Abstract
Conformal field theories (CFTs) with cubic global symmetry in 3D are relevant in a variety of condensed matter systems and have been studied extensively with the use of perturbative methods like the \varepsilonε expansion. In an earlier work, we used the nonperturbative numerical conformal bootstrap to provide evidence for the existence of a previously unknown 3D CFT with cubic symmetry, dubbed “Platonic CFT”. In this work, we make further use of the numerical conformal bootstrap to perform a three-dimensional scan in the space of scaling dimensions of three low-lying operators. We find a three-dimensional isolated allowed region in parameter space, which includes both the 3D (decoupled) Ising model and the Platonic CFT. The essential assumptions on the spectrum of operators used to provide the isolated allowed region include the existence of a stress-energy tensor and the irrelevance of certain operators (in the renormalization group sense).
Highlights
Cubic scalar theories possess global symmetry described by the 48-element group C3 = S3S4 × 2 ⊂ O(3), where Sn is the permutation group of n objects
We study unitarity and crossing symmetry constraints on a system of correlators which consists of 〈φφφφ〉, 〈φφSS〉 and 〈SSSS〉, where S is the scalar singlet with the lowest scaling dimension that appears in the operator product expansion (OPE) of φi with itself
In order to work out the required tensor structures of the global symmetry for the four-point function, it is rather convenient to work with the cubic group as a subgroup of O(3)
Summary
The core constraints of any numerical bootstrap study are crossing symmetry, i.e. the statement of associativity of the operator product expansion (OPE), and unitarity.. The core constraints of any numerical bootstrap study are crossing symmetry, i.e. the statement of associativity of the operator product expansion (OPE), and unitarity.3 Imposing these constraints, one may find bounds on allowed values of scaling dimensions of operators as well as coefficients in the OPE. We study unitarity and crossing symmetry constraints on a system of correlators which consists of 〈φφφφ〉, 〈φφSS〉 and 〈SSSS〉, where S is the scalar singlet with the lowest scaling dimension (besides the obligatory unit operator) that appears in the OPE of φi with itself.
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