Platonic field theories

Abstract

We study renormalization group (RG) fixed points of scalar field theories endowed with the discrete symmetry groups of regular polytopes. We employ the functional perturbative renormalization group (FPRG) approach and the-expansion in d = d c −. The upper critical dimensions relevant to our analysis are d c = 6, 4, 10 /3, 3, 14 /5, 8 /3, 5 /2, 12 /5; in order to get access to the corresponding RG beta functions, we derive general multicomponent beta functionals β V and β Z in the aforementioned upper critical dimensions, most of which are novel. The field theories we analyze have N = 2 (polygons), N = 3 (Platonic solids) and N = 4 (hyper-Platonic solids) field components. The main results of this analysis include a new candidate universality class in three physical dimensions based on the symmetry group D 5 of the Pentagon. Moreover we find new Icosahedron fixed points in d < 3, the fixed points of the 24-Cell, multi-critical O(N) and φ n-Cubic universality classes.

Bootstrapping MN and tetragonal CFTs in three dimensions

“…Examples of critical points examined with the ε expansion in [10,11,23,36] constitute a large unexplored set. The generation of crossing equations for a wide range of finite global symmetry groups was recently automated [37].…”

Multi-Critical Multi-Field Models: A CFT Approach to the Leading Order

“…Similarly to the single field case one can show that the criticality condition as well as the secular equations for the composite operators given above can be obtained from the functional perturbative RG equation at LO for the potential [25,26] Here we briefly discuss a simple application of the results of this section to the critical O(2) Heisenberg model. Like in the example of Section 2.2 we specialize to n = 2 and therefore to a quartic interaction.…”

“…The results are the same as those obtained with perturbative RG methods, which if treated at functional level give rise to a very compact and effective computational framework. Actually for certain models (unitary multi-critical) one can easily reconstruct the functional perturbative RG (FPRG) equations [21][22][23][24][25] starting from the obtained CFT relations. Before discussing which relations are implied by assuming the CFTs and the lagrangian description in the general multi-field case, let us illustrate how to deal with the simpler theories with a single scalar field [13].…”

RG and logarithmic CFT multicritical properties of randomly diluted Ising models

“…First, we have the tetracritical pure Ising fixed point with coordinates {ρ 1 = 0 , ρ 2 = 0 , ρ 3 = 0 , ρ 4 = 0 , ρ 5 = 3 70 }, and LO anomalous dimension given by η = 9 85750 2 , which can be checked against Ref. [36]. 2 , corresponding to the tetracritical SAW universality.…”

“…The functional form of the RG flow in d = 8 3 − can be obtained by promoting the singlefield case of Ref. [38] to arbitrary number of fields, which can be done unambiguously at LO and NLO without further computations [36]. In Sect.…”