Fractional quantum Hall states in two-dimensional electron systems with anisotropic interactions

We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801 (2011)].

Figure 1

Excitation energy gap versus interaction anisotropy for an $N_e=10$ and $\nu =1/3$ FQH system with a square unit cell. The horizontal axis is plotted in the log scale to show the transition at low anisotropy. The system is in a Laughlin-liquid-like state for $A_c<2.0$ (Sec. 3a), and becomes a quasi-1D crystal for $A_c>33.0$ (Sec. 3c), whose excitation gap scales as $1/\sqrt{A_c}$. The intermediate regime is related to a Hall-smectic-like phase, which is unstable towards the quasi-1D crystal and will be discussed in Sec. 3b. The inset shows a linear size-scaling for the minimum gap in the intermediate regime.

Figure 2

(a) 3D and contour plots of the structure factor, (b) the structure factor along the $q_x$ and $q_y$ axes, and (c) the excitation spectrum along the $q_x$ and $q_y$ axes for the $N_e=10$ system with Coulomb interaction anisotropy $A_c=1.8$. To overcome the discrete momentum limitation, we use unit cells with different aspect ratios $R_a=L_x/L_y$ in (b) to obtain more data points and a unit cell with $R_a=0.5$ in (c).

Figure 3

(a) Pair correlation function along the $\widehat{x}$ and $\widehat{y}$ axes for the $N_e=10$ system at interaction anisotropy $A_c=1.8$. The dotted-dashed line represents the correlation function at $A_c=1$ for comparison. The inset reveals the $\alpha r^2$ behavior at small $r$ for the anisotropic case. (b) A linear fit for the prefactor $\sqrt{\alpha }$ at small anisotropy.

Figure 4

(a)–(c) Structure factors for the ground states of the $N_e=10$ system at $A_c=$ 5.0, 11.0, and 20.0, respectively. Note that the locations of the twin peaks for $A_c=5.0$ (CDW-like peaks) differ from those for $A_c=20.0$ (crystal peaks). Both types of peaks coexist at $A_c=11.0$. (d) Peak values of the CDW-like and crystal peaks are plotted as a function of the interaction anisotropy in a range of $1<A_c<50$.

Figure 5

For ground states of the $N_e=10$ system at $A_c=40.0$. (a) Average occupations (scattered squares) of the Landau-level orbital at guiding centers and the charge density (solid line) are plotted along the hard axis. A period of ${\lambda }^{*}=L_x/N_e$ is clearly visible. The density values have been shifted and exaggerated to emphasize a $\sim 2\%$ fluctuation over the background. (b) Structure factor with sharp twin peaks in the $\widehat{q_x}$ direction and periodic oscillation in the $\widehat{q_y}$ direction. (c) Pair correlation function is plotted along two principal axes. The function $g\left(y\right)$ shows an oscillation with the period $L_y/3$.

Figure 6

Wave-function square overlap (lines) and expected value of Coulomb energy (symbols) as a function of the variational parameter $A_L$ for the $\nu =1/3$ FQH system at interaction anisotropy $A_c=1.8$. The energy values have been shifted and enlarged to emphasize that the location of the minimum coincides with that of the largest overlap. The comparison of different sizes of $N_e=6$, 8, 10, and 12 shows a weak size dependence only.

Figure 7

For a Coulomb system with interaction anisotropy $A_c=1.8$. (a) Effective pseudopotentials $V_m$ with a given parameter $A_L=1.47$ is plotted against $m$. (b) The pseudopotential $V_1$ is plotted as a function of $A_L$. The maximum value occurs around $A_L=1.47$.

Figure 8

Contour plot of wave-function square overlap between the Coulomb ground state (parameterized by $A_c$) and the generalized Laughlin state (parameterized by $A_L$) for the $N_e=10$ system. Each dot represents the maximum of the wave-function overlap for a given $A_c$. The solid curve stands for the estimated fit $A_L^{*}$ as a function of $A_c$ using the maximum $V_1$ criterion.