Bootstrapping hypercubic and hypertetrahedral theories in three dimensions

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 $\epsilon =4-d$ expansion is valid when $\epsilon \to 1$. 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 $\epsilon$ 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 $\epsilon$ expansion exists in $d=3$ as well, and a rough estimate of its extent is given.

24 pages, 18 figures. v2: A reference added. v3: A couple of changes. v4: Expanded discussion on structural phase transitions, added some epsilon expansion results. Version to appear in JHEP

References in corpus (5)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Bootstrapping Mixed Correlators in the 3D Ising Model
• Bounds on 4D Conformal and Superconformal Field Theories
• Structural Phase Transitions

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• Anomalous Dimensions in Hypercubic Theories
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