Exact results for the $O(N)$ model with quenched disorder

TL;DR

This paper uses scale invariant scattering theory to exactly identify the renormalization group fixed points for two-dimensional $O(N)$ models with quenched disorder, revealing a rich structure of critical lines and phases.

Contribution

It provides the first exact determination of the fixed point lines for disordered $O(N)$ models in two dimensions, including the characterization of disorder parameters and phase classes.

Findings

Critical lines depend on disorder modulus and phase angle.

Existence of a line of infrared fixed points for $N$ from $ ext{sqrt}(2)-1$ to 1.

Superuniversal energy density operator along a specific fixed line.

Abstract

We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points for $O(N)$-symmetric models with quenched disorder in two dimensions. Random fixed points are characterized by two disorder parameters: a modulus that vanishes when approaching the pure case, and a phase angle. The critical lines fall into three classes depending on the values of the disorder modulus. Besides the class corresponding to the pure case, a second class has maximal value of the disorder modulus and includes Nishimori-like multicritical points as well as zero temperature fixed points. The third class contains critical lines that interpolate, as $N$ varies, between the first two classes. For positive $N$, it contains a single line of infrared fixed points spanning the values of $N$ from $\sqrt{2}-1$ to $1$. The symmetry sector of the energy density operator is superuniversal (i.e. $N$-independent) along this line. For $N=2$ a line of fixed points exists only in the pure case, but accounts also for the Berezinskii-Kosterlitz-Thouless phase observed in presence of disorder.