Fibonacci anyons and charge density order in the 12/5 and 13/5 quantum Hall plateaus

The $\nu =12/5$ fractional quantum Hall plateau observed in $\mathrm{GaAs}$ semiconductor wells is a suspect in the search for non-Abelian Fibonacci anyons. Using the infinite density matrix renormalization group, we find clear evidence that fillings $\nu =12/5$ and $\nu =13/5$ are in the $k=3$ Read-Rezayi phase in the absence of particle-hole symmetry breaking effects. The lowest energy charged excitation is identified as a non-Abelian Fibonacci anyon, distinguished from its Abelian counterpart by its local quadrupole moment. However, several experiments at $\nu =13/5$ observe a reentrant integer quantum Hall effect, implying particle-hole symmetry is broken. We rule out spin polarization as the origin of the asymmetry. Further, we point out extremely close energetic competition between the Read-Rezayi phase and a reentrant integer quantum Hall phase. This competition suggests that even small particle-hole symmetry breaking perturbations can explain the experimentally observed asymmetry between $\nu =12/5$ and $13/5$. We find that at $\nu =12/5$ Landau level mixing favors the Read-Rezayi phase over the reentrant phase.

Figure 1

Orbital entanglement spectra for the two degenerate ground states at circumference $L=32{\ell }_B$, well width $w=3{\ell }_B$. For each ground state, we pick two orbital cuts and plot the entanglement energies vs angular momentum. The low-lying spectra (highlighted in red) agree with the CFT prediction given in Eq. (2), and thus provide an unambiguous identification of the ${\overline{\mathrm{RR}}}_3$ phase.

Figure 2

(a) The splitting of the topological degeneracy $E\left(|{\mathrm{\Omega }}_{\tau }\rangle \right)-E\left(|{\mathrm{\Omega }}_{\mathbb{1}}\rangle \right)$ as a function of cylinder circumference $L$ is consistent with an exponential decrease. $w$ is the width of the well. (b) Scaling of the momentum polarization $M$ with circumference squared $L^2$, which reveals the shift $\mathcal{S}$. Dashed lines indicate the theoretical predictions for the $\mathbb{1}$ and $\tau$ ground states of ${\overline{\mathrm{RR}}}_3$, and dotted lines the competing hierarchy (AH) and Bonderson-Slingerland (BS) states.

Figure 3

Anyon excitation energies $E_{Q,a}$ at various quantum well thicknesses, for the ${\overline{\mathrm{RR}}}_3$ phase in the absence of disorder. Left: The difference between the Abelian and Fibonacci anyons for charges $Q=\pm \frac{e}{5}$. The data show that for a fixed charge $Q$, it is energetically favorable to trap a Fibonacci rather than an Abelian anyon. Right: The energy required to disassociate a pair of $\pm \frac{e}{5}$ quasiparticles. See the main text for caveats in regard to this data.

Figure 4

The density profile of the charge-$e/5$ anyons in the presence of a Gaussian pinning potential of width $\approx 4{\ell }_B$, computed on an $L=21{\ell }_B$ cylinder. We enforce the presence of an anyon in the central region by using various “topological boundary conditions” $[l,r]$, and minimize the energy subject to this constraint. Top: The Abelian $\frac{e}{5}$ anyon. Middle: The Fibonnaci $\frac{e}{5}\tau$ anyon. Bottom: When using topological boundary conditions $[\tau ,\frac{e}{5}\tau ]$, the fusion rule $\tau \times \tau =\mathbb{1}+\tau$ implies that there are two distinct anyon types that could arise: either an Abelian $\frac{e}{5}$ or the non-Abelian $\frac{e}{5}\tau$. We find the non-Abelian type has lower energy. Consequently, the local properties of the excitation (e.g., the charge density and energy) agree between the middle and bottom panels. These quasiparticles are about $15{\ell }_B$ in diameter.

Figure 5

(a) Energies for the ${\overline{\mathrm{RR}}}_3$ states (red and blue triangles) as a function of $L$ and CDO (green circles) as a function of wavelength $\lambda$, at zero well width. For data points denoted by triangles, the energies are computed while enforcing translational invariance of the wave functions. The two ground states $|{\mathrm{\Omega }}_{\mathbb{1}/\tau }\rangle$ are metastable, and their energy approaches $E_{\mathrm{RR}}\approx -0.13717$ in the thermodynamic limit. For data points denoted by circles, a charge density wave of wavelength $\lambda$ is imposed along the length of the cylinder (illustrated in the inset). Here the CDO minimum $E_{\mathrm{CDO}}$ dips below $E_{\mathrm{RR}}$, which suggests that at $w=0$, the quantum Hall phase is unstable to formation of CDO. (b) Orbital (dots) and real-space (line) density profile of the CDO minima at $\lambda \approx 4.8{\ell }_B$. Inset shows the Fourier transform of the orbital density, with an exponential decay. (c) Energy splitting (per flux) between the ${\overline{\mathrm{RR}}}_3$ phase and the CDO phase $(E_{\mathrm{CDO}}-E_{\mathrm{RR}})$, as a function of $w$. The data show that increased well width $w$ tends to stabilize the Read-Rezayi phase.

Figure 6

The real-space density-density correlation function $S\left(\mathbf{r}\right)=\langle \rho \left(\mathbf{r}\right)\rho \left(\mathbf{0}\right)\rangle /{\overline{\rho }}^2$ of the CDO phase at $\nu =\frac{12}{5}$ for $w=1{\ell }_B$ and $L=26.5{\ell }_B$. The point $\mathbf{r}=\mathbf{0}$ is taken to lie at the center of the high-density stripe, and $\overline{\rho }$ is the mean density. The horizontal axis is along the length of the cylinder; the vertical axis is around the circumference of the cylinder. At this circumference the unit cell along the cylinder contains $4\times 5$ flux, corresponding to eight electrons per stripe. The intrastripe correlations have seven local maxima plus a correlation hole. The nearest-stripe correlations have weaker maxima which are out of phase with the intrastripe maxima, which suggests there is a tendency towards forming a body-centered rectangular lattice of single electrons.

Figure 7

The effect of Landau level mixing on energy splitting (per flux) between the CDO and the Read-Rezayi states. The data are given for $w=3{\ell }_B$ and $\kappa =0.4$, where $\kappa$ is the ratio of Coulomb to cyclotron energy. The result is repeated for circumferences $L\sim 18–22$, here expressed at the CDO wavelength ($x$ axis). For reference, the dashed line denotes the energy splitting in the absence of LL mixing. For $\nu =\frac{12}{5}$, the splitting increases with the inclusion of LL mixing, further favoring the ${\overline{\mathrm{RR}}}_3$ phase. For $\nu =\frac{13}{5}$ the data are more ambiguous due to greater numerical uncertainty in the energy of the ${\mathrm{RR}}_3$ phase, requiring a larger $\chi$ with the addition of LL mixing. We conclude that LL mixing stabilizes the ${\overline{\mathrm{RR}}}_3$ phase at $\nu =\frac{12}{5}$, while the best data points show little effect on the $\nu =\frac{13}{5}$ state.

Figure 8

(a) Momentum polarization for the ground states $|{\mathrm{\Omega }}_{\mathbb{1}/\tau }\rangle$. The dashed lines are the predicted values of $h_a-\frac{c}{24}$ for the two ground states. (b) Entanglement entropies $S_{\mathbb{1}}$ and $S_{\tau }$, along with their difference $S_{\tau }-S_{\mathbb{1}}$. The dashed line at $log\left(\phi \right)\approx 0.48$ is predicted for the ${\overline{\mathrm{RR}}}_3$ phase.

Figure 9

Transition between the ${\overline{\mathrm{RR}}}_3$ and hierarchy states as the Haldane pseudopotential $\mathrm{\Delta }{V}_1$ is added to the Coulomb interaction. In the cylinder geometry the transition is first order. For perspective, in the $N=0$ LL, $V_1/V_3=1.6$; in the $N=1$ LL, $V_1/V_3=1.3$; at the observed transition $V_1/V_3=1.43$. Linearly extrapolating the energy of the hierarchy state to the Coulomb point, we obtain a splitting of $E_{\mathrm{AH}}-E_{\mathrm{RR}}\approx 1.5\times {10}^{-4}/\mathrm{flux}$. Data are taken from DMRG at $L=20{\ell }_B, w=2{\ell }_B$.

Figure 10

Phase diagram as a function of modified $V_1$ pseudopotential, for zero width (left) and $w/{\ell }_B=4$ (right). Data are obtained by exact diagonalization of the system with 25 flux quanta through the hexagonal unit cell. We compute the lowest 10 energies per momentum sector and plot them relative to the average energy of the system $\tilde{E}=E-E_{\mathrm{avg}}$. Black symbols denote levels in $\mathbf{K}=0$ sector. For zero width (left), we identify the ${\mathbb{Z}}_3$ ground state degeneracy in a narrow shaded region around the Coulomb point $\delta V_1=0$. In this region the ground state has large overlap $\mathcal{O}$ with the model ${\mathbb{Z}}_3$ wave function (inset). At width $w/{\ell }_B=4$ (right), the ${\mathbb{Z}}_3$ phase widens and becomes more robust. It is surrounded by the hierarchy/composite fermion state for larger positive $\delta V_1$, and several charge density ordered phases for negative $\delta V_1$.

Figure 11

Spin correlator $\langle S_0^{+}{S}_k^{-}\rangle$ for $L=21{\ell }_B, w=3{\ell }_B$ at filling ${\tilde{\nu }}^{\uparrow /\downarrow }=\frac{1}{5}$. Solid curves shows the data at various bond dimensions $\chi$, converging to the expected correlator given as the dashed curve. This is a signature of spontaneously broken spin-rotational symmetry, indicative of a fully spin-polarized ground state at $\nu =\frac{12}{5}$.

Figure 12

Signature of spontaneous spin rotation symmetry breaking. Left: The entanglement spectrum of the spinful system at ${\tilde{\nu }}^{\uparrow /\downarrow }=\frac{1}{5}$, sorted by charge sector $(Q^{\uparrow },Q^{\downarrow })$. Right: The entanglement spectrum for a spinless $\tilde{\nu }=\frac{2}{5}$ system for comparison. Both spectra are taken at $L=21{\ell }_B, w=3{\ell }_B$.

Figure 13

Real-space structure factor $S\left(\mathbf{r}\right)=\langle \rho \left(\mathbf{r}\right)\rho \left(0\right)\rangle /{\langle \rho \rangle }^2$ of the $\nu =\frac{12}{5}$ Read-Rezayi and CDO states at $L=20{\ell }_B, w=1{\ell }_B$. The degenerate ${\overline{\mathrm{RR}}}_3$ states are mostly identical which is a necessary condition for topological order. The ${\overline{\mathrm{RR}}}_3$ states are also nearly rotationally symmetric in contrast to the striped structure of the competing CDO phase; the residual anisotropy is presumably due to the finite circumference of the cylinder.

Figure 14

The effect of finite width $w/{\ell }_B$ on the hexagonal torus threaded by $N_{\varphi }=25$ flux quanta (left) and $N_{\mathrm{\Phi }}=30$ (right). Energy spectrum is obtained by exact diagonalization and plotted relative to the average energy of the system. Black symbols denote levels belonging to $\mathbf{K}=0$ sector.

Figure 15

The effect of changing the geometry (one linear dimension $L_x$ of the torus) on the energy spectrum (top) and the ground state (bottom row) of the system. Energy spectrum is obtained by exact diagonalization of the rectangular torus with $N_{\mathrm{\Phi }}=20$ flux quanta, and black symbols denote levels belonging to $\mathbf{K}=0$ sector. Varying $L_x$ induces charge density order in the ground state, as seen in the pair correlation function $g\left(\mathbf{r}\right)$ for several values of $L_x$ (bottom).