Effective theory of interacting fermions in shaken square optical lattices

We develop a theory of weakly interacting fermionic atoms in shaken optical lattices based on the orbital mixing in the presence of time-periodic modulations. Specifically, we focus on fermionic atoms in a circularly shaken square lattice with near-resonance frequencies, i.e., tuned close to the energy separation between the $s$ band and the $p$ bands. First, we derive a time-independent four-band effective Hamiltonian in the noninteracting limit. Diagonalization of the effective Hamiltonian yields a quasienergy spectrum consistent with the full numerical Floquet solution that includes all higher bands. In particular, we find that the hybridized $s$ band develops multiple minima and therefore nontrivial Fermi surfaces at different fillings. We then obtain the effective interactions for atoms in the hybridized $s$ band analytically and show that they acquire momentum dependence on the Fermi surface even though the bare interaction is contactlike. We apply the theory to find the phase diagram of fermions with weak attractive interactions and demonstrate that the pairing symmetry is $s+d$ wave. Our theory is valid for a range of shaking frequencies near resonance, and it can be generalized to other phases of interacting fermions in shaken lattices.

Figure 1

Band structure of a shaken square lattice from the numerical Floquet analysis. (a) The lowest few bands for lattice depth $V_0/E_R=5$ without shaking. The shaking frequency $\omega$ is shown with a red arrow. It is fixed at $\omega =1.05\mathrm{\Omega }$, where $\mathrm{\Omega }=max{E}_p-min{E}_s$. The first quasienergy Brillouin zone (QeBZ) is indicated by the gray shaded region. (b) The bands folded into QeBZ. The blue circles indicate regions where strong hybridization is expected to occur between $s, p_x, p_y$, and $d_{xy}$ bands. (c) The quasienergy spectrum of a circularly shaken square lattice with shaking amplitude $\beta =0.1$. For clarity only six relevant eigenvalues of the Floquet solution are shown. Band mixing leads to fine features not captured by the folding construction. (d) The close-up details of the hybridized $s$ band. The inset shows the high-symmetry points inside the quasimomentum Brillouin zone. (e) Contour plot of the dispersion ${\mathcal{E}}_s\left(\mathbf{k}\right)$ of the hybridized $s$ band in the Brillouin zone obtained from the numerical Floquet solution for shaking amplitudes $\beta =0.1$ (left), $\beta =0.2$ (middle), and $\beta =0.3$ (right). Dark blue regions correspond to low energies, while bright yellow regions correspond to high energies.

Figure 2

Band structure of the effective four-band model $H_{\mathrm{eff}}$. Top: Hybridization of four lowest Bloch bands by shaking. The dashed red lines are the bare bands folded into same QeBZ [see Eq. (35)] that will be coupled when shaking is turned on. The resulting hybridized energies [${\epsilon }_{\kappa {\kappa }^{\prime }}\left(\mathbf{k}\right)$ in Eq. (44)] are shown by the solid blue lines. Middle: Energy spectrum of the hybridized $s$ band in the first Brillouin zone (left) and the corresponding density of states at the Fermi level for a given lattice filling (right). Bottom: Evolution of Fermi surface with increasing lattice filling. Parameters used are $e_s=-0.56,t_s=-0.07,e_p=2.76,t_p=0.42,d_0=2.63,d_1=-0.28$, corresponding to $V_0/E_R=5$. Shaking frequency $\omega =1.01\mathrm{\Omega }$, whereas the amplitude is $\beta =0.1$.

Figure 3

Momentum dependence of the effective interaction on the Fermi surface $V_{\mathrm{eff}}(\mathbf{k},{\mathbf{k}}^{\prime })\equiv V(\phi ,{\phi }^{\prime })$ for four different fillings $n$. The shaking parameters are $V_0=5E_R, \beta =0.1, \omega =1.01\mathrm{\Omega }$. Here $\phi$ and ${\phi }^{\prime }$ are polar angles of $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ on the Fermi surface (solid lines), respectively, defined with respect to the center of the entire Fermi surface (top) or the Fermi pocket in the first quadrant (bottom). The red and blue dots depict examples for the definitions of angles corresponding to momentum pair $(\mathbf{k},-\mathbf{k})$ for minimum and maximum interaction angles, respectively. Note that the angles are defined from $-\pi$ to $\pi$.

Figure 4

Cooper pairing of fermions in a shaken square lattice. Top: Superfluid critical temperature $k_B{T}_c$ in units of bandwidth $W\equiv max{\epsilon }_1-min{\epsilon }_1$ for lattice depth $V_0=5E_R$, shaking parameters $s_0=0.05, \omega =1.01\mathrm{\Omega }$, and interaction $\left|U\right|=0.2W$ obtained from the solution of the linearized gap equations (66) and (67). The pairing order parameter has ($s+d$)-wave symmetry within the gray shaded region and is predominantly $s$ wave otherwise. Middle: The zero-temperature Fermi surfaces for fillings corresponding to points $A, B$, and $C$. Bottom: Angular dependence of order parameter $\mathrm{\Delta }\left(\phi \right)$ around the Fermi surface. Note that in the weak-coupling BCS limit, the zero-temperature order parameter is proportional to $T_c$. Its magnitudes at points $A, B, C$ are different and can be seen in the top panel.

Figure 5

Interacting fermions in a shaken square lattice with red-detuned shaking frequency. Top: Band mixing between the lowest four orbitals via shaking along the high-symmetry points. Dashed red lines are bare energies, whereas solid blue lines are shaking hybridized energies. Middle: Dispersion of the hybridized $s$ band in the full Brillouin zone on the left and the corresponding density of states at the Fermi level for given filling on the right. Bottom: The angular dependence of interaction on the Fermi surface on the left for the Fermi surface shown on the right. The mean-field order parameter around the Fermi surface with respect to the center $M$ point is shown below the Fermi surface plot. Parameters used are $e_s=-0.56,t_s=-0.07,e_p=2.76,t_p=0.42,d_0=2.63,d_1=-0.28$, corresponding to $V_0/E_R=5$. Shaking frequency is taken as $\omega =0.85\mathrm{\Omega }$, whereas the amplitude is $\beta =0.05$.