Evidence of a fractional quantum Hall nematic phase in a microscopic model

At small momenta, the Girvin-MacDonald-Platzman (GMP) mode in the fractional quantum Hall (FQH) effect can be identified with gapped nematic fluctuations in the isotropic FQH liquid. This correspondence would be exact as the GMP mode softens upon approach to the putative point of a quantum phase transition to a FQH nematic. Motivated by these considerations as well as by suggestive evidence of an FQH nematic in tilted field experiments, we have sought evidence of such a nematic FQHE in a microscopic model of interacting electrons in the lowest Landau level at filling factor 1/3. Using a family of anisotropic Laughlin states as trial wave functions, we find a continuous quantum phase transition between the isotropic Laughlin liquid and the FQH nematic. Results of numerical exact diagonalization also suggest that rotational symmetry is spontaneously broken, and that the phase diagram of the model contains both a nematic and a stripe phase.

Figure 1

Conjectured schematic phase diagram of the $V_1-V_3-V_5$ model: A = isotropic Laughlin liquid, B = stripe/smectic phase, C = FQH nematic.

Figure 2

(a) Value of the variational parameter $\left|Q\right|$ that minimizes the ground-state energy (5) for a system with $N=13$ electrons. A continuous quantum phase transition from an isotropic to a nematic FQH state is seen at $V_3/V_1\approx 0.6$. (b) Amplitude of the microscopic nematic order parameter (7) as a function of $\left|Q\right|$.

Figure 3

Exact diagonalization study of the $V_1-V_3-V_5$ model with $N$ electrons: (a)–(c) gap between ground and first excited state in the zero-momentum sector; (d)–(f) overlap of the ground state with the $\nu =1/3$ Laughlin state; (g)–(i) overlap of the ground state with the $V_3/V_1\to \infty$ ground state.

Figure 4

Dependence of gap to first excited state on system size: comparison between the Laughlin liquid (blue curve, $V_1=1$ and $V_3=V_5=0$) and the pure $V_3$ phase (red curve, $V_3=1$ and $V_1=V_5=0$, i.e., $V_3/V_1\to \infty$). The gap was computed up to $N=14$ (resp. $N=13$) electrons for the pure $V_3$ phase (resp. Laughlin liquid).

Figure 5

Dependence of gap to first excited state on aspect ratio $\delta =L_y/L_x-1$ of the torus in both the zero-momentum sector (blue lines) and at finite momentum (red lines). We have considered both $N=12$ electrons (solid lines) and $N=11$ electrons (dotted lines): (a) Laughlin liquid ($V_3=V_5=0$), (b) pure $V_3$ phase ($V_3/V_1\to \infty$).

Figure 6

Ground-state nematic susceptibility (8), capped for readability at ${\chi }_{\max }=10$.

Figure 7

(a) Gap between ground and first excited state irrespective of its momentum for the $V_1-V_3-V_5$ model with $N=11$ electrons. Note that the scale of the gap in Fig. 3 is slightly different but the gross features of the two plots are mostly identical. (b) Gap as a function of the ratio $V_3/V_1$ for $N=13$ electrons and a square aspect ratio. We show both the gap in the zero-momentum sector (red line) and the gap to the first excited with a nonzero momentum (blue line). Note that in order to compare gaps between different values of $V_3/V_1$, we have set the energy scale to be 1 for the two-particle problem, irrespective of the value of $V_3/V_1$. Inset: Close-up of the gap in the zero-momentum sector around the point $V_3/V_1\simeq 0.500$ where the gap almost closes ($1.5\times {10}^{-4}$).

Figure 8

Gap as a function of the ratio $V_3/V_1$ for $N=12$ electrons and a square aspect ratio. We show both the gap in the zero-momentum sector (red line) and the gap to the first excited with a nonzero momentum (blue line). In order to compare gaps between different values of $V_3/V_1$, we have set the energy scale to be 1 for the two-particle problem, irrespective of the value of $V_3/V_1$. Inset: Overlap of the ground state with the $\nu =1/3$ Laughlin state (red dashed line) and the $V_3/V_1\to \infty$ ground state (blue bashed line) as function of the ratio $V_3/V_1$ for $N=12$ electrons and a square aspect ratio.

Figure 9

Exact diagonalization study of the $V_1-V_3-V_5$ model with $N=12$ electrons: (a) overlap of the ground state with the $\nu =1/3$ Laughlin state; (b) overlap of the ground state with the $V_3/V_1\to \infty$ ground state; (c) gap between ground and first excited state in the zero-momentum sector; (d) ground-state nematic susceptibility (8), capped for readability at ${\chi }_{\max }=10$.