Correction exponents in the chiral Heisenberg model at $1/N^2$: singular contributions and operator mixing

Abstract

We calculate the correction exponents in the chiral Heisenberg model in the $1/N$ expansion. These exponents are related to the slopes of $β$ functions at the phase transition point. We present the results at order $1/N^2$ and check that they agree with the results of the $ε$ expansion near $d = 4$. We find that one of the correction exponents diverges as $d \to 3$. We argue that the appearance of the pole is a rather general phenomenon and is associated with operator mixing involving the system of four-fermion operators. After analyzing the operator mixing structure, we propose a resummation procedure which modifies the exponents already at leading order. We also perform calculations directly in the three-dimensional model and find complete agreement with the resummed exponents.

Figures (8)

• Figure 1: Leading order diagrams.
• Figure 2: An example of a diagram contributing to the $d-3$ pole in the anomalous dimension $\gamma_{22}$.
• Figure 3: Comparison of the resummed and naive scaling dimensions in $1/n$ approximation. The blue and red lines correspond to the solution of \ref{['char-eq']}, while the black line represents the "naive" scaling dimension \ref{['gammaminus1']}. The graph is made for $N = 2$ and $\operatorname{tr}\mathds{1}$ is approximated as $2$ for $2 < d < 3$ and $2d - 4$ for $3< d < 4$.
• Figure 4: The simple box (SB) diagrams.
• Figure 5: Double box diagrams $DB_1$ (top left) - $DB_4$ (bottom right).