Frustration and time-reversal symmetry breaking for Fermi and Bose-Fermi systems

The modulation of an optical lattice potential that breaks time-reversal symmetry enables the realization of complex tunneling amplitudes in the corresponding tight-binding model. For a superfluid Fermi gas in a triangular lattice potential with complex tunnelings, the pairing function acquires a complex phase, so the frustrated magnetism of fermions can be realized. Bose-Fermi mixtures of bosonic molecules and unbound fermions in the lattice also show interesting behavior. Due to boson-fermion coupling, the fermions become enslaved by the bosons and the corresponding pairing function takes the complex phase determined by the bosons. In the presence of bosons the Fermi system can reveal both gapped and gapless superfluidity.

Figure 1

The absolute value $A$ (top) and the complex phase ${\varphi }_J$ (bottom) of the effective tunneling amplitude$J_{\mathrm{eff}}/J=A\exp \left(i{\varphi }_J\right)$, Eq. (6), for a double-harmonic modulation of the optical lattice potential as a function of the dimensionless strength $s_1$ of the $\omega$ component for $s_2=1$ [see Eq. (7)], $\varphi =0.2$ (black solid lines) and $s_2=3$, $\varphi =0.5$ (red dashed lines).

Figure 2

Triangular Bravais lattice points (black circles) and amplitudes $J_{\alpha ,\beta }$ corresponding to tunneling from a lattice point to its nearest neighbors.

Figure 3

Contour plots of the dispersion relation Eq. (10) for $|J_{\alpha }|=|J_{\beta }|$ and ${\varphi }_{\alpha }={\varphi }_{\beta }=\pi /2$ (a) and ${\varphi }_{\alpha }=\pi$ and ${\varphi }_{\beta }=\pi /4$ (c); cool colors indicate regions around energy minima. In the right panels the directions of the arrows indicate the phases $e^{i\mathbf{k}·{\mathbf{r}}_i}$, where $\mathbf{k}$ corresponds to a minimum of the dispersion relation. Specifically $\mathbf{k}a=(\pi /3,\pi /\sqrt{3})$ for the minimum in (a) and $\mathbf{k}a=(+2\pi /3,\pi /2\sqrt{3})$ for one of the two degenerate, nonequivalent minima in (c). The arrows in (b) and (d) relate to (a) and (c), respectively.

Figure 4

Modulus squared of the Fourier transform of the BCS pairing function $|{\Delta }_{\mathbf{k}}{|}^2$, where ${\Delta }_{\mathbf{k}}={\sum }_i{\Delta }_i{e}^{-i\mathbf{k}·{\mathbf{r}}_i}/\sqrt{N_s}$, obtained numerically for a finite system of $60\times 60$ lattice sites for $|J_{\alpha }|=|J_{\beta }|$, ${\varphi }_{\alpha }=\pi$, ${\varphi }_{\beta }=\pi /4$, $U/|J_{\alpha }|=2$, and $\mu =0$. (a) shows $|{\Delta }_{\mathbf{k}}{|}^2$ corresponding to the ground state of the isolated Fermi system. (b) presents similar results but for the Fermi system coupled to a Bose-Einstein condensate, with wave function ${\psi }_i=\sqrt{n_B}{e}^{i{\mathbf{q}}_0·{\mathbf{r}}_i}$, where $\gamma \sqrt{n_B}/\left|J_{\alpha }\right|=2.3$. Note that the peak in (a) is located at $\mathbf{k}a=(0.00,1.78)\approx (0,\pi /\sqrt{3})$ while in (b) it is at $\mathbf{k}a=(2.09,0.89)\approx {\mathbf{q}}_{0}a=(+2\pi /3,\pi /2\sqrt{3})$.