Disorder in Large-N Theories

TL;DR

This paper investigates the effects of quenched disorder on large-N conformal field theories, revealing new fixed points and non-scale-invariant behaviors, with exact calculations and holographic dual descriptions.

Contribution

It provides exact beta functions and correlation functions for disordered large-N theories, extending Imry and Ma's results and exploring holographic mappings of disorder.

Findings

Disordered free fields exist only for dimensions d>4.

Disordered fixed points can be non-scale-invariant.

Holography reproduces field theory results and incorporates 1/N corrections.

Abstract

We consider Euclidean Conformal Field Theories perturbed by quenched disorder, namely by random fluctuations in their couplings. Such theories are relevant for second-order phase transitions in the presence of impurities or other forms of disorder. Theories with quenched disorder often flow to new fixed points of the renormalization group. We begin with disorder in free field theories. Imry and Ma showed that disordered free fields can only exist for d>4. For d>4 we show that disorder leads to new fixed points which are not scale-invariant. We then move on to large-N theories (vector models or gauge theories in the `t Hooft limit). We compute exactly the beta function for the disorder, and the correlation functions of the disordered theory. We generalize the results of Imry and Ma by showing that such disordered theories exist only when disorder couples to operators of dimension \Delta > d/4. Sometimes the disordered fixed points are not scale-invariant, and in other cases they have unconventional dependence on the disorder, including non-trivial effects due to irrelevant operators. Holography maps disorder in conformal theories to stochastic differential equations in a higher-dimensional space. We use this dictionary to reproduce our field theory results. We also study the leading 1/N corrections, both by field theory methods and by holography. These corrections are particularly important when disorder scales with the number of degrees of freedom.