Noninteracting tight-binding models for Fock parafermions

We model $p$-state Fock parafermions on a lattice in one dimension (with occupation per orbital of $0,1 , \ldots ,p-1$). For $p$ a composite number, they may be mapped to $q_m$-state parafermions where $q_m$ are the prime factors of $p$. For a Hamiltonian with a single-particle spectrum, the parafermions decompose into $q_m$-state parafermions. When $p$ is a power of two, the decomposition is into fermions. We use this to construct a parafermionic Hamiltonian for $p=4$ with a single-particle spectrum using a fermionic tight-binding model which is bilinear in creation and annihilation operators. The single-particle levels may be determined by diagonalizing a square matrix whose order scales linearly with system size, and they are the same as those of the fermionic model. We show that the intermediate statistics of the thermodynamic distribution function for the occupation numbers (known as Gentile statistics) are consistent with the mapping to fermions, and we provide an example calculation of the internal energy and heat capacity for a simple linear chain.