Fractional Chern insulators (FCIs), as a platform for the realization of fractionalization phenomena without external magnetic field, have attracted great interest. To realize FCIs, topological flat bands play a significant role. Most recently, FCIs have been extended into two-dimensional hyperbolic systems and the geometric degree of freedom of hyperbolic FCIs has been revealed. In this work, we construct hyperbolic lattices with a conical singularity and demonstrate the existence of topological flat bands. Subsequently, a bosonic $\nu =1/2$ FCI state is investigated on these hyperbolic topological flat band models with a singularity. The edge excitations for these singular hyperbolic FCIs are investigated. We also construct the optimal trial wave functions for the $\nu =1/2$ FCI state and roughly explore the interplay between the $\nu =1/2$ FCI state and geometric factors originated from the hyperbolic geometry and conical singularity.

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• Flat bands
• Fractional quantum Hall effect
• Chern insulators
• Strongly correlated systems
• Exact diagonalization
• Tight-binding model