Ground state search as an optimization problem: discovering unconventional fractional Chern insulators

We introduce a general gradient-based framework for targeting a ground state of interest in parameter space. We define a differentiable function, dubbed "target-phase loss function", which encodes spectral fingerprints of a quantum state, thereby recasting phase search as an optimization problem. The method is broadly applicable to a wide range of symmetry-broken and topological orders. We apply it to spinless fermions on the kagome lattice and discover two fractional Chern insulators (FCIs) in unexpected settings, corroborated through detailed exact diagonalization: (i) at filling $\nu = 1/3$, a "non-ideal" Abelian FCI whose band geometry strongly violates expectations of Landau-level mimicry and all recent generalizations; and (ii) at $\nu = 1/2$, a non-Abelian FCI stabilized purely by finite-range two-body interactions. These results provide the first explicit realization of such types of FCIs and establish a versatile paradigm for systematic quantum-phase discovery.