Bootstrapping MN and Tetragonal CFTs in Three Dimensions

Conformal field theories (CFTs) with MN and tetragonal global symmetry in $d=2+1$ dimensions are relevant for structural, antiferromagnetic and helimagnetic phase transitions. As a result, they have been studied in great detail with the $\varepsilon=4-d$ 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-$N$ expansion in theories with MN symmetry, these are argued to be the CFTs predicted by the $\varepsilon$ expansion. In the tetragonal case no new kinks are found, consistently with the absence of such CFTs in the $\varepsilon$ 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 $O(2)^2\rtimes S_2$ and $O(2)^3\rtimes S_3$, a second kink is found. In the $O(2)^2\rtimes S_2$ 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$_2$ and paramagnetic-helimagnetic transitions of the rare-earth metals Ho and Dy. In the $O(2)^3\rtimes S_3$ 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.


Introduction and discussion of results
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).
Its power has already been showcased in the 3D Ising [3] and O(N ) models [4,5], and recently it has suggested the existence of a new cubic universality class in 3D, referred to as C 3 or Platonic [6,7]. Now that the method has showed its strength, it is time for it to be applied to the plethora of examples of CFTs in d = 3 suggested by the ε = 4 − d expansion [8][9][10]. This is of obvious importance, for the bootstrap gives us nonperturbative information that is useful both for comparing with experiments as well as in testing the validity of field theory methods such as the ε expansion in the ε → 1 limit.
In this work we apply the numerical conformal bootstrap to CFTs with global symmetry groups that are semidirect products of the form K n S n , where K is either O(m) or the dihedral group D 4 of eight elements, i.e. the group of symmetries of the square. These cases have been 1 See [2] for a recent review. analyzed in detail with the ε expansion and other field theory methods due to their importance for structural, antiferromagnetic and helimagnetic phase transitions. This provides ample motivation for their study with the bootstrap, with the hope of resolving some of the controversies in the literature.
One of the cases we analyze in detail in this work is that of O(2) 2 S 2 symmetry. Such theories are relevant for frustrated models with noncollinear order-see [9,Sec. 11.5], [11] and references therein. Monte Carlo simulations as well as the ε expansion and the fixed-dimension expansion have been used in the literature. Disagreements both in experimental as well as theoretical results described in [11] and [12] paint a rather disconcerting picture. In this work we observe a clear kink in a certain operator dimension bound-see Fig. 1 below. Following standard intuition, we attribute this kink to the presence of a CFT with O(2) 2 S 2 symmetry. Using existing results in the literature, namely [4], we can exclude the possibility that this kink is saturated by the CFT of two decoupled O(2) models. Obtaining the spectrum on the kink as explained in [13], we are able to provide estimates for the critical exponents β and ν that are frequently quoted in the literature. 2 We find β = 0.293(3) , ν = 0.566 (6) . (1.1) These results suggest that the ε expansion at order ε 4 perhaps underperforms [14, Table II].
Experimental results for β for XY stacked triangular antiferromagnets are slightly lower and for the helimagnets (spiral magnets) Ho and Dy higher [9, Table 37]. Also, our result for β is below the value measured in the structural phase transition of NbO 2 [15].
Another case of interest is that of CFTs with O(2) 3 S 3 symmetry. Here we again find a kink-see Fig. 2 below-and for the CFT that saturates it we obtain, with a spectrum analysis, β = 0.301(3) , ν = 0.581 (6) . (1.2) Just like in the previous paragraph, we do not find good agreement with results of the ε expansion [14, Table II]. A CFT with O(2) 3 S 3 symmetry is supposed to describe the antiferromagnetic phase transition of Nd [16], but the experimental result for β in [17] is incompatible with our β in (1.2).
In both the O(2) 2 S 2 and O(2) 3 S 3 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. More specifically, in both cases the scaling dimension of S is slightly below four, while marginality is of course at three. Therefore, the ε expansion appears to fail quite dramatically, for it predicts that S , an operator quartic in φ, has dimension slightly above three [14, Table I]. In fact, the purported closeness of the dimension of S to three according to the ε expansion has contributed to controversies in the literature regarding the nature of the stable fixed point, with 2 In terms of the dimensions of the order parameter φ and the leading scalar singlet S it is β = ∆ φ /(3 − ∆S) and arguments that it may be that of decoupled O(2) models-see section 3 below. The bootstrap shows that the fully-interacting O(2) 2 S 2 and O(2) 3 S 3 CFTs are stable.
It is not clear from our discussion so far that our bootstrap bounds are saturated by the CFT predicted by the ε expansion. This is almost certain, however, as we now explain. Our bootstrap results suggest that there is a well-defined large-m expansion in O(m) n S n theories.
This was verified by the authors of [10] for the fully interacting O(m) n S n theory of the ε expansion-see v6 of [10] on the arXiv. The important point here is that since the large-m results of the ε-expansion theory reproduce the behavior we see in our bootstrap bounds, we conclude that the kinks we observe are indeed due to the theory predicted by the ε expansion.
As we alluded to above, experimental results for phase transitions in the helimagnets Ho and Dy as well as the structural phase transition of NbO 2 differ from those in XY stacked triangular antiferromagnets and the helimagnet Tb [12]. Our conclusions contradict the suggestion of [18,12] The action of f h on N is given by conjugation, One can show that, up to isomorphisms, N, H and f uniquely determine G. The multiplication of two elements (n, h) and (n , h ) of G is given by the identity element is (e N , e H ), with e N the identity element of N and e H that of H, and the inverse of (n, h) is given by Note that a direct product is a special case of a semidirect product where f is the trivial homomorphism, i.e. the homomorphism that sends every element of H to the identity automorphism of N .
In this work we analyze CFTs with global symmetry of the form K n S n , where K n denotes the direct product of n groups K and S n the permutation group of n elements. In this case the action of the homomorphism f : S n → Aut(K n ) is to permute the K's in K n , i.e. f σ : . . , k σ(n) ), with σ an element of S n and k i , i = 1, . . . , n, an element of the The first example we analyze is that of the MN m,n CFT. By this we refer to the CFT with global into invariant subspaces in order to derive the set of crossing equations that constitutes the starting point for our numerical analysis. Invariant tensors help us in this task. As far as the OPE is concerned we have where S is the singlet, X, Y, Z are traceless-symmetric and A, B antisymmetric. The first step to doing that is to construct the invariant tensors of the group under study. This way of thinking, in terms of invariant theory, was recently applied to the ε expansion in [10], and it turns out to be very useful when thinking about the problem from the bootstrap point of view.

Invariant tensors
For the MN m,n CFT there are two four-index primitive invariant tensors [10]. They can be defined as follows: The tensor γ is fully symmetric, while the tensor ω satisfies A non-primitive invariant tensor with four indices is defined by which respects O(mn) symmetry. One can verify that (repeated indices are always assumed to be summed over their allowed values) and (2.8)

Projectors and crossing equation
With the help of (2.7) and (2.8) it can be shown that the tensors where d I r is the dimension of the representation indexed by I, with where the sum over I runs over the representations S, X, Y, Z, A, B, x ij = x i − x j , λ 2 O I are squared OPE coefficients and g ∆ I , I (u, v) are conformal blocks 4 that are functions of the conformallyinvariant cross ratios (2.13) The crossing equation can now be derived. With we find 5 The signs that appear as superscripts in the various irrep symbols indicate the spins of the operators we sum over in the corresponding term: even when positive and odd when negative.

MN anisotropy
The MN m,n fixed points were first studied in [20,21,14] and more recently in [10,22]. The relevant In the ε expansion below d = 4 (3.1) has four inequivalent fixed points. They are 1. Gaussian (λ = g = 0), 4 We define conformal blocks using the conventions of [19]. 5 In (2.15) we omit, for brevity, to label the F ∆, 's and λ 2 O 's with the appropriate index I. The appropriate labeling, however, is obvious from the overall sum in each term. 6 Compared to couplings λ, g of [10, Sec. 5.2.2] we have λ here = λ there − m+2 3(mn+2) g there and g here = g there .

O(mn)
These fixed points are known to be physically relevant for m = 2 and n = 2, 3. As already mentioned in the introduction, the MN 2,2 fixed point has applications to frustrated spin systems with noncollinear order [9,Sec. 11.5]. Additionally, it has been argued to describe the structural phase transition of NbO 2 (niobium dioxide) and paramagnetic-helimagnetic transitions in the rare-earth metals Ho (holmium), Dy (dysprosium) and Tb (terbium). The MN 2,3 fixed point is relevant for the antiferromagnetic phase transitions in K 2 IrCl 6 (potassium hexachloroiridate), TbD 2 (terbium dideuteride) and Nd (neodymium) [16].
The MN 2,2 CFT is equivalent to a theory with O(2) 2 /Z 2 symmetry [9,10,22]. An MN 2,2 fixed point independent of the one found in the ε expansion has been suggested in [23], arising after resummation of the perturbative six-loop beta functions.
The stability of the MN m,n fixed point for m = 2 and n = 2, 3 has been supported by higherloop ε expansion calculations [21,14]. However, there exist higher-loop calculations based on the fixed-dimension expansion-see [9] and references therein-indicating that the stable fixed point is actually that of n decoupled O(2) models. As mentioned in the introduction, our numerical results indicate that the MN 2,2 and MN 2,3 theories are both stable.

Tetragonal symmetry
The tetragonal CFT [9,10] has global symmetry R n = D 4 n S n , where D 4 is the eight-element dihedral group. For n = 0 R 0 = {e}, where e is the identity element, and for n = 1 R 1 = D 4 . The order of R n is ord(R n ) = 8 n n!. Note that R n is a subgroup of the hypercubic group C N = Z 2 N S N , N = 2n, whose order is ord(C N ) = 2 N N !. It is easy to see that ord(C N )/ ord(R n ) = (2n − 1)!!, which is an integer for any integer n 0.
In this work we analyze bootstrap constraints on the four-point function of the vector operator φ i . A standard construction of the character table shows that the group R 2 has eight one- 7 Although the theory of n decoupled O(m) models in item 3 on the list also has symmetry MNm,n, we will never characterize it that way; we will reserve that characterization for the fully-interacting case in 4. 8 These numbers have been obtained with the use of the freely available software GAP [24].
dimensional, six two-dimensional and six four-dimensional irreps. 9 In this case we may write 10 S is the singlet. The dimensions of the various irreps are given by the number over their symbol.
W, X, Y, Z are two-index symmetric and traceless, while A, B are two-index antisymmetric.

Invariant tensors
In the tetragonal case there are three primitive invariant tensors with four indices, defined by The tensors δ, ζ are fully symmetric, while the tensor ω is the same as that in (2.4b) for m = 2.
It can be verified that these satisfy and δ ijmn δ klmn = δ ijkl , δ ijmn ζ klmn = 1 3 ζ ijkl + 2 9 ω ijkl , δ ijmn ω klmn = ζ ijkl + 2 3 ω ijkl , ζ ijmn ζ klmn = 1 9 δ ijkl + 4 9 ζ ijkl − 4 27 ω ijkl , To verify that there are only three invariant polynomials of R n made out of the components of the vector φ i , we have computed the Molien series for n = 2, 3, 4. 11 To do this, we think of R n as represented by 2n × 2n matrices acting on the 2n-component vector φ T i . Using those matrices, which represent the group elements g i ∈ G as ρ(g i ), i = 1, . . . , ord(G), we can then explicitly 9 Character tables for a wide range of finite groups can be easily generated using GAP [24]. 10 Of course these S, X, Y, Z, A, B have nothing to do with the ones of section 2. compute the Molien series. The Molien formula is , (4.5) where 1 is the identity matrix of appropriate size. It is obvious that the summands in (4.5) only depend on the conjugacy class, so the sum can be taken to be over conjugacy classes with the appropriate weights. For n = 2, 3, 4 (4.5) gives, respectively,  Thus, we see that we have one quadratic and three quartic invariants. The latter are generated by δ ijkl , ζ ijkl and ξ ijkl , and their form is given in (4.2a,b) and (2.6) with m = 2. The unique quadratic invariant is obviously generated by δ ij and it is given by φ 2 = φ 2 1 + φ 2 2 + · · · + φ 2 2n .

Projectors and crossing equation
If we now define P S ijkl = 1 2n δ ij δ kl , (4.8a) we may verify, using (4.3) and (4.4), the projector relations P I ijmn P J mnkl = P I ijkl δ IJ , whered I r is the dimension of the representation indexed by I, with , n, n − 1, n, 2n(n − 1), n, 2n(n − 1)} . (4.10) The generalization of (4.1), valid for any n 2, is The projectors (4.8a-g) allow us to express the four-point function of interest in a conformal block decomposition in the 12 → 34 channel: where the sum over I runs over the representations S, W, X, Y, Z, A, B. For the crossing equation Let us make a comment about (4.13). We observe that we obtain the same crossing equation if we exchange the second and third line in all vectors and at the same time relabel W + ↔ Y + . This implies, for example, that operator dimension bounds on the leading scalar W operator and the leading scalar Y operator will be identical. Furthermore, if we work out the spectrum on the Wand the Y -bound, then all operators in the solution will have the same dimensions in both cases (except for the relabeling W + ↔ Y + ). The reason for this is that there exists a transformation of φ i that permutes the projectors P W and P X . 13 Indeed, if Under (4.15) we obviously have P W ↔ P Y . Let us remark here that something similar happens in the N = 2 cubic theory studied in [6,Sec. 6], again due to the transformation (4.14) that exchanges two projectors. 14 With the crossing equation (4.13) we can now commence our numerical bootstrap explorations.
Before that, however, let us first summarize results of the ε expansion for theories with tetragonal anisotropy.
14 In the N = 2 cubic case, which corresponds to n = 1 here in which case the ζ tensor does not exist, we can show that δij → δij and δ ijkl → −δ ijkl + 1 2 (δijδ kl + δ ik δ kl + δ il δ jk ). 15 Compared to couplings λ, g1, g2 of [10, Sec. 7] we have λ here = λ there − 2 3(n+1) g there , g here 16 Fixed points physically-equivalent to those in items 2 and 5 on the list are also found in other positions in coupling space, related to the ones given in the list by the field redefinition in (4.14) [9, 10].

Numerical results
The numerical results in this paper have been obtained with the use of PyCFTBoot [19] and SDPB [26]. We use nmax = 9, mmax = 6, kmax = 36 in PyCFTBoot and we include spins up to  [10] realized that the large-m expansion was easy to obtain in the ε expansion and they updated the arXiv 17 The theory of 2n decoupled Ising models in item 2 on the list has symmetry C2n as well. However, we reserve the C2n characterization for the theory in 5. 18 The theory of n decoupled O(2) models in item 3 on the list has symmetry MN2,n as well. However, we reserve the MN2,n characterization for the theory in item 6.
version of [10] to include the relevant formulas. The anomalous dimension of X is equal to ε at leading order in 1/m, and so ∆ ε  Continuing our investigation of the MN 2,2 theory for larger ∆ φ we obtain Fig. 3. There we observe the presence of a second kink. Although not as convincing as the kink for smaller ∆ φ in the same theory, it is tempting to associate this kink with the presence of an actual CFT. This is further supported by the results from our spectrum analysis which give us the critical exponents (1.3) that match experimental results very well as mentioned in the introduction.
Let us mention here that the spectrum analysis consists of obtaining the functional α right at the boundary of the allowed region (on the disallowed side) and looking at its action on the vectors V ∆, of F ± ∆, that appear in the crossing equation all sectors λ 2 O V ∆, = − V 0,0 , where V 0,0 is the vector associated with the identity operator. Zeroes of α · V ∆, appear for (∆, )'s of operators in the spectrum of the CFT that saturates the kink and provide a solution to the crossing equation.
More details for this procedure can be found in [13] and [6,Sec. 3.2]. For the determination of critical exponents we simply find the dimension that corresponds to the first zero of α · V ∆ S ,0 .
For the MN 2,3 theory we also find a second kink-see Another physical quantity one can study in a CFT is the central charge C T , i.e. the coefficient in the two-point function of the stress-energy tensor: where S d = 2π

Tetragonal
The bound on the leading scalar in the singlet sector in the R n theory is identical, for the cases checked, to the bound obtained for the leading scalar singlet in the O(2n) model. The bound on the leading scalar in the X sector is identical, again for the cases checked, to the bound on the leading scalar in the X sector of the MN 2,n theory. Both these symmetry enhancements are allowed, and they show that if a tetragonal CFT exists, then its leading scalar singlet operator has dimension in the allowed region of the bound of the leading scalar singlet in the O(2n) model.
A similar comment applies to the leading scalar X operator and the bound on the leading scalar X operator of the MN 2,n theory.
Let us focus on the bound of the leading scalar in the W sector, shown in Fig. 7. It turns out that the W -bound is the same for all n checked, even for n very large. It is also identical 0  Fig. 7: Upper bound on the dimension of the first scalar W operator in the φ i × φ j OPE as a function of the dimension of φ. The area above the curve is excluded. This bound applies to all R n theories checked.
to the V -bound in [6,Fig. 14]. The coincidence of the W bound with that of [6,Fig. 14] is ultimately due to the fact that the N = 2 "cubic" theory has global symmetry D 4 . Indeed, taking n decoupled copies of the D 4 theory leads to a theory with symmetry R n . The leading scalar operator V , whose dimension is bounded in the D 4 theory in [6, Fig. 14], gives rise to a direct-sum representation that is reducible under the action of R n . That representation splits into two irreps of R n , namely our W and X, and it is easy to see that, if the R n theory is decoupled, the leading scalar operator in the irrep W must have the same dimension as V of [6,Fig. 14]. Hence, the corresponding bounds have a chance to coincide and indeed they do. If a fully interacting R n theory exists, then the dimension of the leading scalar W operator of that theory is in the allowed region of Fig. 7. We point out here that the putative theory that lives on the bound of [6,Fig. 14] is not predicted by the ε expansion. That theory is currently under investigation with a mixed-correlator bootstrap [28].
To see if a fully-interacting R n theory exists, we have obtained bounds for the leading scalar and spin-one operators in other sectors. Unfortunately, our (limited) investigation has not uncovered any features that could signify the presence of hitherto unknown CFTs with R n global symmetry.

Conclusion
In this paper we have obtained numerical bootstrap bounds for three-dimensional CFTs with global symmetry O(m) n S n and D 4 n S n , where D 4 is the dihedral group of eight elements.
The O(m) n S n case displays the most interesting bounds. We have found clear kinks that appear to correspond to the theories predicted by the ε expansion and have observed that the ε expansion appears to be unsuccessful in predicting the critical exponents and other observables with satisfactory accuracy in the ε → 1 limit.
Experiments in systems that are supposed to be described by CFTs with O(2) 2 S 2 symmetry have yielded two sets of critical exponents [12]. Having found two kinks in a certain bound for such CFTs, we conclude that there are two distinct universality classes with O(2) 2 S 2 global symmetry. Our critical-exponent computations in these two different theories, given in (1.1)  For theories with O(2) 3 S 3 symmetry we also find two kinks. The corresponding critical exponents are given in (1.2) and (1.4). The CFT that lives on the second kink, with critical exponents (1.4), is the one with which we can reproduce experimental results. This is not the CFT predicted by the ε expansion. A more complete study of the second set of kinks that appear in our bounds would be of interest. Note that the kinks we find do not occur in dimension bounds for singlet scalar operators, so we consider it unlikely (although we cannot exclude it) that the second kinks correspond to a theory with a different global symmetry group as has been observed in a few other cases [29].  [30,31], where evidence for a CFT not seen in the ε expansion was presented. Such CFTs have been suggested to be absent in perturbation theory but arise after resummations of perturbative beta functions.
Examples have been discussed in O(2) × O(N ) frustrated spin systems [23,32,33]. These examples have been criticized in [34]. However, the results of [31] for the O(2) × O(3) case are in good agreement with those of [23,33], lending further support to the suggestion that new fixed points actually exist. Our results (1.3) are also in good agreement with the corresponding determinations of critical exponents in [23,33].
The study of more examples with numerical conformal bootstrap techniques is necessary in order to examine the conditions under which perturbative field theory methods may fail to predict the presence of CFTs or in calculating the critical exponents and other observables with accuracy.
Examples of critical points examined with the ε expansion in [9,10,22,35] constitute a large unexplored set. The generation of crossing equations for a wide range of finite global symmetry groups was recently automated [36]. This provides a significant reduction of the amount work required for one to embark on new and exciting numerical bootstrap explorations.