Particles, conformal invariance and criticality in pure and disordered systems

TL;DR

This paper reviews recent advances in applying conformal invariance to particle-based field theory, offering exact solutions for critical points in pure and disordered systems, and revealing new insights into critical phenomena and disorder effects.

Contribution

It introduces a novel formalism that classifies critical points using conformal invariance within particle descriptions, including the first exact solutions for disordered systems.

Findings

Exact unitarity equations classify critical points with symmetry.

Superuniversality of some critical exponents in disordered systems.

Disorder softens first-order phase transitions.

Abstract

The two-dimensional case occupies a special position in the theory of critical phenomena due to the exact results provided by lattice solutions and, directly in the continuum, by the infinite-dimensional character of the conformal algebra. However, some sectors of the theory, and most notably criticality in systems with quenched disorder and short range interactions, have appeared out of reach of exact methods and lacked the insight coming from analytical solutions. In this article we review recent progress achieved implementing conformal invariance within the particle description of field theory. The formalism yields exact unitarity equations whose solutions classify critical points with a given symmetry. It provides new insight in the case of pure systems, as well as the first exact access to criticality in presence of short range quenched disorder. Analytical mechanisms emerge that in the random case allow the superuniversality of some critical exponents and make explicit the softening of first order transitions by disorder.