Density-wave patterns for fermionic dipolar molecules on a square optical lattice: Mean-field-theory analysis

We model a system of ultracold fermionic dipolar molecules on a two-dimensional square lattice. Assuming that the molecules are in their nondegenerate hyperfine ground state, and that the dipole moment is polarized perpendicular to the plane (as in the recent experiments on ${}^{40}\mathrm{K}$-${}^{87}\mathrm{Rb}$ molecules), we approximate these molecules as spinless fermions with long-range repulsive dipolar interactions. We use mean-field theory to obtain the restricted phase diagram as a function of the filling, the strength of interaction, and the temperature. We find a number of ordered density-wave phases in the system, as well as phase separation between these phases. A Monte Carlo analysis shows that the higher-period phases are usually suppressed in the exact solution.

Figure 1

Left: Fundamental translation vectors (${\mathrm{a⃗}}_1,{\mathrm{a⃗}}_2$) for possible density-wave phases considered in this work, grouped by the area of the unit cell. Unit cell 1A describes a homogeneous system without order, and 2B corresponds to checkerboard order. Right: Unit cells for the density-wave phases. Only density-wave orders corresponding to unit cells with the solid outline were found to be stabilized in this study.

Figure 2

Reciprocal space for the 3B order. The FBZ is shown in blue in the center of square. The black circles denote $K$ points ($N_c=3$), associated with the density-wave order, and the gray circles are the discretization $\mathrm{k̃}$-points used in the calculation. The open circles denote other possible values of $K+\mathrm{k̃}$ within reciprocal space. The dotted arrows denote other reciprocal space vectors for 3B order used to construct the FBZ.

Figure 3

Chemical potential for $U/t=4$ and $T=0$. The red dashed line shows the MFT solution [which obeys particle-hole symmetry, $\mu (f)+\mu (1-f)=2\mu (0.5)$], but includes regions of unphysical behavior with $\partial \mu /\partial f<0$. The blue line shows the physical solution, which includes regions of phase separation with $\partial \mu /\partial f=0$, accompanied by corresponding gaps at commensurate fillings. Two shaded regions near $f=0.35$ have the same area, according to Maxwell construction.

Figure 4

$U=\infty$ phase diagram without (top) and with (bottom) phase separation. Phase separation is shown as striped regions. The transitions between 4D and 2B (as well as between 2B and 1A) phases are continuous. The rest of the transitions are discontinuous because of phase separation.

Figure 5

$U=4t$ phase diagram without (top) and with (bottom) phase separation. There is no order at low filling, yet for $f>0.25$, the phase diagram qualitatively resembles the $U/t=\infty$ result. Note the reentrant behavior for the checkerboard phase (2B) around $f=0.2$ and the phase separation between the 5C and homogeneous phases.

Figure 6

$T=0$ phase diagram without (top) and with (bottom) phase separation. Decreasing the interaction contracts the range of filling of the ordered phases and progressively eliminates phases commensurate with low values of filling. Only the checkerboard phase survives down to $U=0$. Phase separation replaces the 4D phase near $f=0.28$ and $0.36$ for larger $U/t$. In parts of the phase diagram, 4C and 5C phases show phase separation with the homogeneous state.

Figure 7

$f=0.5$ phase diagram. For large $U/t$, the transition temperature satisfies $T_c=\alpha U/2$, where $\alpha =U[K=(\pi ,\pi )]=1.323$. For small $U/t$, $T_c$ fits perfectly to a semianalytical form. The gap in the density of states (not shown) at $f=0.5$ and $T=0$ shows similar dependence on $U/t$. The entropy per particle (dotted line) at $T=T_c$ reaches $S=2\ln 2$ at $U/t=\infty$.

Figure 8

Structure factors for different phases at representative values of filling and temperature for $U/t=4$. The first column shows the real-space patterns with darker squares denoting larger average density per site. The second column shows the reciprocal space points corresponding to a given density-wave order, followed by the corresponding structure factors at these points, $S(K)$, relative to $S_0=S(K=0)$. Since $S(-K)=S(K)$, only the structure factors for $K_x⩾0$ are shown. The last two columns give the corresponding filling and temperature, respectively.