Thermal one-point functions: CFT's with fermions, large $d$ and large spin

TL;DR

This paper develops a formalism using the OPE inversion formula to compute thermal one-point functions of fermions and higher spin currents in conformal field theories, verifying results in mean field and Gross-Neveu models across various dimensions.

Contribution

It introduces a novel application of the OPE inversion formula to thermal correlators of fermions, providing new insights into higher spin currents and their expectation values in large N models.

Findings

Inversion formula reproduces the spectrum and thermal one-point functions in mean field theory.

Stress tensor from inversion formula matches partition function calculations at criticality.

Ratios of higher spin current expectation values are less than one and depend on spin and dimension.

Abstract

We apply the OPE inversion formula on thermal two-point functions of fermions to obtain thermal one-point function of fermion bi-linears appearing in the corresponding OPE. We primarily focus on the OPE channel which contains the stress tensor of the theory. We apply our formalism to the mean field theory of fermions and verify that the inversion formula reproduces the spectrum as well as their corresponding thermal one-point functions. We then examine the large $N$ critical Gross-Neveu model in $d=2k+1$ dimensions with $k$ even and at finite temperature. We show that stress tensor evaluated from the inversion formula agrees with that evaluated from the partition function at the critical point. We demonstrate the expectation values of 3 different classes of higher spin currents are all related to each other by numerical constants, spin and the thermal mass. We evaluate the ratio of the thermal expectation values of higher spin currents at the critical point to the Gaussian fixed point or the Stefan-Boltzmann result, both for the large $N$ critical $O(N)$ model and the Gross-Neveu model in odd dimensions. This ratio is always less than one and it approaches unity on increasing the spin with the dimension $d$ held fixed. The ratio however approaches zero when the dimension $d$ is increased with the spin held fixed.