Phase transition and intrinsic metric of dipolar fermions in the quantum Hall regime

For the fast rotating quasi-two-dimensional dipolar fermions in the quantum Hall regime, the rotational symmetry in the two-body interaction breaks when the dipole moment has an in-plane component that can be tuned by an external field. Assuming that all the dipoles are polarized in the same direction, we perform the numerical diagonalization for finite size systems on a torus. We find that while $\nu =1/3$ Laughlin state is stable in the lowest Landau level (LLL), it is not stable in the first Landau level (1LL); instead, the most stable Laughlin state in the 1LL is the $\nu =2+1/5$ Laughlin state. These FQH states are robust against moderate introduction of anisotropy, but large anisotropy induces a transition into a compressible phase in which all the particles are attracted and form a bound state. We show that such phase transitions can be detected by the intrinsic geometrical properties of the ground states alone. The anisotropy and the phase transition are systematically studied with the generalized pseudopotentials and characterized by the intrinsic metric, the wave function overlap, and the nematic order parameter. We also propose simple model Hamiltonians for this physical system in the LLL and 1LL, respectively.

Figure 1

The sketch map for the dipolar fermions in a fast rotating harmonic trap. While the rotating frequency $\mathrm{\Omega }$ is approaching the trap frequency $\omega$, the system is equivalent to the two-dimensional electron gas in a strong magnetic field. $\alpha$ is the angle between the direction of the dipole moment and the $z$ axis.

Figure 2

The effective interaction $V_{\text{eff}}(\mathbf{k},\alpha )$ in the LLL at different $\alpha \mathrm{s}$ with thickness $q=0.01$. (a) $\alpha =0^{\circ }$; (b) $\alpha ={20}^{\circ }$; (c) $\alpha =54.7^{\circ }$; (d) $\alpha ={90}^{\circ }$.

Figure 3

Energy spectrum for 8 electrons in 24 orbitals on a torus at different tilting angles. The aspect ratio of the torus is set to be one. (a) $\alpha =0^{\circ }$; (b) $\alpha ={30}^{\circ }$; (c) $\alpha ={52}^{\circ }$; (d) $\alpha ={90}^{\circ }$.

Figure 4

(a) Contour plot of the ground state overlap $\mathcal{O}(\varphi ,\gamma )$. Here dipoles are $\alpha ={40}^{\circ }$ tilted with thickness $q=0.01$. The maximum point at $\gamma =1.3$ and $\varphi =\pi /2$ corresponds to the intrinsic metric. (b) The PPs $V_1^g$ and $|V_{1,2}^g|$ as a function of $\gamma$ at the same parameter as that in (a). The maximum $V_1^g$ corresponds to $V_{1,2}^g=0$. The $V_1^g$ is optimized at $\gamma \simeq 1.3$ which is the same as that for max ($\mathcal{O}(\varphi ,\gamma )$) in (a).

Figure 5

Comparison of the intrinsic metric for different systems and the analytic results for $q=0.01$. The finite size results gradually approach the analytic one in the thermodynamic limit.

Figure 6

The overlap ${\mathcal{O}}_0$ and $max\left({\mathcal{O}}_g\right)$ as a function of the $\alpha$ for $q=0.01$ (a) and $q=0.5$ (b). The system is for 10 electrons in 30 orbitals on the torus with aspect ratio one. (c) and (d) show the nematic order parameter $N_{x^2-y^2}$ as a function of the $\alpha$ for systems with $5–10$ electrons at $\nu =1/3$ filling.

Figure 7

PPs as a function of the tilted angle $\alpha$ in the LLL. (a) and (b) show the three dominant diagonal and off-diagonal terms with odd $m$, respectively.

Figure 8

The same plot as in Fig. 7 for the 1LL.

Figure 9

The effective interaction $V_{\text{eff}}(\mathbf{k},\alpha )$ for 1LL at $\alpha ={90}^{\circ }$. In (a), we can see the squeezing direction of the contours has a ${90}^{\circ }$ rotation as compared to that in a large $\left|k\right|$ range plot shown in (b).

Figure 10

The energy spectrum for 7 electrons at $\nu =1/5$ in the 1LL for different $\alpha \mathrm{s}$ with $q=0.01$. (a) $\alpha =0^{\circ }$; (b) $\alpha ={30}^{\circ }$; (c) $\alpha ={45}^{\circ }$; (d) $\alpha ={46}^{\circ }$.

Figure 11

The comparison of the overlap and intrinsic metric for the 7-electron system at $\nu =2+1/5$ for different thicknesses in the 1LL. (a) $q=0.01$; (b) $q=0.5$; (c) $q=0.01$; (d) $q=0.5$.

Figure 12

The fitting function for the model Hamiltonian in the (a) LLL and (b) 1LL, respectively. The model Hamiltonian is written as $H=\text{sgn}(\alpha -{\alpha }_c)(V_1^g+{\lambda }_0{V}_{1,2}^g)$ for the LLL and $H=\text{sgn}(\alpha -{\alpha }_c)(V_1^g+{\lambda }_1{V}_3^g+{\lambda }_2{V}_{1,2}^g+{\lambda }_3{V}_{3,2}^g)$ for the 1LL. The parameter ${\lambda }_i(\alpha ,q)$ can be written as ${\lambda }_i(\alpha ,q)=h\left(\alpha \right)f_i\left(q\right)$ where $h\left(\alpha \right)=\frac{{sin}^2\left(\alpha \right)}{\frac{1}{2}(3{cos}^2\left(\alpha \right)-1)}$ and $f_i\left(q\right)$ are fitted as $f_0\left(q\right)\simeq -0.61-1.1q-0.10q^2+0.05q^3,f_1\left(q\right)\simeq 1.5-0.95q+1.09q^2-0.42q^3,f_2\left(q\right)\simeq 1.58-0.1q+0.37q^2-0.1q^3$, and $f_3\left(q\right)\simeq -0.35-0.71q+0.2q^2-0.11q^3$.