This paper presents an exact solution of a two-dimensional quantum spin model on a honeycomb lattice, where spins interact via XX, YY, or ZZ couplings depending on the link direction. The model is solved exactly by reducing it to free fermions in a static Z₂ gauge field. The phase diagram in parameter space is analyzed, revealing two phases: one with an energy gap and Abelian anyons, and another gapless phase that becomes gapped in a magnetic field, hosting non-Abelian anyons with Ising model braiding rules. The anyonic properties are characterized by a spectral Chern number ν, with Abelian and non-Abelian phases corresponding to ν = 0 and ν = ±1, respectively. The paper also provides a general theory of free fermions with a gapped spectrum, including mathematical background on anyons and Chern numbers. The model is shown to represent a universality class of topological order, similar to the resonating valence bond (RVB) state. The paper discusses the implications of topological order, including its role in quantum computing through topological quantum computation (TQC), where braiding of anyons enables universal quantum computation. The study highlights the importance of topological phases in condensed matter physics and their potential applications in quantum information science.This paper presents an exact solution of a two-dimensional quantum spin model on a honeycomb lattice, where spins interact via XX, YY, or ZZ couplings depending on the link direction. The model is solved exactly by reducing it to free fermions in a static Z₂ gauge field. The phase diagram in parameter space is analyzed, revealing two phases: one with an energy gap and Abelian anyons, and another gapless phase that becomes gapped in a magnetic field, hosting non-Abelian anyons with Ising model braiding rules. The anyonic properties are characterized by a spectral Chern number ν, with Abelian and non-Abelian phases corresponding to ν = 0 and ν = ±1, respectively. The paper also provides a general theory of free fermions with a gapped spectrum, including mathematical background on anyons and Chern numbers. The model is shown to represent a universality class of topological order, similar to the resonating valence bond (RVB) state. The paper discusses the implications of topological order, including its role in quantum computing through topological quantum computation (TQC), where braiding of anyons enables universal quantum computation. The study highlights the importance of topological phases in condensed matter physics and their potential applications in quantum information science.