Bootstrapping hypercubic and hypertetrahedral theories in three dimensions

TL;DR

This paper uses the non-perturbative numerical conformal bootstrap to study hypercubic and hypertetrahedral theories in three dimensions, revealing new insights into their phase structure and potential physical applications.

Contribution

It provides the first non-perturbative bootstrap analysis of hypercubic and hypertetrahedral theories in three dimensions, identifying key features like kinks and the non-conformal window.

Findings

Bound with a kink in the cubic case suggests non-perturbative solutions.

Evidence of the non-conformal window in hypertetrahedral theories in 3D.

Implications for cubic magnets and structural phase transitions.

Abstract

There are three generalizations of the Platonic solids that exist in all dimensions, namely the hypertetrahedron, the hypercube, and the hyperoctahedron, with the latter two being dual. Conformal field theories with the associated symmetry groups as global symmetries can be argued to exist in $d=3$ spacetime dimensions if the $\varepsilon=4-d$ expansion is valid when $\varepsilon\to1$. In this paper hypercubic and hypertetrahedral theories are studied with the non-perturbative numerical conformal bootstrap. In the $N=3$ cubic case it is found that a bound with a kink is saturated by a solution with properties that cannot be reconciled with the $\varepsilon$ expansion of the cubic theory. Possible implications for cubic magnets and structural phase transitions are discussed. For the hypertetrahedral theory evidence is found that the non-conformal window that is seen with the $\varepsilon$ expansion exists in $d=3$ as well, and a rough estimate of its extent is given.