# Fractional Chern Insulators in Singular Geometries

## TL;DR

This paper explores how fractional Chern insulator states behave on 2D singular lattices with various symmetries, revealing geometry-dependent properties and exotic edge excitation degeneracies.

## Contribution

It introduces a generalized framework for FCI/FQAH states on singular lattices with arbitrary rotational symmetry, using trial wave functions and effective projection methods.

## Key findings

- High wave-function overlaps confirm the geometric dependence of FCI/FQAH states.
- Discovery of exotic degeneracy sequences of edge excitations.
- Explanation of mixed edge excitation branches in singular geometries.

## Abstract

The fractional quantum anomalous Hall (FQAH) states or fractional Chern insulator (FCI) states have been studied on two-dimensional (2D) flat lattices with different boundary conditions. Here, we propose the geometry-dependent FCI/FQAH states that interacting particles are bounded on 2D singular lattices with arbitrary $n$-fold rotational symmetry. Based on the generalized Pauli principle, we construct trial wave functions for the singular-lattice FCI/FQAH states with the aid of an effective projection approach, and compare them with the exact diagonalization results. High wave-function overlaps show that the singular-lattice FCI/FQAH states are certainly related to the geometric factor $\beta$. More interestingly, we observe some exotic degeneracy sequences of edge excitations in these singular-lattice FCI/FQAH states, and provide an explanation that two branches of edge excitations mix together.