High-gradient operators in perturbed Wess–Zumino–Witten field theories in two dimensions

Abstract

Many classes of non-linear sigma models (NL σMs) are known to contain composite operators with an arbitrary number 2 s of derivatives (“high-gradient operators”) which appear to become strongly relevant within renormalization group (RG) calculations at one (or fixed higher) loop order, when the number 2 s of derivatives becomes large. This occurs at many conventional fixed points of NL σMs which are perturbatively accessible within the usual ϵ-expansion in d = 2 + ϵ dimensions. Since such operators are not prohibited from occurring in the action, they appear to threaten the very existence of such fixed points. At the same time, for NL σMs describing metal-insulator transitions of Anderson localization in electronic conductors, the strong RG-relevance of these operators has been previously related to statistical properties of the conductance of samples of large finite size (“conductance fluctuations”). In this paper, we analyze this question, not for perturbative RG treatments of NL σMs, but for two-dimensional Wess–Zumino–Witten (WZW) models at level k, perturbatively in the current–current interaction of the Noether current (“non-Abelian Thirring/Gross–Neveu models”). WZW models are special (“Principal Chiral”) NL σMs on a Lie Group G with a WZW term at level k. In these models the role of high-gradient operators is played by homogeneous polynomials of order 2 s in the Noether currents, whose scaling dimensions we analyze. For the Lie Supergroup G = GL ( 2 N | 2 N ) and k = 1 , this corresponds to time-reversal invariant problems of Anderson localization in the so-called chiral symmetry classes, and the strength of the current–current interaction, a measure of the strength of disorder, is known to be completely marginal (for any k). We find that all high-gradient (polynomial) operators are, to one loop order, irrelevant or relevant depending on the sign of that interaction.