Application of non-commutative algebra to a soluble fermionic model

We explore the properties of the non-commutative Grassmann algebra to obtain the high-temperature expansion of the grand canonical partition function for self-interacting fermionic systems. As an application, we consider the anharmonic fermionic oscillator, the simplest model in Quantum Mechanics with self-interacting fermions that is exactly soluble. The knowledge of the exact expression for its grand canonical partition function enables us to check the β-expansion obtained using our Grassmann-algebra-based technique.