Single-mode approximation for quantum Hall states with broken rotational symmetry

A topological phase can often be represented by a corresponding wave function (exact eigenstate of a model Hamiltonian) that has a higher underlying symmetry than necessary. When the symmetry is explicitly broken in the Hamiltonian, the model wave function fails to account for the change due to the lack of a variational parameter. Here we exemplify the case by an integer quantum Hall system with anisotropic interaction. We show that the single-mode approximation can introduce a variational parameter for a better description of the ground state, which is consistent with the recently proposed geometric description of the quantum Hall phases.

Figure 1

The mean total angular momentum of the ground state $\overline{M}=\langle {\Psi }^{\left(N\right)}(\alpha ,\theta )|L^z|{\Psi }^{\left(N\right)}(\alpha ,\theta )\rangle$ (in units of $\hslash$) in a system with $N=6$ particles for various confining potential strength $\alpha$ and tilt angle $\theta$. The blue region to the top and right of the orange dashed line represents the anisotropic IQH state, while the rest is loosely referred to as the FQH regime, with the brown region in the lower left corner representing the $\nu =1/3$ Laughlin state.

Figure 2

Low-energy excitations for the isotropic IQH state with $\alpha =0.1$ and $\theta =0^{\circ }$. The blue thick levels can be identified as ${\Phi }_0$ [Eq. (1)], ${\Phi }_2^2$ [Eq. (20)], and ${\Phi }_4^2$ [Eq. (21)]. Note that their excitation energies $\Delta E$ scale with their momenta $\Delta M$. These states are the main components of the anisotropic IQH ground state for $\theta \ne 0^{\circ }$.

Figure 3

(a) Comparison of $1-|\langle {\Psi }^{\left(6\right)}(\alpha ,\theta )|{\tilde{\Phi }}_0\rangle {|}^2$ and the leading contribution $|\langle {\Psi }^{\left(6\right)}(\alpha ,\theta )|{\tilde{\Phi }}_2^2\rangle {|}^2$ for a six-particle system. (b) Comparison of $|\langle {\Psi }^{\left(6\right)}(\alpha ,\theta )|{\tilde{\Phi }}_4^2\rangle |$ obtained from exact diagonalization (ED) and that from the evaluation of Eq. (24), a consequence of the single-mode approximation. (c) $|\langle {\Psi }^{\left(N\right)}(\alpha ,\theta )|{\tilde{\Phi }}_2^2\rangle /\langle {\Psi }^{\left(N\right)}(\alpha ,\theta )|{\tilde{\Phi }}_0\rangle |$, a measure of the anisotropy parameter $\gamma$, for $N=5$–7. In all three panels we keep $\alpha =0.1$ fixed and vary $\theta$ only.

Figure 4

(a) Comparison of $1-|\langle {\Psi }^{\left(6\right)}(\alpha ,\theta )|{\tilde{\Phi }}_0\rangle {|}^2$ and the leading contribution $|\langle {\Psi }^{\left(6\right)}(\alpha ,\theta )|{\tilde{\Phi }}_2^2\rangle {|}^2$ for a six-particle system. (b) Comparison of $|\langle {\Psi }^{\left(6\right)}(\alpha ,\theta )|{\tilde{\Phi }}_4^2\rangle |$ obtained from exact diagonalization (ED) and that from the evaluation of Eq. (24), a consequence of the single-mode approximation. (c) $|\langle {\Psi }^{\left(N\right)}(\alpha ,\theta )|{\tilde{\Phi }}_2^2\rangle /\langle {\Psi }^{\left(N\right)}(\alpha ,\theta )|{\tilde{\Phi }}_0\rangle |$, a measure of the anisotropy parameter $\gamma$, for $N=5$–7. In all three panels we keep $\theta ={70}^{\circ }$ fixed and vary $\alpha$ between 0.01 (weak confinement) and 0.1 (strong confinement).

Figure 5

The increase of the mean angular momentum $▵M=\langle {\Psi }_{\gamma }|L^z|{\Psi }_{\gamma }\rangle -M_0$ with the anisotropy parameter $\gamma$. The inset (a) shows the elliptical density profile [in units of $1/\left(2\pi {\ell }^2\right)$] on the $x$-$y$ plane when $\gamma =0.01$ for an $N=6$ system. The inset (b) shows the density profiles along $x$ and $y$ axes.