Modulated pairing states of mass-imbalanced Fermi superfluids: multi-wave-vector competition with the uniaxial Larkin-Ovchinnikov state and the full phase diagram

The long-sought finite-momentum pairing (FFLO) states of imbalanced Fermi superfluids face an unresolved structural question when the two pairing partners have unequal masses, as in the $^6\mathrm{Li}$-$^{40}\mathrm{K}$ Fermi-Fermi mixture: which modulated (Larkin-Ovchinnikov-type) pairing states are actually favored, and does the pairing field modulate along a single wave vector or several? The available static-auxiliary-field Monte Carlo phase diagram of the two-dimensional mass- and population-imbalanced attractive Hubbard model compared only single-wave-vector trial states (uniaxial, diagonal, and two-dimensional) and excluded multi-wave-vector, supersolid-like configurations for numerical simplicity, leaving the modulated-state phase diagram as a function of interaction strength, filling, population imbalance, and mass ratio unexplored. No existing study settles the multi-wave-vector competition or maps these phase boundaries, and the problem remains open.

Background

When fermions of two species pair into a superfluid, any mismatch between their Fermi surfaces frustrates ordinary zero-momentum Bardeen-Cooper-Schrieffer (BCS) pairing. Such a mismatch arises from unequal populations, represented by an effective Zeeman field $h=(\mu_L-\mu_H)/2$, or from unequal masses. One route to accommodating the mismatch is Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing: Cooper pairs with finite center-of-mass momentum. The Fulde-Ferrell (FF) state has a pairing field $\Delta(\mathbf{r})=\Delta e^{i\mathbf{q}\cdot\mathbf{r}}$ with a single wave vector; the Larkin-Ovchinnikov (LO) state superposes opposite wave vectors, $\Delta(\mathbf{r})=\Delta\cos(\mathbf{q}\cdot\mathbf{r})$, preserving time-reversal symmetry. The minimal LO state uses one pair $\pm\mathbf{q}$, but a pairing field modulated along several distinct wave vectors (a multi-$Q$ state) is in principle possible; because such a state combines superfluid order with density modulation in several directions, it is often called supersolid-like. Energy differences between competing LO variants are known to be small, so which modulation pattern wins is a genuine quantitative question, and mean-field lattice studies (e.g. Baarsma and Törmä on population-imbalanced square lattices) have had to compare candidate wave-vector orientations explicitly.

Mass imbalance brings additional physics: even at equal chemical potentials the two species have different Fermi wave vectors, so a mismatch exists already in a population-balanced mixture (in the weak-coupling, low-filling limit mass and Zeeman imbalance are related by a change of variables). This is experimentally relevant: the $^6\mathrm{Li}$-$^{40}\mathrm{K}$ Fermi-Fermi mixture has been realized, and state-dependent optical lattices can engineer different hoppings $t_\sigma\propto 1/m_\sigma$ for the two species, realizing the mass-imbalanced attractive Hubbard model. Continuum mean-field and $T$-matrix (Gaussian pair-fluctuation) studies (Baarsma, Gubbels, and Stoof; Wang, Che, Zhang, and Chen) found that sufficiently large mass imbalance promotes modulated pairing, including a Lifshitz point — where the second-order normal-state transition line meets the line toward inhomogeneous order — and a supersolid instability at unitarity for $^6\mathrm{Li}$-$^{40}\mathrm{K}$; however, these treat only single-wave-vector FF/LO ansätze.

The most complete lattice treatment is the static auxiliary field (SAF) Monte Carlo study of Karmakar [Phys. Rev. A 97, 033617 (2018)]: classical Monte Carlo over spatially varying, static pairing fields (temporal fluctuations neglected) on the two-dimensional mass- and population-imbalanced attractive Hubbard model, at interaction $U=4t_L$ near lattice unitarity and chemical potential $\mu=-0.2t_L$ near half filling, with mass ratio $\eta=m_L/m_H$ ($L$ and $H$ denoting the light and heavy species) and imbalance field $h$ as control parameters. It mapped the finite-temperature phase diagram — uniform superfluid, breached-pair phase (one species partially unpaired), two pseudogap regimes, and an LO phase at large $h$. For the LO phase, the variational minimization compared single-wave-vector trial states (uniaxial, diagonal, and two-dimensional modulations) and found the uniaxial one optimal; multi-wave-vector configurations were explicitly excluded, in the author's words, for the sake of numerical simplicity. The full phase diagram of modulated states in mass-imbalanced mixtures — which modulated states are actually favored, and how their boundaries move with interaction strength, filling, and mass ratio — was thereby left unresolved. Neighboring results frame but do not answer it: at half filling the mass-imbalanced attractive Hubbard model shows coexistence of charge-density-wave and superfluid order (a commensurate supersolid), in one dimension FFLO-type correlations survive mass imbalance, and at extreme mass ratios a distinct quartet superfluid with momentum-space crystallization of the pairing field appears.

Problem Statement

For mass-imbalanced two-species Fermi systems — the two-dimensional attractive Hubbard model (two fermion species with an attractive on-site interaction and species-dependent hoppings $t_\sigma\propto 1/m_\sigma$, so the hopping ratio is fixed by the mass ratio $\eta=m_L/m_H$ of the light and heavy species), together with its dilute continuum limit relevant to mixtures such as $^6\mathrm{Li}$-$^{40}\mathrm{K}$, with or without population imbalance, represented by an effective Zeeman field $h=(\mu_L-\mu_H)/2$ in terms of the two species' chemical potentials — determine which modulated, finite-center-of-momentum pairing states are thermodynamically favored (free energies being compared at the relevant temperature; as in the source, both the finite-temperature phase diagram and the zero-temperature variational comparison of modulated states are in scope), and construct the phase diagram of these modulated states as a function of interaction strength, filling (density), population imbalance, and mass ratio. In particular: (i) establish whether pairing-field configurations modulated along multiple distinct wave vectors — supersolid-like generalizations of the Larkin-Ovchinnikov state — have lower free energy than the uniaxial single-wave-vector LO solution anywhere in this parameter space, or whether the uniaxial assumption used in existing calculations is justified; and (ii) locate the phase boundaries separating modulated states from the uniform superfluid, the breached-pair phase (uniform pairing with one species partially unpaired) or the polarized normal state, and other competing phases over the same parameters. The question is posed at the source's natural generality — not restricted to the single interaction, filling, or mass-ratio slice already studied — and answers may come from any controlled method. Because the source does not bound that space, a resolving answer must state the ranges of the axes it covers — interaction strength, filling, population imbalance, mass ratio, and temperature — and each of sub-claims (i) and (ii) may be resolved over any declared, physically relevant region of that space rather than over the whole of it, with whatever lies outside that region left explicitly open.

References

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