Dipolar fermions in a two-dimensional lattice at nonzero temperature

We examine density-ordered and superfluid phases of fermionic dipoles in a two-dimensional square lattice at nonzero temperature. The critical temperature of the density-ordered phases is determined and is shown to be proportional to the coupling strength for strong coupling. We calculate the superfluid fraction and the Berezinskii-Kosterlitz-Thouless transition temperature of the superfluid phase. We finally analyze the effects of an external harmonic trapping potential.

Figure 1

Dipoles move in a 2D square lattice with lattice constant $a$. They are aligned, forming an angle ${\theta }_P$ with the $z$ axis perpendicular to the lattice plane and the azimuthal angle ${\phi }_P$ with the $x$ axis, which is parallel to a lattice vector.

Figure 2

The critical temperature of the striped phase for $({\theta }_P,{\phi }_P)=(\pi /2,0)$ (circles) and of the checkerboard phase for ${\theta }_P=0$ (crosses) as a function of coupling strength for half filling obtained from a numerical calculation on $30\times 30$ lattice sites. The dashed lines give the strong-coupling results $T_c^{\mathrm{st}}=-{\tilde{V}}_D(0,\pi /a)/4$, with $({\theta }_P,{\phi }_P)=(\pi /2,0)$, and $T_c^{\mathrm{cb}}=-{\tilde{V}}_D(\pi /a,\pi /a)/4$, with ${\theta }_P=0$. The top left inset shows the striped order parameter $M$ for $g/t=3.3$ as a function of $T$ for $({\theta }_P,{\phi }_P)=(\pi /2,0)$. The bottom right inset shows the checkerboard order parameter $M$ for $g/t=1$ as a function of $T$ for ${\theta }_P=0$.

Figure 3

The critical temperature in units of $g$ of the striped and checkerboard phases as a function of the dipole orientation (${\theta }_P$, ${\phi }_P$) for half filling. A black dashed line marks the boundary between the stripe and checkerboard phases, and the region with no density order is bounded by a white dashed line.

Figure 4

The critical temperature for the superfluid phase (stars) and the striped phase (circles) for $({\theta }_P,{\phi }_P)=(\pi /2,0)$ as a function of coupling strength for one-third filling obtained from a numerical calculation on a $27\times 27$ lattice site. The crosses give the mean-field superfluid transition temperature $T^{*}$. For illustrative purposes, we plot the critical temperature of the superfluid phase even for $g/t>1.15$, where stripe order suppresses superfluidity.

Figure 5

The superfluid fraction ${\overline{\rho }}_s$ and the nearest-neighbor pairing $|\langle {\widehat{c}}_{i+e_x}{\widehat{c}}_i\rangle |$ as a function of $T$ for various coupling strengths. $|\langle {\widehat{c}}_{i+e_x}{\widehat{c}}_i\rangle |$: pink crosses for $g/t=0.7$, pink stars for $g/t=0.8$, and pink triangles for $g/t=1.5$. ${\overline{\rho }}_s$: blue inverted triangles for $g/t=0.7$, blue circles for $g/t=0.8$, and blue diamonds for $g/t=1.5$. The numerical calculations are performed on a $27\times 27$ lattice with one-third filling. The inset shows the nearest-neighbor pairing $|\langle {\widehat{c}}_{i+e_x}{\widehat{c}}_i\rangle |$ (green circles) and the superfluid fractions ${\rho }_{s,x}$ (pink crosses) and ${\rho }_{s,y}$ (red stars) as a function of $g$ for $T=0$.

Figure 6

(top) The density for $T/t=0$ (left), $T/t=0.3$ (middle), and $T/t=0.4$ (right) for $g/t=1$, $\tilde{\omega }=0.24$, ${\theta }_P=0$, and 207–210 dipoles trapped. (bottom) Crosses are the checkerboard order parameter $|\langle {\widehat{n}}_i-{\widehat{n}}_{i+e_y}\rangle |$ in the center of the trap as a function of $T$, and circles are the checkerboard order parameter performed on the untrapped system at half filling with the same parameters.

Figure 7

(top) The density for $T/t=0$ (left), $T/t=0.1$ (middle), and $T/t=0.3$ (right) for $g/t=1$, $\tilde{\omega }=0.11$, $({\theta }_P,{\phi }_P)=(\pi /2,0)$, and 179–190 dipoles trapped. (bottom) Crosses are the stripe order parameter $|\langle {\widehat{n}}_i-{\widehat{n}}_{i+e_y}\rangle |$ in the center of the trap as a function of $T$, and circles are the stripe order parameter of an untrapped system at half filling with the same parameters.

Figure 8

The pairing order parameter $\left(\right|\langle {\widehat{c}}_{i+e_x}{\widehat{c}}_i\rangle |+|\langle {\widehat{c}}_{i-e_x}{\widehat{c}}_i\rangle \left|\right)/2$ in the center for $\mu =-1.72$ with ${\theta }_P=\pi /2$, $g/t=0.85$, and $\tilde{\omega }=0.11$ as a function of $T$. There are 205–207 dipoles trapped. The inset shows a diagonal cross section of the pairing order parameter. Green circles are $T/t=0.05$, red crosses are $T/t=0.08$, pink stars are $T/t=0.09$, and blue solid line is for $T/t=0.11$.