One-dimensional impenetrable anyons in thermal equilibrium: I. Anyonic generalization of Lenard's formula

We have obtained an expansion of the reduced density matrices (or, equivalently, correlation functions of the fields) of impenetrable one-dimensional anyons in terms of the reduced density matrices of fermions using the mapping between anyon and fermion wavefunctions. This is the generalization to anyonic statistics of the result obtained by Lenard (1966 J. Math. Phys. 7 1268) for bosons. In the case of impenetrable but otherwise free anyons with statistics parameter κ, the anyonic reduced density matrices in the grand canonical ensemble are expressed as Fredholm minors of the integral operator with the complex statistics-dependent coefficient γ = (1 + e^±iπκ)/π. For κ = 0 we recover the bosonic case of Lenard γ = 2/π. Due to nonconservation of parity, the anyonic field correlators 〈Ψ†_A(x′)Ψ_A(x) 〉 are different depending on the sign of x′ - x.