Composite Fermion Theory of Fractional Chern Insulator Stability

Abstract

We develop a mean-field theory of the stability of fractional Chern insulators based on the dipole picture of composite fermions (CFs). We construct CFs by binding vortices to Bloch electrons and derive a CF single-particle Hamiltonian that describes a Hofstadter problem in the enlarged CF Hilbert space, with the trace-condition term emerging naturally in the small-$q$ limit as part of the CF Hamiltonian. Going beyond the small-$q$ limit, we apply our theory to twisted MoTe$_2$ and calculate its CF band structures. The resulting CF phase diagram matches closely with that from exact diagonalization, and the projected many-body wavefunctions achieve exceptionally high overlaps with the latter. Our theory provides both a microscopic understanding and a computationally efficient tool for identifying fractional Chern insulators.

Figures (2)

• Figure 1: CF phase diagram (a), $6 \times 4$ ED phase diagram (b), geometric fluctuations (c), and typical CF bands in twisted MoTe$_2$ (d--f). The color maps in (a) and (b) represent the CF band gap and the many-body gap obtained from ED, respectively. Panel (c) shows the dimensionless standard deviation of the effective magnetic field $\Omega_{\bm k}$, traditional trace condition, and the effective scalar potential of the CF Hamiltonian \ref{['eq:CF single particle Hamiltonian']} at $\epsilon=15$, normalized by the moiré unit cell volume $A_{\text{u.c.}}$ and the effective CF mass $m_*\approx4.87$ meV measured from the calculated CF spectrum close to the ideal limit $\theta=3.55^\circ$. Panels (d)--(f) show the two lowest CF bands for three representative points labeled in (a), computed on a $9 \times 9$ electronic sample, with colors represents the band widths. Note: For visualization, the CF bands are unfolded into the electronic Brillouin zone (BZ), rather than plotted within the CF’s original BZ, which occupies only one-third of the electronic BZ at filling $\nu = 1/3$. The extra threefold degeneracy within CF's reduced BZ, known as the symmetry fractionalization pattern due to the interplay between topological orders and magnetic translation symmetries barkeshli2019symmetrychen2017symmetryessin2014spectroscopichu2024hyperdeterminants, is also explicitly visible in our calculation.
• Figure 2: Wavefunction overlap with ED results (a) and typical $6 \times 4$ ED spectra (b–c) before and after FCI breakdown in twisted MoTe$_2$. Data are shown along the blue line cut in Fig.\ref{['fig:combined_CF_phase_diagram_and_CF_bands']}(b), spanning twist angles $\theta = 3.5^\circ$ to $4.2^\circ$ at dielectric constant $\epsilon = 18$, crossing the FCI phase boundary near $\theta_c \approx 4.01^\circ$. The projected wavefunction is constructed via \ref{['eq:wavefunction overlap']}, using CF states $\ket{\psi_\text{CF}}$ obtained from the single-particle Hamiltonian $H^P_\text{single}$. We compute the overlap between the projected wavefunction and the three ED eigenstates highlighted by red circles in (b) and (c), located at the center-of-mass momenta expected for the $6 \times 4$ FCI ground states. Shaded bands in (a) indicate statistical errors from variational Monte Carlo samplings.