Microscopic Model for Feshbach Interacting Fermions in an Optical Lattice with Arbitrary Scattering Length and Resonance Width

We numerically study the problem of two fermions in a three-dimensional optical lattice interacting via a zero-range Feshbach resonance and display the dispersions of the bound states as a two-particle band structure with unique features compared to typical single-particle band structures. We show that the exact two-particle solutions of a projected Hamiltonian may be used to define an effective two-channel, few-band model for the low-energy, low-density physics of many fermions at arbitrary $s$-wave scattering length. Our method applies to resonances of any width and can be adapted to multichannel situations or higher-$\ell$ pairing. In strong contrast to usual Hubbard physics, we find that pair hopping is significantly altered by strong interactions and the presence of the lattice, and the lattice induces multiple molecular bound states.

Figure 1

Schematic of the FRH transformation. (a) In a broad Feshbach resonance, all two-particle scattering continua (thick gray-shaded areas) are strongly coupled to bare molecular bands (solid lines). Thus, all scattering continua are virtually strongly coupled. (b) By correctly dressing the molecular bands, one obtains a single scattering continuum (thick gray-shaded area) plus well-separated dressed molecular bands (thin green areas), with much simpler couplings. This is our efficient, numerically tractable FRH.

Figure 2

Exact two-particle band structures for various $a_s$ in a strong optical lattice. The bound state energies for $a_s/a=-5$ (purple boxes), $-0.1$ (red pluses), 0.1 (green crosses), and 5 (blue asterisks) as a function of the total quasimomentum $\mathbf{K}$ along a path connecting the high-symmetry points in the irreducible BZ $\Gamma =(0,0,0)$, $X=(-\pi /a,0,0)$, $M=(-\pi /a,-\pi /a,0)$, $R=(-\pi /a,-\pi /a,-\pi /a)$ for a lattice with $V/E_R=12$. The near-resonant points $a_s/a=\pm 5$ lie nearly on top of one another, demonstrating universality.

Figure 3

Hubbard parameters for the FRH. (a) The detunings and tunnelings of the completely even-parity dressed molecule in the lowest sheet as a function of $a_s/a$. The detuning is negative for $a_s<0$ and positive otherwise. The solid black horizontal line is the tunneling of a single open channel fermion in the lowest band. The nearest neighbor dressed molecular tunneling is nearly two orders of magnitude larger than the open channel tunneling near resonance. (b) The effective atom-dressed molecule couplings of the completely even parity-dressed molecule in the lowest sheet as a function of $a_s/a$. Schematics of the spatial dependence of the various coupling processes are shown in the boxes.