Multi-component ultracold Fermi gases in optical lattices realize the multi-component Fermi-Hubbard model, in which pairing is renormalized by induced (particle-hole screening) interactions between components. Weak-coupling mean-field theory gives explicit exponential corrections to s- and d-wave gaps and mean-field transition temperatures, but it fails precisely where the physics is richest: across the BCS-BEC crossover, near half filling in two dimensions where the screening interaction becomes singular, in the strongly correlated regime relevant to d-wave pairing, and for the true (Berezinskii-Kosterlitz-Thouless) transition temperature. The open problem is a controlled, quantitative many-body treatment of induced-interaction effects on pairing in multi-component lattice Fermi gases valid in these regimes, including the interplay with antiferromagnetic and Mott-insulating orders, which existing partial results (two-component lattice calculations of the Gorkov-Melik-Barkhudarov correction, continuum crossover treatments, and attractive-Hubbard quantum Monte Carlo) leave unresolved.

Background

In a dilute two-component Fermi gas, Bardeen-Cooper-Schrieffer (BCS) theory predicts a superfluid transition temperature TcT_c set by an exponentially small pairing gap. Gorkov and Melik-Barkhudarov showed that particle-hole fluctuations screen the attractive interaction between the two pairing components, reducing TcT_c by a factor (4e)1/3≈2.2(4e)^{1/3}\approx 2.2; these 'induced interactions' are the leading beyond-mean-field correction in the weak-coupling limit. In a mixture of ν\nu fermion species (hyperfine states of 6^{6}Li, or the nuclear-spin states of alkaline-earth atoms such as 173^{173}Yb and 87^{87}Sr that realize an approximate SU(NN) symmetry, under which the NN spin states interact identically), the extra components j≥3j\ge 3 contribute additional screening loops that can either suppress or enhance pairing. An optical lattice - a standing-wave light field creating a periodic potential for the atoms - reduces the problem to the Fermi-Hubbard model with on-site couplings UijU_{ij} between components i,ji,j and hopping tt. On a lattice the pairing gap acquires an exponential induced-interaction correction Δ=Δ0exp⁡[−χˉ+∑j≥3(U1jU2j/U122)(N(μj)/N(μ1))χˉ(μ1,μj)]\Delta=\Delta_0\exp[-\bar\chi+\sum_{j\ge3}(U_{1j}U_{2j}/U_{12}^2)(N(\mu_j)/N(\mu_1))\bar\chi(\mu_1,\mu_j)], where N(μ)N(\mu) is the density of states and χˉ\bar\chi is an angular average of the Lindhard screening function (the standard density response of a Fermi sea); near half filling (one particle per site) in two dimensions, Umklapp processes (momentum transfers that include a reciprocal-lattice vector) make the Fermi surface square-shaped (nested), and χˉ2D\bar\chi^{2D} diverges, so the correction is no longer perturbatively small. Repulsive on-site interaction with a weak nearest-neighbor attraction VV favors d-wave pairing (gap changing sign under 90∘90^\circ rotation, as in cuprate superconductors), for which mean-field expressions exist near half filling; but at strong coupling the same lattice realizes Mott-insulating (interaction-localized) and antiferromagnetic (AF) phases that mean field ignores. Strong attraction, tunable by Feshbach resonances (magnetic-field-tuned scattering resonances), drives the BCS-BEC crossover from weakly bound Cooper pairs to tightly bound molecules. Finally, in two dimensions the true transition is of Berezinskii-Kosterlitz-Thouless (BKT) type - vortex-antivortex unbinding at TBKTT_{BKT} below the mean-field TMFT_{MF} - which the lattice mean-field calculation does not evaluate. Heiselberg's 2009 analysis (arXiv:0905.2107) computed gaps, TMFT_{MF}, and induced interactions to leading orders for multi-component lattice gases, finding, for example, that ν=6\nu=6 components enhance the 2D gap by eν−3e^{\nu-3} in the dilute regime while near half filling the screening becomes singular; the present problem is everything that analysis explicitly could not do.

Problem Statement

Obtain a controlled, quantitative many-body treatment of induced (screening) interactions and their effect on superfluid pairing in multi-component Fermi gases in optical lattices - i.e., in the multi-component Fermi-Hubbard model with ν≥3\nu\ge 3 components - beyond the weak-coupling mean-field regime in which current results are derived. This means a theory, with a controlled approximation whose validity is established in the regime it treats, not a re-derivation at the same weak-coupling mean-field level. The single overarching objective is a theory of induced-interaction renormalization of pairing valid where the existing perturbative mean-field framework fails, and its linked scope comprises: (i) the behavior of the induced interaction and the resulting gap and transition-temperature renormalization from weak coupling through the BCS-BEC crossover (strong attraction, up to the lattice unitary threshold ∣Uij∣=∣Uc∣|U_{ij}|=|U_c| where two-body bound states appear) - as motivating context, the source notes that for the continuum uniform gas Nozieres-Schmitt-Rink-type estimates (fluctuation corrections beyond mean field) reduce pairing by only ~20% whereas experiments indicate a factor ~2-3; (ii) the non-perturbative regime near half filling in two dimensions, where Umklapp-enhanced screening makes the induced interaction comparable to or larger than the on-site coupling and the exponential weak-coupling correction breaks down; (iii) the correction of the near-half-filling d-wave mean-field gap by strong on-site correlations, antiferromagnetic exchange, and Mott physics, including how the induced-interaction enhancement or suppression of d-wave pairing survives correlations; and (iv) the true two-dimensional transition temperature TBKTT_{BKT} (vortex-unbinding BKT transition) including induced interactions and lattice effects, not only the mean-field TMFT_{MF}. Each item is a regime or observable of the same induced-interaction renormalization that the weak-coupling mean-field baseline explicitly identifies as beyond its validity; they are not independent questions, and controlled progress in any of them advances the overarching objective. Relative terms are determinate by field convention: 'weak coupling' means ∣U12∣≪t|U_{12}|\ll t and induced interactions small compared to on-site couplings, 'strong coupling' the approach to ∣U12∣∼∣Uc∣|U_{12}|\sim|U_c| or ∣U12∣/t|U_{12}|/t of order several, and 'near half filling' the filling range around n≃1n\simeq1 per site where the 2D screening function diverges.

References

  1. H. Heiselberg, Induced interactions and superfluidity in optical lattices with multi-component Fermi gases, arXiv:0905.2107 (2009), https://arxiv.org/abs/0905.2107 (primary source of the open problem)
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