portion of the phase-separated regime.

ii.2 Summary

Before turning to details, I summarize the salient features of the isotropically-trapped collinear class of FFLO states, beyond their mean-field approximation. The key observation(66); (67) is that such striped states spontaneously break continuous rotational symmetry, and as a result exhibit phonon-like Goldstone modes that are qualitatively (energetically) softer than the solid-state analogs (where only discrete rotational symmetry can be broken) and all other superfluids. Namely, striped FF(138) and LO classes of states are characterized by highly anisotropic (with the modulation wavevector and transverse to it) collective spectra, with


rather than by the usual linear-in-momentum Bogoluibov sound mode. The more experimentally relevant striped LO state, also exhibits a quantitatively anisotropic Bogoluibov linear-in- sound mode, but with the superfluid stiffness ratio that vanishes as the transition to the normal imbalanced Fermi gas is approached from below.

As a result, the fluctuations in such ”soft” superfluid smectic states are qualitatively stronger. Although the states are stable to quantum fluctuations, in 3d the LO and FF long-range orders are marginally unstable at any nonzero temperature. Consequently, (seemingly paradoxically) inside the LO state the average LO order parameter


vanishes in the thermodynamic limit (of a large cloud with atom number and cloud size ), suppressed to zero by thermal phonon fluctuations. The LO state is therefore strictly speaking homogeneous on long scales, exhibiting “algebraic topological”, but no long-ranged translational order. Namely, the mean-field approximation fails qualitatively and the state instead is characterized by power-law order-parameter correlations, distinguished from the spatially short-ranged disordered phase by confined topological defects (bound dislocations), not by a nonzero LO order parameter. It is therefore a 3d analog of the more familiar quasi-long-range ordered superfluid film, a 2d easy-plane ferromagnet and a 2d crystal(139); (141); (142); (143); (144).

As a consequence, a 3d LO state is characterized by a static structure function and momentum distribution function with universal anisotropic quasi-Bragg peaks (around ), akin to the Landau-Peierls(139); (140) behavior of films of a conventional superfluid and 2d crystals(141); (142); (143); (144). Such novel behavior is not, however, exhibited by 3d crystalline FFLO states with multiple non-collinear ordering wavevectors(119); (118), that, in contrast are characterized by the long-range positional order and a nonzero pair-condensate, that is stable to thermal fluctuations.

Another fascinating feature that arises because the LO order parameter, is a product (rather than the sum, as it is in a conventional supersolid) of the superfluid and density-wave component, is the unusual topological excitation that is a half-vortex bound to a half-dislocation – in addition to integer vortices and dislocations.

In 2d, at nonzero the LO state is even more strongly disordered, characterized by short-range positional order with Lorentzian structure function peaks, and unstable to proliferation of dislocations(145). The state that results from such dislocated superfluid smectic is either a “charge”-4 (paired Cooper pairs) nematic superfluid(66); (37) or a nematic (possibly “fractionalized”) Fermi liquid(35); (47), latter qualitatively the same as the deformed Fermi surface state (52).

Furthermore, a consideration of states that arise due to unbinding of various combination of topological defects (illustrated in the flow-chart in Fig.4) leads to a rich array of LO descendent states, that generically must intervene between the LO superfluid and a fully-paired conventional (isotropic and homogeneous) superfluid and a conventional polarized Fermi liquid. If indeed, as argued above, the 3d LO state is energetically stable, these novel states are expected to appear in the region collectively denoted “LO liquid crystals” of the detuning-polarization phase diagram of Fig.3. They include a nonsuperfluid smectic (, driven by an unbinding of integer -vortices), and a superfluid (, driven by a proliferation of integer -dislocations) and a nonsuperfluid (, driven by an unbinding of both vortices and dislocations) nematics, and the corresponding isotropic states, when disclinations also condense. In addition, a variety of topologically-ordered isotropic and nematic “fractionalized” Fermi-liquid states (, , , and others) were predicted(66); (67), that are distinguished from their more conventional fully-disordered forms by gapped (bound) half-integer defects. These phases are summarized by a flowchart Fig.4 and a schematic phase diagram illustrated in Fig.5.

Figure 4: A flowchart of superfluid () and nonsuperfluid () phases, exhibiting smectic () and nematic () conventional orders as well as topological orders (indicated by and ), induced by a proliferation of various combination of topological defects, , , and .

Finally, 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. Consequences of the interplay between these fermionic and Goldstone mode degrees of freedom remains an open problem.

Figure 5: A schematic imbalance-chemical potential (Zeeman energy), vs detuning (interaction strength), phase diagram, illustrating the 3d LO smectic phase () and its descendant (described in the text), driven by a proliferation of various combinations of topological defects. The inset shows the global imbalance-interaction BCS-BEC phase diagram, illustrating the location of these putative phases.

ii.3 Microscopic model of imbalanced Fermi gas

The physics of imbalanced atomic Fermi gases interacting through a broad Feshbach resonance is well captured by the one-channel model (3); (4); (5); (146), characterized by a grand-canonical Hamiltonian


with the single-particle energy . The separately conserved number of atomic species (hyperfine states) ) are imposed by two chemical potentials, .

The key feature that distinguishes this Fermi system from those familiar from solid state contexts is the attractive resonant interaction parameterized by a short-range s-wave pseudopotential, . Through an exact T-matrix scattering calculation(3), controls the magnetic-field tunable (148) scattering length


that diverges above a critical attraction strength, ( is the short-scale pseudo-potential cutoff set by the extent of the molecular bound state), corresponding to a formation of a two-atom bound state.

The many-body thermodynamics of the resonant Fermi gas as a function of , (i.e., the extension of the BEC-BCS crossover to a finite imbalance, ) at large presents a formidable challenge. However, much progress has been made in mapping out its qualitative (and in some regimes quantitative) phenomenology through a variety of approximate theoretical techniques, including quantum Monte Carlo (76), mean-field theory (79); (101); (80); (100); (103), the large- (fermion flavor) (109); (110) and -expansions (107).

The simplest of these is the standard mean-field analysis(79); (100) that gives a satisfactory qualitative description (quantitatively valid deep in the weakly-coupled BCS regime, ), as a starting point of more sophisticated treatments. To this end we assume the existence of a condensate


corresponding to pair-condensation at momenta , with the set of amplitudes and to be self-consistently determined by the minimizing the ground state energy subject to the constraints of fixed total atom number and the atom species number imbalance (“polarization”) , imposed by the average and difference chemical potentials , latter corresponding to the pseudo Zeeman energy.

Specializing to the simplest case of a single of the Fulde-Ferrell state(50) this reduces the Hamiltonian to a quadratic Bogoluibov form, that can be easily diagonalized. This gives(79); (100) the ground state energy


and the excitation spectrum


with and . (and its generalizations to finite-temperature free energy(103)), then gives all the thermodynamics, including the phase behavior summarized by the phase diagram in Fig.1.

In particular, this analysis predicts the existence of the FF state, stable only over a narrow sliver of imbalance, closing down for  (79); (100). As mentioned in the Introduction, there are compelling arguments suggesting that this fragility is specific to the single planewave FF condensate, and that the more generic PDW states are far more stable because they allow energetically important amplitude modulation (134); (135); (136); (137); (67). However, general PDW states are difficult to analyze at the transition from the fully gapped, balanced paired superfluid, near the lower critical field (at vanishing imbalance). In contrast, an analytic analysis near the upper-critical chemical potential difference at the transition from the normal state is indeed possible as a controlled Ginzburg-Landau expansion in the small pair-density wave amplitude, (51); (66); (147); (67). While not quantitatively accurate away far below the transition (where PDW order parameter is large and is not limited to a single Fourier component ) such Landau expansion is expected to be qualitatively correct and is a good starting point for a more complete analysis of fluctuations and phase transitions into the PDW LO state.

ii.4 Order-parameter theory of FFLO states

Ginzburg-Landau expansion near

The analytical treatment of the FFLO states near relies on the Ginzburg-Landau expansion in , that is small near the (in mean-field) continuous normal-to-FFLO transition (51); (149). This expectation is supported by the exact 1d BdG solution(134) at high fields, where is indeed well-approximated by a single sinusoid, with an amplitude that vanishes continuously near .

Consistent with these general arguments, by integrating out the atomic degrees of freedom, near the Ginzburg-Landau expansion for the ground-state energy takes a familiar form

where the dispersion is given by(51); (79); (100); (67)


whose minimum at a finite (near ) captures the imbalanced atomic Fermi system’s energetic tendency to pair at a finite momentum, and thereby to form a pair-density wave characterized by a reciprocal lattice vector with magnitude and a spontaneously chosen orientation. The value of at which vanishes determines the corresponding mean-field N-FFLO transition point. While at quadratic order, all Fourier modes with are degenerate, becoming unstable simultaneously, the form of the FFLO state is dictated by the interaction vertex function, that has been explicitly computed. Near the transition the physics of a unidirectional pair-density wave (Cooper-pair stripe) order, characterized by a collinear set of ’s is well captured by focusing on long-wavelength fluctuations of these most unstable modes, well described by a Ginzburg-Landau Hamiltonian density


where deep in the BCS limit, near the the model parameters are given by


and the inclusion of the current-current interaction, is necessary for a complete description. More generally, away from the weak-coupling BCS limit these couplings can be taken as phenomenological parameters to be determined experimentally, but the general form of the Ginzburg-Landau model has broader range of applicability, capturing all the qualitative features of the transition and the PDW state.

Larkin-Ovchinnikov state near

However, the derivation and expressions for the associated couplings, Eqs.(11) are limited to the weak coupling BCS regime and near the high chemical potential imbalance (Zeeman field) normal-to-FFLO transition at .

In a complementary, low chemical potential imbalance regime, just above the transition from the fully-paired (BCS-BEC) superfluid to the LO state at , a phenomenological analysis is possible(67). It treats the LO state as a periodic array of fluctuating domain-walls (stripes) in , akin to the lyotropic phases in soft condensed matter(48); (49).

However, such approach implicitly assumes that as the domain-wall surface energy becomes negative(134); (135); (136); (137) for , their interaction remains repulsive, and so the domain-walls proliferate continuously as a periodic array inside the LO state. Under this assumption (that warrants further study) the domain-wall density and the associated species imbalance () is then set by a balance between the negative surface energy and the domain-wall repulsion, growing continuously as a function of according to the Pokrovsky-Talapov’s commensurate-incommensurate (CI) transition phenomenology(150). This behavior is clearly exhibited in 1d(134); (152); (153) through an exact solution and bosonization methods, and has been argued to persist in higher dimensions (134); (135); (136); (137). The CI route for a transition to the LO state contrasts sharply with the Landau theory(51); (79); (100) of two independent order parameters , , that always predicts a first-order BCS-LO transition. The latter corresponds to the case of an attractive domain-wall interaction, that therefore proliferate discontinuously above , leading to the ubiquitous phase separation found in mean-field theory(79); (100). It is currently unclear what dimensionless microscopic parameter, analogous to Abrikosov’s (distinguishing between type I and type II superconductors)(154); (155), controls these two alternatives of the macro-phase separation (a first-order transition) and the micro-phase separated LO state (a continuous transition out of the gapped SF state)(149). A detailed analysis of such low-imbalance approach to the SF-LO transition and the LO state is sorely missing and is a subject of current research.

A semi-phenomenological local density approximation (LDA) model that assembles all known ingredients is given by

where and is the effective potential derived within a BdG analysis for a uniform . It fully captures the double-minimum structure and the associated 1st-order normal to (fully-gapped BCS) superfluid transition that skips the interesting intermediate e.g., the FFLO states. This LDA potential is supplemented by the gradient energy functional, inherited from a microscopic GL analysis (valid only near ). Its use near is supported by the fact that it is the simplest form that incorporates the underlying symmetries and encodes the expected energetics for the system to order at a finite -dependent momentum even near . Work is under way to derive this functional through a controlled Moyal (semi-classical) gradient expansion on the BdG Hamiltonian.

The functional has a double-well structure, with an additional normal state minimum (at ) developing for . It thus allows periodic soliton structure in , corresponding to oscillations between the minima at .

Figure 6: Ground state energy as a function of the order parameter and Zeeman energy , indicating a first-order transition at (thick red curve) between the paired superfluid state () and the normal state (), at fixed density and/or imbalance exhibiting phase-separated coexistence. As argued in the text, analogously to the Pokrovsky-Talapov systems and type-II superconductors, this first-order transition can be preempted by a continuous CI-like transition to a striped LO superfluid and its descendent quantum liquid-crystal states, illustrated in Fig.2.

ii.5 Goldstone modes in striped FFLO states

Using the Ginzburg-Landau model one can develop a low-energy Goldstone modes description of the striped paired states and use it to analyze their stability to fluctuations. We focus on the unidirectional (striped) pair-density wave states, with FF and LO states as simplest representatives of two qualitatively distinct universality classes. The corresponding order parameter is given by


where are two complex scalar order parameters, the dominant Fourier coefficients and amplitudes distinguishing between the FF and LO states. Using this representation inside and minimizing the ground-state energy for a simple analysis shows that indeed it is the LO state with that is most stable inside the BCS regime; the FF state is characterized by only one nonvanishing order parameter, (51). More generally, either state can be stabilized depending on the relative magnitudes of and .

Fulde-Ferrell state

The FF state is characterized by a single (independent) nonzero complex order parameter,


that is a plane-wave with the momentum and a single Goldstone mode corresponding to the local superconducting phase. The state carries a nonzero, uniform spontaneously-directed supercurrent and thereby breaks the time-reversal and rotational symmetry, chosen spontaneously along , as well as the global gauge symmetry, corresponding to the total atom conservation. Although the FF order parameter itself is not translationally invariant, it is invariant under a modified transformation of an arbitrary translation followed by a gauge transformation, with all gauge-invariant observables thus translationally invariant. Thus, the FF state is a uniform orientationally-ordered polar superfluid. The underlying rotational invariance also demands that it is invariant under a rotation of by an angle that generates a nontrivial, spatially-dependent phase . Simple algebra demonstrates that the fully nonlinear form of the longitudinal current ensures that it and the corresponding energy vanish for , as required by the underlying rotational invariance.

The analysis of the GL functional lead to a Goldstone mode Hamiltonian(138); (66); (67)


where , is the superfluid stiffness along and . The Hamiltonian form, is valid beyond its weak-coupling microscopic derivation and is familiar from studies of conventional smectic liquid crystals(48); (49); (156), despite the fact that FF state is a translationally-invariant polar superfluid not a smectic. The necessity of keeping the higher order gradients and nonlinearities in is due to the identical vanishing of the transverse superfluid stiffness, (guaranteed by the underlying rotational invariance unique to the FFLO striped states, absent in solid state contexts) that leads to fluctuations that are otherwise infrared-divergent in a purely harmonic model.

Larkin-Ovchinnikov state

The LO state is instead described by two independent nonzero PDW amplitudes (growing below ), that lead to a standing wave pair-density wave order parameter,


that is a product of a superfluid and a unidirectional density wave striped order parameters. These are respectively characterized by two Goldstone modes , corresponding to the superfluid phase and the smectic phonon of the striped state. This also contrasts with the conventional smectic(48) (e.g., in liquid crystal materials, where one instead is dealing with a real mass density not a pair condensate wavefunction), characterized by a single phonon Goldstone mode, .

The mean-field LO order parameter, simultaneously exhibits the ODLRO (superfluid) and the smectic (unidirectional density wave) orders. It thus spontaneously breaks the rotational, translational, and global gauge symmetries, and therefore realizes a form of a paired supersolid. However, it is distinguished from a conventional purely bosonic supersolid (125); (126); (127); (128), where homogeneous superfluid order and periodic density wave coexist, by the vanishing of the (“charge”-2 two-atom) zero momentum () superfluid component in the LO condensate(129).

Similarly to the FF state, the underlying rotational symmetry of the LO state strongly restricts the form of the Goldstone-mode Hamiltonian. Namely, its sector must be invariant under a rotation of , that defines the spontaneously-chosen orientation of the pair-density wave, and therefore must be described by a smectic form(48); (49); (156). On the other hand because a rotation of the LO state leaves the superconducting phase, unchanged, the superfluid phase sector of the Hamiltonian is therefore expected to be of a conventional -model type. Consistent with these symmetry-based expectations the LO Goldstone-mode Hamiltonian was indeed found(66); (67) to be given by


with the nonlinear strain tensor ensuring the full rotational invariance. In the weakly-coupled BCS limit the smectic elastic moduli and the superfluid stiffnesses are given by


Thus, the LO state is a highly anisotropic superfluid (though less so than the FF state, where ), with


a ratio that vanishes for .

We stress that while the detailed expressions for the moduli above are specific to the weak-coupling BCS limit near the general form of , (17), including the structure of the symmetry-enforced nonlinearities in the (smectic) sector is valid beyond our microscopic derivation, and holds throughout the LO phase.

By extending the Hamiltonian to include density fluctuations, , canonically conjugate to and integrating them out in an imaginary time () coherent-state path integral, leads to a Lagrangian density


with given by the in the LO ground state (FF state is treated similarly using and a single Goldstone mode). For the LO state, this analysis then predicts the existence of two anisotropic low-frequency modes with dispersions


where is the compressibility of the Fermi gas. These modes respectively correspond to the zeroth sound (the Bogoliubov mode as in a conventional superfluid) and smectic phonon, unique to the LO state. In cold atomic gases, these should in principle be measurable via the Bragg spectroscopy technique(157); (158); (159)

With the Goldstone-mode Lagrangian in hand, the effects of quantum and thermal fluctuations as well as equilibrium correlation and response functions can be calculated(160).

Goldstone modes fluctuations

Armed with the action for the Goldstone modes, the low-energy quantum and thermal fluctuations in the FFLO striped states are straightforwardly computed. Despite smectic like softness of these modes, it is easy to show that quantum fluctuations remain finite for and therefore the FF and LO states remain stable at zero temperature.

Harmonic approximation:

In contrast, at finite temperature the root-mean-squared fluctuations of the smectic phonon modes in the LO state (and phase fluctuations in the FF state) diverge with trap size (in a purely harmonic Goldstone-modes theory) according to


where the fluctuations are evaluated in a trap with an aspect ratio , with its extent along the ordering axis (162); (48); (49). These fluctuations lead to an emergence of important crossover length scales that characterize the finite-temperature LO state,


defined as scales at which LO phonon fluctuations are comparable to LO stripe period .

The thermal connected correlation function of LO phonons


is also straightforwardly worked out, in 3d giving the logarithmic Caillé form(162)


In 2d it is instead given by(145)


Above finding of the divergence of smectic phonon fluctuations at nonzero temperature have immediate drastic implications for the properties of the LO (and FF) states. The most important of these is that the thermal average of the Landau’s LO order parameters (16) vanishes in thermodynamic limit

with the thermally suppressed order parameter amplitude given by


where accounts for the finite quantum and thermal superconducting phase fluctuations, and is the Caillé exponent(162).

Akin to 2d xy-model systems, the vanishing of the LO order parameter does not imply the instability of the phase, but that the true fluctuating state contrasts qualitatively with perfectly periodically-ordered mean-field description.

Figure 7: The finite momentum pairing at and divergent 3d smectic phonon fluctuations in the LO state are reflected in the Cooper-pair center-of-mass momentum distribution function, (observable via the time-of-flight measurements), displaying power-law Bragg peaks, characteristic of spatial quasi-long-range order.
Figure 8: The structure function, for the 3d LO state, displaying power-law Bragg peaks, characteristic of the LO superfluid’s spatial quasi-long-range order.

These divergent LO phonon fluctuations also qualitatively modify the Cooper-pair momentum distribution function and structure function (respectively measurable via time-of-flight and Bragg spectroscopy(157); (158); (159) imaging of pair-condensate) from their true Bragg (-function peaks) form characteristic of the mean-field long-range periodic order. They are highly anisotropic () and exhibit quasi-Bragg peaks (see Fig.8) around the ordering wavevector (and its harmonics, ), reminiscent of (1+1)d Luttinger liquids and two-dimensional crystals(139); (140); (141); (142); (143); (144)


These predictions are a reflection of the well-known(48); (49) and experimentally tested(163) behavior of conventional smectic liquid crystals. In two dimensions, the LO order is even more strongly suppressed by thermal fluctuations down to short-ranged correlations, strongly suggesting true instability of the LO order. Because it is the “soft” smectic Goldstone mode that is responsible for these interesting properties they are necessarily also shared by the FF state(138).

I emphasize that analogous Kosterlitz-Thouless phase fluctuation physics has been observed in conventional 2d trapped superfluids, despite the finite trap size(164); (165). Thus, I am hopeful that they can be similarly seen in the 3d LO state. A more detailed analysis of the trap effects is necessary for direct comparison with experiments.

Another fascinating consequence of the vanishing of , (28) is that the leading nonzero Landau order parameter characterizing the LO state is the translationally-invariant “charge”-4 (4-atom pairing) superconducting order parameter,


Thus in the presence of thermal fluctuations the LO phase corresponds to an exotic state in which the off-diagonal order is exhibited by pairs of Cooper pairs, i.e., a bound quartet of atoms, rather than by the conventional 2-atom Cooper pairs(37); (66). In 2d and 3d this higher order pairing is driven by arbitrarily low- fluctuation, rather than by a fine-tuned attractive interaction between Cooper pairs, and therefore has no simple mean-field description. A microscopic formulation of such a state and its detailed properties remain an open problem.

Nonlinear elasticity beyond Gaussian fluctuations:

As discussed in the context of conventional smectics(156) and more recently for the FFLO state(66); (67), above predictions neglect the effects of Goldstone mode nonlinearities , in Eq.(15),(17), that modify the asymptotics on scales longer than the crossover scales ,


The behavior on scales beyond can be obtained using renormalization-group analysis for (156); (67), with an exact solution in 2d(166). The finite-temperature asymptotics is well-approximated by a correlation function


with moduli and that display a universal singular wavevector-dependence, that is asymptotically exact logarithmic(156) in 3d


and power-law in 2d


with universal scaling functions and exact(166). In 3d this translates into an equal-time LO order parameter correlations given by(156)


Although these 3d anomalous effects are less dramatic and likely to be difficult to observe in practice, theoretically they are quite significant as they represent a qualitative breakdown of the mean-field and harmonic descriptions of the FFLO striped states.

ii.6 Phases and transitions

Topological defects

Associated with its two compact Goldstone modes, (equivalently , ) the LO state admits two types of topological defects, characterized by integers defined by . These equivalently correspond to superfluid vortices and edge dislocations in the striped PDW(167). These are characterized by multiples of half-integers and therefore allow four types of elementary defects: integer vortex, integer dislocation and two half-integer vortex-dislocation composites . The latter composite fractional defects are allowed because a sign change in due to a -dislocation in is compensated by a -vortex in (thereby preserving a single-valuedness of )(40); (37); (66); (67). In terms of the two coupled Goldstone modes, these correspond to an integer vortex in one and no vortex in the other superfluid phase.

Their thermodynamics and correlations can be treated via a mapping on a multi-component Coulomb gas




with the Fourier transforms of the vortex and dislocation densities. Equivalently, defects thermodynamics can be analyzed via duality transformation. In two dimensions it leads to a sine-Gordon-like model for dual fields characterizing fractional defects, with the dual Hamiltonian(67); (37); (171)


It is convenient for analyzing the effects of defects on the LO state, particularly for a computation of their screening on long scales, unbinding, and for the analysis of the resulting disordered state. From the form (38) it is clear that (aside from an inconsequential anisotropy) the dual vortex sector described by has a standard sine-Gordon form. In contrast, the dual dislocation sector, described by is qualitatively modified by the highly nonlocal and qualitatively anisotropic smectic kernel, .

A standard analysis gives the relative energetics of these defects, in the thermodynamic limit () given by


Based on this energetics one may be tempted to conclude that in this limit (unless preempted by a first-order transition) it is the integer dislocation loop defects that proliferate first and the LO smectic preferentially disorders into a nematic superfluid, . However, in contrast to the 2d KT mechanism(144), the 3d disordering transitions take place when the relevant stiffness, renormalized by quantum and thermal fluctuations is continuously driven to zero at the transition, or takes place at a finite (rather than a vanishing) defects fugacity. For a thermal transition this roughly corresponds to a transition temperature set by the corresponding stiffnesses, and . Thus, in principle by tuning these stiffnesses via imbalance and resonant interaction, a variety of phases can be accessed.

Conventional phases and transitions

By considering all possible basic combinations of spontaneously “broken” subset of spatial and gauge symmetries leads to an array of partially spatially-ordered paired superfluids and Fermi-liquid states, that are descendants of the smectic LO (SF) state. These isotropic (I), nematic (N) and smectic (Sm) SF and FL states are summarized in Table I. It is notable that the isotropic superfluid, exhibits a finite species imbalance and off-diagonal long-range order, symmetry-wise isomorphic to the polarized superfluid, (79); (100), latter confined to the BEC side of the BCS-BEC crossover. In contrast, (as a descendant of the LO state expected to be stabilized by Fermi surfaces imbalance) the state is realized in the BCS regime, something that has been searched for dating back to Sarma(117), but has not been possible within mean-field treatments, that instead predict an instability to phase separation(74); (79); (100). The isotropic Fermi liquid, is isomorphic to the conventional normal state. Together these intermediate fluctuation-induced phases naturally interpolate between the fully gapped singlet (homogeneously and isotropic) BCS superconductor at zero imbalance and low temperature, and a polarized Fermi liquid at large imbalance and/or high temperature.

U(1) U(1) U(1)
Table 1: Five phases that naturally emerge as disordered descendants of the LO (superfluid smectic, ) state.
Table 2: A summary of LO liquid crystal Fermi-liquid (FL) and superfluid (SF) phases, and corresponding order parameters and broken symmetries, indicated by X’s. Unbroken symmetries (gauge , translational , rotational ) are marked by check marks. The subscripts respectively indicate the Isotropic, Nematic and Smectic orders.

Topological phases via defects unbinding

The phases discussed above can be complementarily characterized through unbinding of different combinations of topological defects. The smectic (whether SF LO state or FL smectic) to nematic transition is driven by unbinding of defects with edge dislocation charge, followed by transition into the isotropic state driven by proliferation of disclinations. The superfluid and Fermi-liquid version of these liquid crystal states are distinguished by unbinding defects with superfluid vortex charge.

However, a characterization in terms of topological defects also allows a distinction between topologically distinct phases with the same conventional order, where a Landau order parameter is insufficient to distinguish them. In fact because of the vanishing LO order parameter, description in terms of topological order is necessary even for the smectic LO state, distinguished from its more disordered descendants by the absence of unbound topological defects, in direct analogy with the quasi-long-range ordered state of the 2d xy model.

A rich variety of possible phases and transitions is displayed in a schematic imbalance-detuning phase diagram, Fig.5. Increasing the imbalance suppresses the superfluid stiffness and drives the system toward a conventional Fermi liquid state, at . Conversely, a reduction in species imbalance primarily reduces the elastic moduli of the smectic pair-density wave by increasing its period