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

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.

Figure 1

A mean-field zero-temperature phase diagram from Refs. [4, 44] of an imbalanced Fermi gas, as a function of the inverse scattering length and normalized species imbalance $P=(N_{\uparrow }-N_{\downarrow })/N\equiv \Delta N/N$, showing the magnetized (imbalanced) superfluid (SF${}_{\mathrm{M}}$), the FFLO state (approximated as the simplest FF state, confined to a narrow red sliver bounded by $P_{\mathrm{FFLO}}$ and $P_{\mathrm{c}2}$), and the imbalanced normal Fermi liquid.

Figure 2

An illustration of a continuous commensurate-incommensurate (CI) transition at $h_{c1}$ 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 $n_{\mathrm{dw}}\left(h\right)$ is then proportional to the imbalance $P\left(h\right)$ and grows continuously with the chemical potential difference (Zeeman energy) $h-h_{c1}$.

Figure 3

A proposed $\Delta N/N$ vs $1/\left(k_{\mathrm{F}}{a}_s\right)$ phase diagram for an imbalanced resonant Fermi gas, showing the more stable LO liquid-crystal phases (discussed in the text and illustrated in detail in Fig. 4) replacing a portion of the phase-separated regime.

Figure 4

A schematic imbalance-chemical potential (Zeeman energy) $h=\left({\mu }_{\uparrow }-{\mu }_{\downarrow }\right)/2$ vs detuning (interaction strength) $-1/k_{F}a$ phase diagram, illustrating the 3D LO smectic phase (${\mathit{SF}}_{\mathrm{Sm}}$) 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.

Figure 5

An illustration of longitudinal and transverse (to $\mathbf{q}$) LO nonzero supercurrent configuration controlled by the corresponding superfluid stiffnesses ${\rho }_s^{\parallel }$,${\rho }_s^{\perp }$. Because there is no change in the effective period associated with the transverse current excitation, it is qualitatively energetically “cheaper,” scaling as ${\Delta }_{q_0}^4$, as compared to the longitudinal excitation that scales as ${\Delta }_{q_0}^2$. As indicated in Eq. (82), the ratio therefore vanishes as $h$ approaches the transition into a nonsuperfluid phase at $h_{c2}$.

Figure 6

Feynman graph that renormalizes the elastic moduli $K$ and $B$ of the LO superfluid.

Figure 7

Renormalization group flow for a LO state in $d<3$ dimensions, illustrating that at low $T$ it is a “critical phase” displaying universal power-law phenomenology, controlled by a nontrivial infrared stable fixed point.

Figure 8

A $2\pi$-vortex defect in a LO state. The 2D arrows indicate the corresponding complex condensate LO wave function ${\Delta }_{\mathit{LO}}^v\left(\mathbf{r}\right)$, with orientations characterizing the phase $\phi \left(\mathbf{r}\right)$.

Figure 9

An integer $a$-dislocation defect in the LO layered structure, with no defects in its superfluid phase.

Figure 10

Two complementary forms illustrating a half-integer vortex-dislocation $(\pi ,a/2)$ defect in the LO state. The 2D arrows indicate the corresponding complex condensate LO wave function ${\Delta }_{\mathit{LO}}^{v-d}\left(\mathbf{r}\right)$, with orientations characterizing the phase $\phi \left(\mathbf{r}\right)$.

Figure 11

Anisotropic superfluid flow around a $2\pi$-vortex near $h_{c2}$, predicted to be characterized by an anisotropic superfluid ratio ${\rho }_s^{\parallel }/{\rho }_s^{\perp }$ that diverges at $h_{c2}$.

Figure 12

A flowchart of superfluid ($\text{SF}$) and nonsuperfluid ($\text{FL}$) phases, exhibiting smectic ($\text{Sm}$) and nematic ($N$) conventional orders as well as topological orders (indicated by $*$ and $**$), induced by a proliferation of various combinations of topological defects $(0,a)$, $(2\pi ,0)$, and $(\pi ,\pm a/2)$.

Figure 13

An illustration of Fermi pockets (full curve) of the gapless Bogoliubov quasiparticles characteristic of the Larkin-Ovchinnikov ground state. The periodic array of domain walls in ${\Delta }_{\mathit{LO}}\left(z\right)$, the associated wave vector $\mathbf{q}$, and the Fermi surface of the underlying normal state (dashed circle) are also indicated.

Figure 14

Lowest-order Feynman diagram of a fermionic contribution that corrects the superfluid stiffness.

Figure 15

Lowest-order Feynman diagram of a fermionic contribution that corrects the smectic compressional modulus.

Figure 16

Lowest-order subdominant Landau-damping contribution to the superfluid stiffness coming from finite fermion density.

Figure 17

The collinear Larkin-Ovchinnikov state, compressed by an isotropic trap, $V_t\left(r\right)$ (on the right), showing the enhanced imbalance ($P\sim -{\partial }_{z}u$) confined to the edge of the trap.

Figure 18

Possible alternative forms of the LO state confined in a tight isotropic trap. The form on the right trades off the cost of the curvature energy for the dislocation energy.

Figure 19

Illustration of the $q_z=0$ ($z$-independent) phonon modes, even in an isotropic trap (indicated by a blue circle) dominating over the (scalingwise) anisotropic $q_z\sim \lambda q_{\perp }^2$ bulk smectic modes (indicated by an ellipse).

Figure 20

The finite momentum pairing at $q_0$ and divergent 3D smectic phonon fluctuations in the LO state are predicted to be reflected in the Cooper-pair center-of-mass momentum distribution function $n_{\mathbf{k}}$, displaying power-law Bragg peaks, characteristic of the spatial quasi-long-range order. We expect these to be observable through time-of-flight measurements.

Figure 21

The structure function $S\left(\mathbf{q}\right)$ for the 3D LO state, displaying power-law (as opposed to $\delta$-function) Bragg peaks, characteristic of the LO superfluid’s spatial quasi-long-range order.