Hyperdeterminants and composite fermion states in fractional Chern insulators

Fractional Chern insulators (FCI) were proposed theoretically about a decade ago. These exotic states of matter are fractional quantum Hall states realized when a nearly flat Chern band is partially filled, even in the absence of an external magnetic field. Recently, exciting experimental signatures of such states have been reported in twisted ${\mathrm{MoTe}}_2$ bilayer systems. Motivated by these experimental and theoretical progresses, in this paper, we develop a projective construction for the composite fermion states (either the Jain's sequence or the composite Fermi liquid) in a partially filled Chern band with Chern number $C=\pm 1$, which is capable of capturing the microscopics, e.g., symmetry fractionalization patterns and magnetoroton excitations. On the mean-field level, the ground states' and excited states' composite fermion wave functions are found self-consistently in an enlarged Hilbert space. Beyond the mean field, these wave functions can be projected back to the physical Hilbert space to construct the electronic wave functions, allowing direct comparison with FCI states from exact diagonalization on finite lattices. We find that the projected electronic wave function corresponds to the combinatorial hyperdeterminant of a tensor. When applied to the traditional Galilean invariant Landau level context, the present construction exactly reproduces Jain's composite fermion wave functions. We apply this projective construction to the twisted bilayer ${\mathrm{MoTe}}_2$ system. Experimentally relevant properties are computed, such as the magnetoroton band structures and quantum numbers.