Liquid and Crystal Phases of Dipolar Fermions in Two Dimensions

The liquid and crystal phases of a single-component Fermi gas with dipolar interactions are investigated using quantum Monte Carlo methods in two spatial dimensions and at zero temperature. The dipoles are oriented by an external field perpendicular to the plane of motion, resulting in a purely repulsive $1/r^3$ interaction. In the liquid phase we calculate the equation of state as a function of the interaction strength and other relevant properties characterizing the Fermi-liquid behavior: effective mass, discontinuity at the Fermi surface, and pair correlation function. In the high density regime we calculate the equation of state of the Wigner crystal phase and the critical density of the liquid to solid quantum phase transition. Close to the freezing density we also search for the existence of a stripe phase, but such a phase is never found to be energetically favorable.

Figure 1

Equation of state of the liquid and solid phase in units of the Hartree-Fock energy (2). Red circles refer to the liquid and green triangles to the solid. The red dashed line corresponds to the second-order expansion in Ref. 10. The purple dashed horizontal line and solid line correspond, respectively, to the classical energy of the Wigner crystal and to the result of Ref. 29 including the first correction arising from the zero-point motion of phonons. Inset: Energy difference between the solid and the liquid (blue circles) and between the stripe phase and the liquid (black circles). The blue solid line is obtained from a best fit to the equation of state of the liquid and solid phase. Error bars are smaller than the size of the symbols and are comparable in the three phases.

Figure 2

Finite size effects in a Fermi liquid at the density $k_F{r}_0=0.22$. The black symbols and line correspond to TABC energies and linear fit. The green symbols and line correspond to PBC energies $E_N-(m/m^{*})\Delta T_N$ and linear fit. Red symbols correspond instead to PBC energies $E_N-\Delta T_N$. The values extrapolated to the TL are shown with the corresponding error bars.

Figure 3

Effective mass and renormalization factor in the liquid phase as a function of the interaction strength. The line corresponds to the perturbation expansion for $m^{*}/m$ of Ref. 10. Inset: Momentum distribution corresponding to $k_F{r}_0=20$ for different system sizes.

Figure 4

Pair correlation function in the liquid and in the crystal phase. The pair correlation function of the noninteracting gas is also shown.

Figure 5

Static structure factor in the liquid and in the crystal phase ($N=56$). In the liquid phase, solid lines correspond to the Fourier transform of $g(r)$ while symbols correspond to the direct calculation of $S(k)$. The static structure factor of the noninteracting gas is also shown.