Newton's Identity in Finite-Bead Fermionic Partition Function

Abstract

For non-interacting fermions in a harmonic trap, the partition function at any discrete number of imaginary time slices (or beads) and for any choice of short-time propagator admits an exact recursion relation derived directly from the contracted determinant form of the path integral. This finite-bead recursion is distinct from earlier continuum-limit recursions, which do not apply to the discrete time partition functions. By identifying a direct correspondence between this recursion and Newton's identity, application of a closed-form result from the theory of partitions provides an exact expression for the one-dimensional $n$-fermion finite-bead partition function. From this, the Thermodynamic and Hamiltonian energies and specific heats are analytically calculated for any $n$, $N$, $τ$, and propagator choice.