Four-Fermion Theory and the Conformal Bootstrap

TL;DR

This paper uses the conformal bootstrap approach to analyze the critical behavior of four-Fermion theories at strong coupling fixed points, providing non-perturbative insights into their conformal properties.

Contribution

It demonstrates the application of the conformal bootstrap to four-Fermion theories, deriving equations for critical exponents and confirming results with large N expansions.

Findings

Confirmed the existence of nontrivial fixed points in four-Fermion theories.

Derived equations for scaling dimensions and critical couplings.

Matched bootstrap results with large N expansions in 3D.

Abstract

We employ the conformal bootstrap to re-examine the problem of finding the critical behavior of four-Fermion theory at its strong coupling fixed point. Existence of a solution of the bootstrap equations indicates self-consistency of the assumption that, in space-time dimensions less than four, the renormalization group flow of the coupling constant of a four-Fermion interaction has a nontrivial fixed point which is generally out of the perturbative regime. We exploit the hypothesis of conformal invariance at this fixed point to reduce the set of the Schwinger-Dyson bootstrap equations for four-Fermion theory to three equations which determine the scale dimension of the Fermion field $\psi$, the scale dimension of the composite field $\bar{\psi}\psi$ and the critical value of the Yukawa coupling constant. We solve the equations assuming this critical value to be small. We show that this solution recovers the fixed point for the four-fermion interaction with $N$-component fermions in the limit of large $N$ at (Euclidean) dimensions $d$ between two and four. We perform a detailed analysis of the $1/N$-expansion in $d=3$ and demonstrate full agreement with the conformal bootstrap. We argue that this is a useful starting point for more sophisticated computations of the critical indices.