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

TL;DR

This paper computes thermal one point functions of higher spin currents in large N vector models at finite chemical potential, revealing simplifications at large spin and dimension, and identifying critical behavior at specific points.

Contribution

It provides explicit calculations of one point functions in critical vector models at finite chemical potential, highlighting their behavior at large spin and dimension, and analyzing critical phenomena in the Gross-Neveu model.

Findings

One point functions simplify at large spin, resembling free theory behavior.

At large dimension, one point functions are exponentially suppressed.

Critical point in 3D exhibits a branch cut with a 3/2 critical exponent.

Abstract

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.