# Bootstrapping MN and Tetragonal CFTs in Three Dimensions

## TL;DR

This paper applies the conformal bootstrap to three-dimensional CFTs with MN and tetragonal symmetries, identifying potential new universality classes and providing critical exponent estimates relevant for material phase transitions.

## Contribution

It initiates the nonperturbative bootstrap study of these symmetries, discovering sharp kinks indicating possible new CFTs and proposing their relevance to specific material phase transitions.

## Key findings

- Sharp kinks in MN bounds suggest existence of new CFTs.
- No new kinks found in tetragonal case.
- Identification of potential new universality classes for certain phase transitions.

## Abstract

Conformal field theories (CFTs) with MN and tetragonal global symmetry in $d=2+1$ dimensions are relevant for structural, antiferromagnetic and helimagnetic phase transitions in a wide class of materials. The study of these theories with the nonperturbative numerical conformal bootstrap is initiated in this work. Bounds for operator dimensions are obtained and they are found to possess sharp kinks in the MN case, suggesting the existence of full-fledged CFTs. In the tetragonal case, no new kinks are found. Estimates for critical exponents are provided for a few cases describing phase transitions in various materials. In two particular MN cases, corresponding to theories with global symmetry groups $O(2)^2\rtimes S_2$ and $O(2)^3\rtimes S_3$, a second kink is found. In the $O(2)^2\rtimes S_2$ case it is argued to be saturated by a CFT that belongs to a new universality class relevant for the structural phase transition of NbO$_2$ and paramagnetic-helimagnetic transitions of the rare-earth metals Ho and Dy. In the $O(2)^3\rtimes S_3$ case it is suggested that the CFT that saturates the second kink belongs to a new universality class relevant for the paramagnetic-antiferromagnetic phase transition of the rare-earth metal Nd.