The Role of Quantum Geometry in Stabilizing Fractional Chern Insulators: Coupled-Wires Approach

In the presence of strong electronic interactions, a partially filled Chern band may stabilize a fractional Chern insulator (FCI) state, the zero-field analogue of the fractional quantum Hall phase. While the FCI phase has been long hypothesized, feasible solid-state realizations only recently emerged with the rise of moir'e materials. In these systems, the quantum geometry of the electronic bands plays a critical role in stabilizing the FCI over competing correlated phases. Whereas in the limit of ``ideal'' quantum geometry this role is well understood, the role of quantum geometry far from ideality is supported only by empiric numerical evidence, without clear analytical understanding. We analyze an anisotropic model of a $left|C ight|=1$ Chern insulator. Upon partial filling of one of its bands, an FCI phase may be stabilized over a certain parameter regime. We incorporate strong electronic interaction analytically by employing the coupled-wires approach, and analyze the FCI stability, and its relation to a "chiralness" parameter which controls the quantum metric. We identify a competing anti-FCI phase benefiting from non-ideal metric, which generically does not become favorable over the FCI, however its presence hinders the FCI stabilization in favor of metallic or CDW phases. We thus establish an analytical connection between quantum geometry and FCI stability far from ideal conditions.