One point functions in large $N$ vector models at finite chemical potential

We evaluate the thermal one point function of higher spin currents in the critical model of $U(N)$ complex scalars interacting with a quartic potential and the $U(N)$ Gross-Neveu model of Dirac fermions at large $N$ and strong coupling using the Euclidean inversion formula. These models are considered in odd space time dimensions $d$ and held at finite temperature and finite real chemical potential $\mu$ measured in units of the temperature. We show that these one point functions simplify both at large spin and large $d$. At large spin, the one point functions behave as though the theory is free, the chemical potential appears through a simple pre-factor which is either $\cosh\mu$ or $\sinh\mu$ depending on whether the spin is even or odd. At large $d$, but at finite spin and chemical potential, the 1-point functions are suppressed exponentially in $d$ compared to the free theory. We study a fixed point of the critical Gross-Neveu model in $d=3$ with 1-point functions exhibiting a branch cut in the chemical potential plane. The critical exponent for the free energy or the pressure at the branch point is $3/2$ which coincides with the mean field exponent of the Lee-Yang edge singularity for repulsive core interactions.