Beyond the Lowest Landau Level: Unlocking More Robust Fractional States Using Flat Chern Bands with Higher Vortexability

Abstract

Enhancing the many-body gap of a fractional state is crucial for realizing robust fractional excitations. For fractional Chern insulators, existing studies suggest that making flat Chern bands closely resemble the lowest Landau level (LLL) seems to maximize the excitation gap, providing an apparently optimal platform. In this work, we demonstrate that deforming away from the LLL limit can, in fact, produce substantially larger FQH gaps. Using moiré flat bands with strongly non-Landau-level wavefunctions, we show that the gap can exceed that of the LLL by more than two orders of magnitude for short-range interactions and by factors of two to three for long-range interactions. This enhancement is generic across Abelian FCI states and follows a universal enhancement factor within each hierarchy. Using the Landau level framework, we identify the amplification of pseudopotentials as the microscopic origin of the observed enhancement. This finding demonstrates that pseudopotential engineering can substantially strengthen fractional topological phases. We further examined non-Abelian states and found that, within finite-size resolution, this wavefunction construction method can also be used to manipulate and enhance the gap for certain interaction parameters.

Figures (13)

• Figure 1: Many-body gap enhancement in moiré model. (a) Single particle band structure of the moiré Hamiltonian Eq. \ref{['eq:moireHamiltonian']} with four-fold degenerate exact flat bands. The vertical axis in (a) is normalized energy $\tilde{E}=E/(4 \pi/ \sqrt{3} a)^2$. (b) The ratio between the average trace of the quantum metric $\langle\mathrm{Tr}[g(\mathbf{k})]\rangle$ and the berry curvature $\braket{|F_{xy}(\mathbf{k})|}$ for the higher vortexable band $\tilde{\Psi}_{\mathbf{k},2}$ given in the main text. (c–f) Many-body gap of FCI states (blue dots) as a function of $\theta$. The gap increases markedly as $\theta$ deviates from the limit of ideal quantum geometry ($\theta=0$). The lower and upper horizontal red dashed lines in (c–f) mark, respectively, the gap of the corresponding FQH states in the LLL and the maximal many-body gap achieved in the LL-hybridization model with the same interaction strength. Notably, in all cases the moiré model produces a gap larger than the LLL, and even exceeds the best LL-hybridization model values for $\nu=1/3$. The gap size here is measured in units of the Coulomb scale, $U^{\rm LL}_{\mathrm{int}}=e^2/\varepsilon l_B$, where $l_B=3^{1/4}a/(4\pi)^{1/2}$ is the magnetic length, $a$ is moiré lattice constant, and $\varepsilon$ is the dielectric constant. The parameters used in ED are (c) $\nu=1/3$, $d_s/l_B=0.037$, $N_s=21$; (d) $\nu=1/3$, $d_s/l_B=0.74$, $N_s=21$; (e) $\nu=1/5$, $d_s/l_B=0.23$, $N_s=25$; (f) $\nu=1/5$, $d_s/l_B=0.74$, $N_s=25$.
• Figure 2: Gap enhancement via Landau level hybridization and pseudopotential decomposition. (a–c) Many-body gap and pseudopotentials as functions of the hybridization parameter $\theta$, for LL01, LL02, and LL03, respectively. As in Fig. \ref{['fig:moire']}, the energy is measured in units of the Coulomb energy scale $U^{\rm LL}_{\rm int}$. (d) shows the average trace of quantum metric and Berry curvature normalized by $l_B^2$ for the three models. Gap values are obtained for multiple finite clusters SM2025, using screened Coulomb interactions with screening length $d_s=0.037\,l_B$. The gap reaches its maximum near $\theta \!\approx\! \pi/4$, showing up to $\sim500$-fold enhancement for LL01 and LL02, with peak gap values of $\sim0.2\%$ of the Coulomb scale (compared to $0.0003\%$ for the LLL). For LL03, the enhancement is smaller--about $\sim200$-fold, yielding a maximum gap $\sim0.06\%$ of the Coulomb scale.
• Figure 3: Many-body gap ratio along Jain sequences. Normalized many-body gap $\Delta_{\mathrm{mb}}(\theta)/\Delta_{\mathrm{mb}}(0)$ for the LL01 hybridization model. (a) $1/3$ sequence:$\nu=1/3,\,2/5,\,3/7,\,4/9$ at short-distance screening $d_s/l_B=0.037$. The enhancement exhibits a universal dome-like profile across the entire sequence, following the pseudopotential $\!c_1-c_3$ and peaking near $\theta=\pi/4$. (b) $1/5$ sequence:$\nu=1/5,\,2/9,\,3/13$ at $d_s/l_B=0.19$. Here, the $\theta$ dependence tracks the pseudopotential $\!c_3-c_5$ throughout the sequence, with the maximum occurring near $\theta=\pi/2$.
• Figure A1: (a-b) Many-body energy spectra at filling $\nu=1/3$, $d_s/l_B=0.037$ and $\nu=1/5$, $d_s/l_B=0.23$ respectively. The energy is normalized by $U^{\rm LL}_{\mathrm{int}}$ as in Fig. 1. (c-d) Particle entanglement spectra of the ground states in (a-b) with particle cuts $N_A=3,N_B=4$ for $\nu=1/3$ and $N_A=2,N_B=3$ for $\nu=1/5$, respectively. The number of low lying states below the red dashed lines (as written in red) match the quasi-hole counting of Laughlin states at the respective filling fractions Regnault2011fractional. $\theta=\pi/4$ was used for all cases.
• Figure A2: Many-body gap enhancement in a moiré model with significant single-particle quantum geometric fluctuations. The single particle Hamiltonian is the same as in Eq. \ref{['eq:moireHamiltonian']} with $\tilde{A}(\mathbf{r}) = -\alpha \sum_{n=1}^3 e^{i(1-n)\phi}\cos(\mathbf{G}_n\cdot\mathbf{r})/2$, where $\mathbf{G}_n$ and $\phi$ were defined below Eq. \ref{['eq:moireHamiltonian']}, and $\alpha = 0.62|G|^2$. Similar to Fig. \ref{['fig:moire']}(a), there are four exact flatbands, two per sublattice. One of these two sublattice polarized bands in vortexable with wavefunction $\tilde{\Psi}_{\mathbf{k},1}(\mathbf{r})$, the other one higher vortexable $\tilde{\Psi}_{\mathbf{k},2}(\mathbf{r})$ as discussed in the main text. (a) and (b) show the fluctuations of Berry curvature and quantum metric for the higher vortexable band $\tilde{\Psi}_{\mathbf{k},2}(\mathbf{r})$, respectively. Clearly, for this choice of $\tilde{A}(\mathbf{r})$, the fluctuations in Berry curvature and quantum metric are much higher than the one in Fig. \ref{['fig:moire']}. (c-f) Many-body gap of FCI states (blue dots) from ED as a function of $\theta$. As in Fig. \ref{['fig:moire']}, the energy is measured in units of the Coulomb energy scale $U^{\rm LL}_{\rm int}$. The parameters used in ED for (c-f) are same as the parameter used in Fig. \ref{['fig:moire']}(c-f), respectively. The lower and upper horizontal red dashed lines in (c–f) mark, respectively, the gap of the corresponding FQH states in the LLL and the maximal many-body gap achieved in the LL-hybridization model with the same interaction strength. Gap enhancement trends in this case remain the same as in Fig. \ref{['fig:moire']}, although the maximum gap enhancements are significantly lower than those in Fig. \ref{['fig:moire']}; the gap sizes do not follow the variation in $\text{tr}(g)$ and $F_{xy}$. In (d), a phase transition occurs at $\theta\approx 1.18$; our analysis here focuses on the maximum gap of the FCI phase.