Gradient-based search of quantum phases: discovering unconventional fractional Chern insulators

Abstract

The discovery and understanding of new quantum phases has time and again transformed both fundamental physics and technology, yet progress often relies on slow, intuition-based theoretical considerations or experimental serendipity. Here, we introduce a general gradient-based framework for targeted phase discovery. We define a differentiable function, dubbed "target-phase loss function", which encodes fingerprints of a quantum state, thereby recasting phase search as a tractable optimization problem in Hamiltonian space. The method is broadly applicable to a wide range of symmetry-broken and topological orders and can be interfaced with most many-body numerical solvers. As a demonstration, we apply it to spinless fermions on the kagome lattice using exact diagonalization and discover two distinctive fractional Chern insulators (FCIs): (i) at filling $\nu = 1/3$, a "non-ideal" Abelian FCI whose band geometry lies far beyond the Landau-level mimicry paradigm 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.