Abstract
Conformal field theories (CFTs) with MN and tetragonal global symmetry in d=2+1d=2+1 dimensions are relevant for structural, antiferromagnetic and helimagnetic phase transitions. As a result, they have been studied in great detail with the \varepsilon=4-dε=4−d expansion and other field theory methods. 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. Based on the existence of a certain large-NN expansion in theories with MN symmetry, these are argued to be the CFTs predicted by the \varepsilonε expansion. In the tetragonal case no new kinks are found, consistently with the absence of such CFTs in the \varepsilonε expansion. Estimates for critical exponents are provided for a few cases describing phase transitions in actual physical systems. In two particular MN cases, corresponding to theories with global symmetry groups O(2)^2\rtimes S_2O(2)2⋊S2 and O(2)^3\rtimes S_3O(2)3⋊S3, a second kink is found. In the O(2)^2\rtimes S_2O(2)2⋊S2 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_22 and paramagnetic-helimagnetic transitions of the rare-earth metals Ho and Dy. In the O(2)^3\rtimes S_3O(2)3⋊S3 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.
Highlights
Introduction and discussion of resultsIn recent years it has become clear that the numerical conformal bootstrap as conceived in [1]1 is an indispensable tool in our quest to understand and classify conformal field theories (CFTs)
Its power has already been showcased in the 3D Ising [3, 4, 6] and O(N ) models [5, 7, 8], and recently it has suggested the existence of a new cubic universality class in 3D, referred to as C3 or Platonic [9,10]
In this work we apply the numerical conformal bootstrap to CFTs with global symmetry groups that are semidirect products of the form K n Sn, where K is either O(m) or the dihedral group D4 of eight elements, i.e. the group of symmetries of the square. These cases have been analyzed in detail with the expansion and other field theory methods due to their importance for structural, antiferromagnetic and helimagnetic phase transitions
Summary
In recent years it has become clear that the numerical conformal bootstrap as conceived in [1]1 is an indispensable tool in our quest to understand and classify conformal field theories (CFTs). In this work we apply the numerical conformal bootstrap to CFTs with global symmetry groups that are semidirect products of the form K n Sn, where K is either O(m) or the dihedral group D4 of eight elements, i.e. the group of symmetries of the square These cases have been analyzed in detail with the expansion and other field theory methods due to their importance for structural, antiferromagnetic and helimagnetic phase transitions. A CFT with O(2) S3 symmetry is supposed to describe the antiferromagnetic phase transition of Nd [19,20,21,22,23], but the experimental result for β in [24], namely β = 0.36(2), is incompatible with our β in (2) In both the O(2) S2 and O(2) S3 cases we just discussed, we find that the stability of our theory, as measured by the scaling dimension of the next-to-leading scalar singlet, S , is not in question.
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