$d_{xy}$-density wave in fermion-fermion cold-atom mixtures

We study density wave instabilities in a doubly degenerate Fermi-Fermi mixture with $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)$ symmetry on a square lattice. For sufficiently large on-site inter-species repulsion, when the two species of fermions are both at half-filling, two conventional ($s$ wave) number density waves are formed with a $\pi$-phase difference between them to minimize the interspecies repulsion. Upon moving one species away from half-filling, an unconventional density wave with $d_{xy}$-wave symmetry emerges. When both species are away from the vicinity of half-filling, superconducting instabilities dominate. We present results of a functional renormalization-group calculation that maps out the phase diagram at weak couplings. Also, we provide a simple explanation for the emergence of the $d_{xy}$-density wave phase based on a four-patch model. We find a robust and general mechanism for $d_{xy}$-density-wave formation that is related to the shape and size of the Fermi surfaces. The density imbalance between the two species of fermions in the vicinity of half-filling leads to a phase-space discrepancy for different interspecies umklapp couplings. Using a phase-space argument for leading corrections in the one-loop renormalization-group approach to fermions, we show that the phase-space discrepancy in our system causes opposite flows for the two leading intraspecies umklapp couplings and that this triggers the $d_{xy}$-density-wave instability.

Figure 1

(a) Contour plot of a square optical lattice with a fermion-fermion mixture, where lighter regions correspond to lower potential energy and darker to higher energy, and real-space picture of the $d_{xy}$-CDW state for one species of fermions. (b) The phase diagram is parameterized by the chemical potential of $c$ fermions ${\mu }_c$, and the ratio of interspecies and intraspecies interactions $U_{\mathrm{cf}}/U_{\mathrm{ff}}$. (c) The FS patches used in this study ($M=36$ on each FS), as well as the FS for ${\mu }_f=0$ and ${\mu }_c=-0.1$, and (d) corresponding $d_{xy}$ order parameter for $f$ and $c$ fermions (orange and green dots, respectively).

Figure 2

(a) A four-point vertex function $g({\mathit{k}}_1,{\mathit{k}}_2,{\mathit{k}}_3)$. (b)–(f) One-loop diagrams that contribute to the renormalization of a four-point vertex.

Figure 3

Sketches of two interspecies couplings, $g_1^{\mathrm{cf}}=g_{\mathrm{cffc}}(1,1,3,3)$ (left) and $g_2^{\mathrm{cf}}=g_{\mathrm{cffc}}(1,4,2,3)$ (right), for two different fillings of $c$ fermions. In panel (b), the deviation of the FS for ${\mu }_c=-0.1$ from the one at half-filling is intentionally enlarged for better visualization.

Figure 4

The flows of the most dominant couplings in the RG procedure $g_{\mathrm{ffff}}(5,4,24,23)$ and $g_{\mathrm{ffff}}(5,32,14,23)$ are illustrated for ${\mu }_c=0$ (dashed and solid blue lines): Note that these two lines are on top of each other, indicating that the two couplings have the same magnitude in this case and ${\mu }_c=-0.1$ (dashed and solid orange lines). The two couplings are illustrated in the insets.

Figure 5

Snapshots of interaction vertices during RG flow for $f$ fermions at half-filling (${\mu }_f=0$). (a) $g_{\mathrm{cffc}}(i_1,i_2=1,i_3)$ and (b)–(d) $g_{\mathrm{ffff}}(i_1,i_2=1,i_3)$. The other parameters are set to ${\mu }_c=0$, $U_{\mathrm{cf}}/U_{\mathrm{ff}}=2.0$ for (a) and (b), where the $s_{\pm }$-CDW instability occurs. In panel (c), ${\mu }_c=0.1t$ and $U_{\mathrm{cf}}/U_{\mathrm{ff}}=2.0$, this is in the regime where unconventional $d_{xy}$-wave CDW dominates. (d) ${\mu }_c=0.1t$, $U_{\mathrm{cf}}/U_{\mathrm{ff}}=0.5$, the SDW will be the instability if interspecies interaction is weak. intraspecies interactions are set to $U_{\mathrm{cc}}=U_{\mathrm{ff}}=1.0t$ in all cases shown here.

Figure 6

The phase diagram parameterized by ${\mu }_c$ and ${\mu }_f$. Note the large difference in scales between ${\mu }_c$ and ${\mu }_f$.

Figure 7

Critical temperatures in units of the Fermi temperature $T_{\mathrm{F}}$ for the dominant instabilities SDW and $d_{xy}$-CDW as a function of the interaction ratio $U_{\mathrm{cf}}/U_{\mathrm{ff}}$, with ${\mu }_f=0$ and ${\mu }_c=0.2t$.