Critical lines in the pure and disordered $O(N)$ model

TL;DR

This paper uses scale invariant scattering theory to map out the critical lines of the two-dimensional $O(N)$ model with quenched disorder, revealing a unified pattern of fixed points and superuniversality phenomena.

Contribution

It provides an exact determination of the RG fixed points in the disordered $O(N)$ model, connecting pure and disordered critical lines within a unified framework.

Findings

Critical lines form a connected pattern in parameter space.

Disorder introduces additional parameters affecting critical behavior.

Superuniversality of some critical exponents emerges in the disordered case.

Abstract

We consider replicated $O(N)$ symmetry in two dimensions within the exact framework of scale invariant scattering theory and determine the lines of renormalization group fixed points in the limit of zero replicas corresponding to quenched disorder. A global pattern emerges in which the different critical lines are located within the same parameter space. Within the subspace corresponding to the pure case (no disorder) we show how the critical lines for non-intersecting loops ($-2\leq N\leq 2$) are connected to the zero temperature critical line ($N>2$) via the BKT line at $N=2$. Disorder introduces two more parameters, one of which vanishes in the pure limit and is maximal for the solutions corresponding to Nishimori-like and zero temperature critical lines. Emergent superuniversality (i.e. $N$-independence) of some critical exponents in the disordered case and disorder driven renormalization group flows are also discussed.