Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 81 / 4 / b / Solution

At $U=0$, use fermionic Matsubara frequencies ${\omega }_r=(2r+1)\pi /\beta$. The fermionic Gaussian integral has inverse propagator $-i{\omega }_r+{\xi }_{\mathbf{k}}$. With its thermal normalization, the determinant for one single-particle level is $1+e^{-\beta \xi }$, as also follows from its empty and occupied states. Two spin species therefore giveThe expression labeled Helmholtz free energy in the PDF is the grand potential, since the trace contains $H-\mu \hat{N}$. The thermodynamic Helmholtz free energy at a prescribed electron number $N_e$ is $F={\Omega }_0+\mu N_e$, with $\mu$ fixed by $N_e=2{\sum }_{\mathbf{k}}{n}_F({ϵ}_{\mathbf{k}})$. The Matsubara determinant requires a normalization prescription; a raw infinite sum of logarithms is not an ordinary convergent sum.