Perturbative and Nonperturbative Studies of CFTs with MN Global Symmetry
LLA-UR-21-20310
Perturbative and Nonperturbative Studiesof CFTs with MN Global Symmetry
Johan Henriksson a,b,c and Andreas Stergiou da Dipartimento di Fisica E. Fermi, Universit`a di Pisa, and
INFN, Sezione di Pisa,Largo Bruno Pontecorvo 3, 56127 Pisa, Italy b Lincoln College, University of Oxford, Turl Street, Oxford, OX1 3DR, UK c Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK d Theoretical Division, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Fixed points in three dimensions described by conformal field theories with MN m,n = O ( m ) n (cid:111) S n global symmetry have extensive applications in critical phenomena. Associated experimentaldata for m = n = 2 suggest the existence of two non-trivial fixed points, while the ε expansionpredicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrapstudy has found two kinks for small values of the parameters m and n , with critical exponentsin good agreement with experimental determinations in the m = n = 2 case. In this paper weinvestigate the fate of the corresponding fixed points as we vary the parameters m and n . Wefind that one family of kinks approaches a perturbative limit as m increases, and using largespin perturbation theory we construct a large m expansion that fits well with the numerical data.This new expansion, akin to the large N expansion of critical O ( N ) models, is compatible withthe fixed point found in the ε expansion. For the other family of kinks, we find that it persistsonly for n = 2, where for large m it approaches a non-perturbative limit with ∆ φ ≈ .
75. Weinvestigate the spectrum in the case MN , and find consistency with expectations from thelightcone bootstrap.January 2021 a r X i v : . [ h e p - t h ] J a n ontents1. Introduction 22. Review 5 MN m,n symmetry . . . . . . . . 52.2. Results from previous studies in the ε expansion . . . . . . . . . . . . . 7
3. The perturbative fixed point at large m ε expansion . . . . . . . . . . . . . . . . . . . . 12
4. The non-perturbative fixed point at large m m theory . . . . . . . . . . . . . 16
5. Discussion 21Appendix A. Large m results for general d . Introduction Second-order phase transitions display scale-invariant physics and are widely believed to bedescribed by conformal field theories (CFTs), which arise at fixed points of the renormalizationgroup (RG) flow. Due to universality, the physics at these phase transitions is independent ofthe underlying microscopic degrees of freedom, which means that the same CFT may describea variety of systems. Many important applications of three-dimensional CFTs arise for non-zerotemperature phase transitions, for instance critical liquid-vapor transitions, transitions betweenmagnetic phases and structural phase transitions.The observables of a conformal field theory, such as the critical exponents, can be extractedfrom the CFT data, which are the scaling dimensions and structure constants (OPE coefficients)of the local operators in the theory. One principal goal in the theory of critical phenomena istherefore to make precise determinations of the CFT data. A useful tool in studying CFTs relevantfor three-dimensional systems is the Landau–Ginzburg–Wilson description, in which one writesdown a quantum field theory with quartic interactions preserving a given global symmetry group.By tuning mass parameters (equivalent to tuning the temperature in experiments), the field theoryflows under the RG to a fixed point preserving the same (or larger) global symmetry. Methodswithin this paradigm, such as the ε expansion [1], produce in many cases values of the criticalexponents that match well with experiments; see [2] for an extensive review.Despite the remarkable success of the mentioned paradigm for many systems, including thosewith emergent O ( N ) symmetry, one cannot rule out the existence of additional CFTs, not capturedby the Landau–Ginzburg–Wilson description but still relevant for experimental realizations. Aninteresting case is systems with global symmetry group MN m,n = O ( m ) n (cid:111) S n . The most generalLagrangian that preserves MN m,n symmetry is L = ∂ µ φ i ∂ µ φ i + λ ( φ ) + g [( φ + · · · + φ m ) + · · · + ( φ m ( n − + · · · + φ mn ) ] , (1.1)where φ i is an mn -dimensional vector under MN m,n , φ = φ i φ i , and we keep only the kinetic termand the quartic interaction terms. For the two-coupling theory (1.1), the ε expansion predictsonly one fully-interacting fixed point with this global symmetry. For the experimentally accessiblecase of m = n = 2, when the O (2) (cid:111) S theory in (1.1) is equivalent to the more commonlydiscussed O (2) / Z theory [3], the critical exponents derived from this fixed point have not beensuccessful in matching those measured in experiments with helimagnets and XY stacked triangularantiferromagnets, which cluster in two distinct regions; see Table 1. Additionally, the fixed point inthe ε expansion appears to have g <
0, which is inconsistent with the expected chiral universalityclass that should describe phase transitions in these systems [2, 5]. Evidence for the existence of In addition to this fixed point, one finds also the free theory with g = λ = 0, the O ( mn ) symmetric theory with g = 0 and n decoupled O ( m ) models with λ = 0. In much of the literature, e.g. [2,5], a coupling v is used instead of our g in (1.1). The coupling v and our coupling able 1: Experimental results for phase transitions described by MN , theory. These values are compiledfrom [2, 4] and references therein. MN , β ν XY STAs 0 . . . . . . . +0 . − . further non-perturbative fixed points in the chiral region has been offered [6, 7], but this has beendisputed by other authors [4, 8, 9].This contradictory set of observations motivated the recent study of MN symmetric theoriesusing the non-perturbative (numerical) conformal bootstrap [10]. This method, proposed in [11]and extensively reviewed in [12], makes no assumptions on the underlying microscopic descriptionand studies CFTs based only on global and conformal symmetry, unitarity, and consistency withthe operator algebra (crossing symmetry). In agreement with the experimental data, the conformalbootstrap study in [10] found evidence for the existence of two distinct CFTs with MN , symmetry,as can be seen from Fig. 1. This figure displays bounds on operator dimensions in the (∆ φ , ∆ X )plane, where φ and X respectively denote the smallest dimension operators transforming in thevector and a certain rank-two representation of the MN symmetry group. The region below thecurves is the allowed parameter space in the respective theories. In the conformal bootstrap, akink in the boundary of the allowed region signals the existence of a conformal field theory withparameters near the location of the kink, and from the two kinks of the MN , curve in Fig. 1the following values for the critical exponents were derived [10]:kink 1: β = 0 . , ν = 0 . , kink 2: β = 0 . , ν = 0 . . (1.2)While the critical exponents corresponding to the second kink show reasonable agreement, withinuncertainties, with the results of [7, 14], neither set of values is compatible with the predictionsfrom the ε expansion [15]. Although it was speculated in [10] that the first kink may be related tothe ε expansion through the large m limit, the results of that study were not sufficient to makeany conclusive statements.The purpose of the present paper is to make a more systematic study of the hypothetical CFTsliving at the two kinks, by investigating the behavior of numerical bounds for various ( m, n ). Morespecifically, we perform numerical conformal bootstrap studies for varying values of m , keeping n fixed (for most of the work we keep n = 2, but we also obtain a bound for m = 100 and n = 3 , n = 2 we find that, as we increase m , both kinks continue to exist, and for large m the bound g have opposite signs, so the chiral region, defined by v > g < . .
55 0 . .
65 0 . .
75 0 . . MN , MN , ∆ φ ∆ X Fig. 1:
Bounds in the (∆ φ , ∆ X ) plane for 3D CFTs with MN m,n global symmetry with ( m, n ) = (2 , m, n ) = (20 , PyCFTBoot [13] with parameters n max=9 , m max=6 , k max=36 and l max=26 . in the (∆ φ , ∆ X ) plane attain a profile similar to the case MN , displayed in Fig. 1. The large m behavior of the CFT at each kink is subsequently studied.First, we focus on the first kink, which approaches the value (∆ φ , ∆ X ) = (0 . ,
2) as m → ∞ .In [10], it was observed that this limit is compatible with the results in the ε expansion [3, 15–17]expanded at large m , where indeed ∆ φ → d − and ∆ X → m − corrections. We showhere that the m − corrections can be computed in a perturbative expansion similar to the usuallarge N expansion of the O ( N ) model (see e.g. [18]), where now the operator X acts as theHubbard–Stratonovich auxiliary field. Specifically, we use the analytic conformal bootstrap methodof large spin perturbation theory, developed in [19–22], to compute m − corrections to scalingdimensions and OPE coefficients. This expansion is valid for all spacetime dimensions d ∈ (2 , d = 4, the results agree with the ε expansion and in d = 3 we get good agreement withthe non-perturbative bootstrap results for the first kink. This gives substantial evidence that weshould view the first kink as describing a perturbative CFT with MN symmetry, existing for arange of ( m, n ) and d and connected to the ε expansion via the large m limit.Second, we study the second kink for n = 2 and increasing m . The bounds in Fig. 1 revealthat, as we increase m , the values of the scaling dimensions ∆ φ and ∆ X corresponding to this kink4ove far away from the free theory values. We perform a single correlator numerical bootstrapstudy where we search for bounds in the (∆ φ , ∆ X ) plane for increasing values of m , with the hopeof finding a limit point at infinite m for the position of the kink. The results show that the secondkink continues to exist for all values of m studied, and that the position in the ∆ φ directionappears to stabilize near the value 0 .
75. The position in the ∆ X direction takes a value ∆ X (cid:38) φ and ∆ X show that, if there is a CFT corresponding to the second kink, it must be of a non-perturbativetype.To investigate further the potential CFT corresponding to the second kink, we focus on the case MN , and increase the numerical precision. We use the extremal functional method, developedin [23], to extract information about the spectrum of operators in the φ × φ OPE. The resultsgive some hints of an organization of the leading twist operators in twist families, as must be thecase according to the lightcone bootstrap [24, 25]. For fixed m = 100 we also study MN ,n for n = 3 and n = 4, but for these values we find no second kink in the (∆ φ , ∆ X ) bound.This paper is organized as follows. In section 2 we explain how to study MN symmetric CFTsin the bootstrap approach using crossing symmetry and unitarity. We introduce relevant notationand review the known perturbative results of the ε expansion. In section 3 we show that the CFTcorresponding to the first kink can be matched with a perturbative large m expansion, which weconstruct using large spin perturbation theory. Further, we comment on the connection to the ε expansion. In section 4 we use the non-perturbative numerical bootstrap to study the second kink,and discuss the twist families of the spectrum for the representative case MN , . We finish witha discussion, and include some explicit results in an appendix.For our numerical computations we have used PyCFTBoot [13], qboot [26] and
SDPB [27].
2. Review
In this section we briefly review the constraints from unitarity and crossing symmetry on conformalfield theories, with emphasis on theories with global MN symmetry. We then summarize theresults from previous studies in the ε expansion. MN m,n symmetry We consider the four-point correlator of φ i , i = 1 , . . . , mn , transforming in the vector representation V of the MN m,n = O ( m ) n (cid:111) S n global symmetry. More precisely, O ( m ) acts by rotating the fieldswithin each of the n groups of m fields, and S n permutes these groups. In terms of the conformalcross-ratios u = x x x x and v = x x x x , with x ij = | x i − x j | , the correlator takes the form (cid:104) φ i ( x ) φ j ( x ) φ k ( x ) φ l ( x ) (cid:105) = 1 x φ x φ (cid:88) R = S,X,Y,Z,A,B T ijklR G R ( u, v ) , (2.1)5here T R are projection tensors for the representations R in the tensor product V ⊗ V = S ⊕ X ⊕ Y ⊕ Z ⊕ A ⊕ B . S denotes the singlet representation with T ijklS = mn δ ij δ kl , while theremaining symbols denote rank-two symmetric ( X , Y and Z ) and antisymmetric ( A and B )representations. We will not need the precise form of these projection tensors, which can be foundin [10]. Each function G R ( u, v ) admits a decomposition in conformal blocks, G R ( u, v ) = (cid:88) O∈ R ( − (cid:96) O λ O g ∆ O ,(cid:96) O ( u, v ) , (2.2)where the sum runs over conformal primary operators of dimension ∆ O and spin (cid:96) O transformingin the representation R . The conformal blocks, denoted by g ∆ ,(cid:96) ( u, v ), sum up the contribution tothe correlator of a given primary and all its descendants, and are functions depending only on thecross-ratios and the dimension and spin of the primary.We will adopt a notation where we denote by R , R (cid:48) , R (cid:48)(cid:48) etc. the smallest dimension scalars inthe representation R , and likewise by R (cid:96) , R (cid:48) (cid:96) , R (cid:48)(cid:48) (cid:96) etc. the smallest dimension spin (cid:96) operators inthe representation R . In any unitary CFT, the S representation will contain the identity operator with ∆ = (cid:96) = 0, and the stress-energy tensor T = S with ∆ T = d . In the presence of acontinuous global symmetry, a CFT contains in addition a conserved Noether current J µ with∆ J = d −
1, which in our case resides in the A representation: J = A .Unitarity imposes constraints on the decomposition (2.2). Reality of the three-point functions (cid:104) φ ( x ) φ ( x ) O ( x ) (cid:105) implies positivity of the expansion coefficients λ O , and positivity of two-pointfunctions of descendants implies unitarity bounds, namely∆ R (cid:62) ( d − , ∆ R (cid:96) (cid:62) d − (cid:96) , (2.3)where the inequalities are saturated only for a free scalar and a conserved current respectively.Moreover, a non-trivial consequence of unitarity is Nachtmann’s theorem [28], which states thatthe twists of the leading singlet operators, τ S,(cid:96) = ∆ S (cid:96) − (cid:96) , form an upward convex function for allspin (cid:96) above some (cid:96) , see [29] for a recent discussion.Crossing symmetry follows from the invariance of the correlator (2.1) under exchanging pairsof insertion points. The invariance under x ↔ x is satisfied by each conformal block, whilethe invariance under x ↔ x leads to a non-trivial crossing equation which we will use. In thepresence of global MN m,n symmetry, the crossing equation takes the form G R ( u, v ) = (cid:16) uv (cid:17) ∆ φ (cid:88) (cid:101) R M R (cid:101) R G (cid:101) R ( v, u ) , (2.4) This means we will denote by S the smallest dimension operator different from the identity. M R (cid:101) R in the basis { S, X, Y, Z, A, B } is given by M RR (cid:48) = mn mn m − − n − mn n − mn n − m − − n ( m − m +2)2 mn ( m − m +2)2 mn m − m m +22 n − n − n − m − mn − m − mn m − n − n n . (2.5)We refer to the left-hand side of (2.4) as the direct channel, and to the right-hand side as thecrossed channel. In the analytic bootstrap approach in section 3.1, the crossing equation isexpanded in the double lightcone limit u (cid:28) v (cid:28)
1, and the operators in the crossed channel willsource corrections to the CFT data of the operators in the direct channel.In the numerical bootstrap approach in section 4.1, the crossing equation is re-written in aform that treats the channels symmetrically. This form is given explicitly in [10], and the technicaldetails can be found in that paper. The principles are the standard ones of the numerical conformalbootstrap [11, 12]: by acting on the crossing equation with a family of functionals, positivity ofthe squared OPE coefficients λ O is turned into rigorous inequalities which rule out large regionsof the space of allowed operator dimensions. For the theory at hand, we identify the potentialfixed points by observing kinks in the bound in the (∆ φ , ∆ X ) plane, following [10]. To gain moreinformation about the CFT at the position of the kink, we apply the extremal functional methoddeveloped in [23], which uses the fact that a functional in the vicinity of the CFT should vanishwhen applied to the conformal blocks of the operators present in the spectrum. ε expansion Conformal field theories with MN symmetry have been studied in the d = 4 − ε expansion over along time, [15–17, 30, 31], most recently in section 5.2.2. of [3]. From the beta functions of thecouplings λ and g in the Lagrangian (1.1), four fixed points are found in the ε expansion: mn free fields, n decoupled critical O ( m ) models, the critical O ( mn ) model, and the perturbative MNCFT. We focus on the MN CFT, which has λ, g (cid:54) = 0. At this fixed point, the scaling dimensionsof the leading scalar operators have an expansion of the form∆ φ = 1 − ε m ( n − m + 2) mn − m + 16]4 C mn ε + γ (3) φ ε + O ( ε ) , (2.6)∆ S = 2 − ε + 6 m ( n − C mn ε + γ (2) S ε + γ (3) S ε + + O ( ε ) , (2.7)∆ X = 2 − ε + m (( m + 2) n − C mn ε + γ (2) X ε + γ (3) X ε + O ( ε ) , (2.8)∆ Y = 2 − ε + 2 m ( n − C mn ε + γ (2) Y ε + γ (3) Y ε + O ( ε ) , (2.9)7 Z = 2 − ε − m − C mn ε + γ (2) Z ε + γ (3) Z ε + O ( ε ) , (2.10)where C mn = ( m + 8) mn − m − ε and ε corrections, which were derived in [3] and are available by an email requestto the authors. Moreover, the eigenvalues of the stability matrix are [3] ω = ε + ω (2)1 ε + ω (3)1 ε + O ( ε ) , (2.11) ω = − ( m − mn − C mn ε + ω (2)2 ε + ω (3)2 ε + O ( ε ) , (2.12)which correspond to ω i = ∆ i − d for the singlet operators of φ type in the theory.For the specific case of m = 2, the order ε renormalization was performed in [15, 31], giving∆ φ , ∆ S , ω and ω to this order. For the physically relevant cases n = 2 and n = 3, a Borel–Leroyresummation was performed to give estimates for the critical exponents γ , ν , η in three dimensions;see section 3.2 below.
3. The perturbative fixed point at large m In this section we will derive a large m expansion for MN m,n symmetric CFTs, and show that itgives predictions that match well with those found in the numerical bootstrap for the first kink.The existence of this expansion establishes the perturbative nature of the corresponding family ofCFTs . Expanding the expressions (2.7)–(2.10) for the scalar operators at large m we observe that∆ X = 2 + O ( m − ) , (3.1)whereas the scalar operators in the S , Y and Z representations all satisfy ∆ = 2 − ε + O ( m − ).This observation indicates that there exists, for all d ∈ (2 , m expansion where ∆ X =2 + O ( m − ), and ∆ R = d − O ( m − ) for R = S, Y, Z . These values are consistent with adescription in terms of Hubbard–Stratonovich auxiliary fields, similar to the large N expansion ofthe critical O ( N ) model.In [22], based on [20, 21], it was described how to use large spin perturbation theory to extractproperties of φ theories with Hubbard–Stratonovich auxiliary fields. We will follow this approach,which means that we assume that at large m the operator spectrum is that of mean field theory The leading m dependence is given in equation (5.104) and (5.105) in the arXiv submission of [3], where N = mn and σ = S , ρ = X , ρ = Z and ρ = Y . φ with ∆ φ = d − + O ( m − ), but with the bilinear scalar in the X representation replaced bya Hubbard–Stratonovich field X of dimension ∆ X = 2 + O ( m − ). The framework of [22] will thenshow what operator dimensions, in the large m expansion, are consistent with these assumptions.Let us briefly review the method of large spin perturbation theory, which in the present casewill be a perturbation of mean field theory, i.e. we assume that each representation R in thetensor product V ⊗ V contains operators of spin (cid:96) and scaling dimensions 2∆ φ + 2 k + (cid:96) + O ( m − ).The operators with k > /m expansion, andwe can therefore focus on the leading twist operators, which we denote by R (cid:96) . These are bilinearoperators of the schematic form φ∂ (cid:96) φ and acquire individual anomalous dimensions∆ R (cid:96) = 2∆ φ + (cid:96) + γ R (cid:96) , (3.2)where γ R,(cid:96) is of order O ( m − ). Symmetry under x ↔ x constrains the leading twist operators suchthat those in the S, X, Y, Z representations have even spin, and those in the
A, B representationshave odd spin.In large spin perturbation theory, the OPE coefficients λ φφR (cid:96) and the anomalous dimensions γ R (cid:96) of spinning operators in the direct channel are computed using the Lorentzian inversionformula [33]. The integrand of the inversion formula is proportional to the double-discontinuitydDisc[ G R ( u, v )], defined as the difference between the correlator and its two analytic continuationsaround v = 0. In the limit u (cid:28) v (cid:28)
0, and in an expansion in m − , the double-discontinuitycan be computed from crossed-channel operators, i.e. those appearing in the conformal blockdecomposition of the right-hand side of (2.4). The contribution to the R (cid:96) from an operator O inthe ˜ R representation is proportional todDisc | O ∼ M R ˜ R λ φφ O sin (cid:2) π ( τ O − φ ) (cid:3) , (3.3)where the argument of the squared sine is derived from the v → v − ∆ φ g ∆ O ,(cid:96) O ( v, u ) ∼ v τ O − ∆ φ with τ O = ∆ O − (cid:96) O . The appearance of the sin factor means that the contribution frommean field theory operators will be suppressed by their squared anomalous dimension.In order to apply the framework of [22], we assume that the operator X has an OPE coefficientof the form λ φφX = a X m + O ( m − ) , (3.4)for some constant a X depending on n and spacetime dimension d . The contribution to the CFTdata in the direct channel is then computed using the inversion formula from double-discontinuities(3.3) of crossed-channel operators. The identity operator generates the leading OPE coefficientsof R (cid:96) , and the operator X gives a leading order contribution to γ R (cid:96) in all representations R . In the This follows immediately from the expression for the mean field theory OPE coefficients derived in [32], uponinserting ∆ φ = d − + O ( m − ). Y, Z, A, B the crossed-channel operators and X provide the only contributionsat order m − , and, using the formulas given in [22], we can write down the scaling dimensions ∆ Y (cid:96) = ∆ A (cid:96) = (cid:96) + 2∆ φ − a X (cid:96) + 1 / (cid:96) − / m + O ( m − ) , (3.5)∆ Z (cid:96) = ∆ B (cid:96) = (cid:96) + 2∆ φ + a X (cid:96) + 1 / (cid:96) − / n − m + O ( m − ) . (3.6)Recall that the spin (cid:96) is even for Y and Z , and odd for A and B .Due to the combined m dependence of all three factors in (3.3), the anomalous dimensions inthe S and X representations will get leading order contributions from X as well as from the R (cid:96) inthe other four representations. In the language of [22], we therefore have group I = { Y, Z, A, B } ,and group II = { S, X } . Evaluating the formulas of that paper gives∆ S (cid:96) = (cid:96) + 2∆ φ − a X (cid:96) + 1 / (cid:96) − / m − π (cid:96) a X n (cid:96) + 1 / (cid:96) − / n − m + O ( m − ) , (3.7)∆ X (cid:96) = (cid:96) + 2∆ φ − a X (cid:96) + 1 / (cid:96) − / m − π (cid:96) a X n ( n − (cid:96) + 1 / (cid:96) − / n − m + O ( m − ) . (3.8)The expressions (3.5)–(3.8) depend on two unknowns: the constant a X introduced in (3.4), and theleading anomalous dimension γ (1) φ defined by ∆ φ = d − + γ (1) φ /m + O ( m − ). However, conservationof the global symmetry current and the stress-energy tensor gives the two equations ∆ A = d − S = d . The latter equation is quadratic in a X , and we get two solutions: a X = γ (1) φ = 0 and a X = 4( n − π n , γ (1) φ = 4( n − π n . (3.9)Choosing the non-trivial solution, we have fixed the leading order anomalous dimensions of allleading twist spinning operators in the theory.We have also computed the corrections to the OPE coefficients of these operators, definedwith respect to the leading order result given by M RS times the OPE coefficients of mean fieldtheory. In general, these results are not particularly illuminating, but specifying to the conservedoperators we extract the corrections to the central charge and the current central charge, C T C T, free = 1 − n − n π m + O ( m − ) , (3.10) C J C J, free = 1 − n − n π m + O ( m − ) . (3.11)The most important class of observables is the dimensions of the leading scalar operators ineach representation. Unfortunately, spin zero is beyond the guaranteed region of convergence of We present only the values in d = 3 dimensions, results for generic d are given in Appendix A. The observant reader may note that these results agree with those of one free and n − O ( m ) models.This agreement is broken at higher orders in m − , which can be seen from the ε expansion, where C T /C T, free =1 − γ (2) φ ε / O ( ε ) [22]. a priori clear how to extract these values. However,in the large N expansion of theories with O ( N ) [21] and O ( m ) × O ( N ) [22] global symmetry, ithas been observed that evaluating ∆ R (cid:96) for (cid:96) = 0 correctly reproduces the dimensions of the scalaroperators as computed by independent methods. If we assume that this is the case also for our MNsymmetric theory, we can find the scalar operator dimensions by demanding that ∆ X = d − ∆ X (cid:96) =0 and that ∆ R = ∆ R (cid:96) =0 for R = S, Y, Z . Further support for this assumption is that the expressionsderived from this assumptions, evaluated for d = 4 − ε , agree with the expressions (2.7)–(2.10)in the overlap of the orders: ε /m . In three dimensions we find the following dimensions of thescalar operators: ∆ φ = 12 + 4( n − n π m + O ( m − ) , (3.12)∆ S = 1 + 32( n − n π m + O ( m − ) , (3.13)∆ X = 2 − n − n π m + O ( m − ) , (3.14)∆ Y = 1 + 32( n − n π m + O ( m − ) , (3.15)∆ Z = 1 + 8( n − n π m + O ( m − ) , (3.16)and results for generic spacetime dimension d are presented in Appendix A. The results (3.12)–(3.16) can now be compared with the numerical bootstrap results for the firstkink. In Fig. 2 we display the bounds in the (∆ φ , ∆ X ) plane from Fig. 2 of [10], together withour new large m results (3.12), (3.14), as well as the ε results (2.6), (2.8) from the literature.We see that the agreement is good between all three methods for m (cid:29)
1, and that the large m expansion better captures the finite m behavior than does the ε expansion.Our new results in the large m expansion can also be used to derive predictions for the criticalexponents using the relations η = 2 − d + 2∆ φ , ν − = d − ∆ S , α = 2 − νd , (3.17) β = ν ∆ φ , γ = ν ( d − φ ) , φ κ = ν ( d − ∆ Z ) . (3.18)For the physically relevant case MN , , our new predictions for β and ν from the large m expansionare closer to numerical bootstrap results, as well as experiments and Monte Carlo results, thanpredictions from the ε expansion, as can be seen in Table 2. Also for the chiral cross-over exponent In producing this graph, as well as the values in Table 2, we have used the truncated results of the ε expansion atorder ε . We comment on this in section 3.3. . .
505 0 .
51 0 .
515 0 .
52 0 .
525 0 .
53 0 . . . . . . MN , MN , MN , MN , ∆ φ ∆ X Fig. 2:
Bounds and corresponding locations of fixed points given as dots for large m and crosses for ε expansionresults. The lines connecting dots and crosses are drawn to help illustrate results pertaining to thesame theory. φ κ , the value 1 . m results compares favorably with the Monte Carlovalue 1 . ε expansion As we mentioned in the introduction, results derived in the ε expansion have not been successful inmatching the experimental values observed in the cases MN , and MN , . This is in contrast tocritical phenomena described by CFTs with several other symmetry groups, where the results in the ε expansion give surprisingly good agreement with experimental data as well as non-perturbativeresults from Monte Carlo simulations and numerical conformal bootstrap.The lack of agreement between bootstrap and ε expansion results in the MN , case may betaken as a sign that the fixed point found of the ε expansion, as discussed in section 2.2, is notconnected to the CFT describing the critical phenomena in three dimensions. Our results stronglyindicate the contrary, and that the connection is manifest through the large m expansion derivedabove. Specifically, near four dimensions our new analytic results agree with the ε expansion, andin three dimensions, the large m expansion is connected to the finite m CFTs through the familyof kinks displayed in Fig. 2.We note that for the larger values of m there is good agreement between all three methods: We first estimated ∆ S and ∆ Z by evaluating the truncated expansions (3.13) and (3.16) for m = n = 2, beforeusing (3.18) to find φ κ . able 2: Comparison of data for MN symmetric theories across various methods. The truncated series denotetruncation to orders ε and m − respectively for the scaling dimensions. These numeric values arethen used in (3.17) and (3.18) to give estimates for β and ν . MN , ∆ φ ∆ S ∆ X β ν XY STA 0 . . . . . . . . ε resummation [31] 0 . . ε expansion (trunc.) 0 . . . . . m expansion (trunc.) 0 . . . . . MN , ∆ φ ∆ S ∆ X β ν Numerical bootstrap [10] 0.518(1) 1.279(20) 1.590(10) 0 . . ε resummation [31] 0 . . ε expansion (trunc.) 0 . . . . . m expansion (trunc.) 0 . . . . . MN , ∆ φ ∆ S ∆ X β ν Numerical bootstrap 0.5032(1) 1.025(20) 1.965(10) 0.2548(26) 0.506(5) ε expansion (trunc.) 0 . . . . . m expansion (trunc.) 0 . . . . . m expansion, ε expansion, and numerical conformal bootstrap. For the lower values of m , our new large m expansion evaluates at a point closer to the corresponding kink than doesthe ε expansion. For the latter, we have simply used direct truncation of the order ε results,alternatively one could use Pad´e approximants or various resummation techniques. In Table 2we extended this comparison to more observables, and again we get an improved agreement withthe numerical bootstrap compared to the ε expansion. Note, for instance, that for small m the ε expansion predicts that ∆ X < ∆ S , which is inconsistent with the bootstrap results.While the values for the critical exponent ν show good agreement across experiments, bootstrap,Monte Carlo and large m expansion, the situation for the exponent β is more concerning. Infact, as already pointed out in the literature [2, 4], the experimental and Monte Carlo valuesdo not satisfy the constraint 2 β − ν (cid:62) φ . Thisinconsistency could be explained by unknown systematic errors of these methods, or that they We find that the Pad´e approximants constructed from the order ε results contain spurious poles in the region ε (cid:54)
1, and since the ε expansion is not the focus of this paper we have not attempted any resummation methods. Notethat the resummed ε expansion of [15], included in Table 2, does not give any improvement compared to a directtruncation.
4. The non-perturbative fixed point at large m In this section we study, using the numerical bootstrap, the large m limit of the second kink for MN m,n symmetric theories. While two kinks are clearly visible for MN , in [10, Fig. 4], we findthat the second kink only persists at large m for the case n = 2, and we focus our attention tothe cases MN m, for various m . Our first set of results consists of bootstrap bounds for MN m, theories for large values of m ; seeFig. 3. These bounds show that the kink persists at large m and that its position stabilizes closeto ∆ φ = 0 .
75. However, the kink still moves significantly in the ∆ X direction.0 .
71 0 .
72 0 .
73 0 .
74 0 .
75 0 .
76 0 .
77 0 .
78 0 . . . . . MN , MN , MN , MN , MN , MN , ∆ φ ∆ X Fig. 3:
Bounds for 3D CFTs with MN m, global symmetry for various values of m . The allowed region is belowthe curves in the corresponding theories. These bounds are obtained with the use of PyCFTBoot [13]with parameters n max=9 , m max=6 , k max=36 and l max=26 . .
74 0 .
742 0 .
744 0 .
746 0 .
748 0 .
75 0 .
752 0 .
754 0 .
756 0 .
758 0 . . . . . . . . . lambda=31lambda=35lambda=41lambda=45lambda=51 ∆ φ ∆ X Fig. 4:
Bounds for 3D CFTs with MN , global symmetry with increasing numerical strength (top tobottom). For these bounds we used qboot [26]. These bounds are all stronger than the correspondingbound in Fig. 3. The position for ∆ φ of the kink is not stable upon increasing the number of derivatives; seeFig. 4. It appears, however, that ∆ φ is fairly stable near the value 0 .
75. We conjecture that∆ φ | m →∞ ,n =2 = 0 . . (4.1)For our strongest numerics we used qboot [26] with parameters prec=1300 , n Max=560 , lambda=51 , numax=30 and the set of spins {
0, . . . , 80, 85, 86, 89, 90, 93, 94, 97, 98, 101, 102, 105, 106, 109,110, 111, 112, 115, 116, 119, 120 } .We also obtained bounds for m = 100 with n = 3 ,
4. As we see in Fig. 5, the kink is clearlypresent only for n = 2. This suggests that there exists a critical line n c ( m ) below which we havetwo distinct CFTs with MN m,n global symmetry.Subsequently, we focused on the values ( m, n ) = (100 , φ = 0 . PyCFTBoot [13] with parameters n max=13 , m max=10 , k max=50 and l max=40 .From Fig. 4 we we estimate the scalar operator dimensions, for ( m, n ) = (100 , φ = 0 . , (4.2)∆ X = 6 . . (4.3)From Figs. 6 and 7 we then read off the dimensions of the leading scalar operators to∆ S ≈ . , (4.4)15 .
74 0 .
742 0 .
744 0 .
746 0 .
748 0 .
75 0 .
752 0 .
754 0 .
756 0 .
758 0 . . . . . MN , MN , MN , ∆ φ ∆ X Fig. 5:
Bounds for 3D CFTs with MN ,n global symmetry for various values of n . The allowed region isbelow the curves in the corresponding theories. For these bounds we used qboot [26] with parameters prec=1200 , n Max=520 , lambda=45 , numax=26 and the set of spins {
0, . . . , 60, 63, 64, 66, 67, 73, 74,77, 78, 81, 82, 85, 86, 89, 90, 93, 94, 97, 98 } . ∆ Y ≈ . , (4.5)∆ Z ≈ . . (4.6)It is interesting to note that these results have ∆ Z < ∆ φ . The small values for ∆ Y and ∆ Z suggest that a mixed correlator bootstrap involving the operators Y and/or Z may give resultsthat are quite constraining. m theory The results from our numerical bootstrap show that the second kink continues to exist for allvalues of m (cid:62)
2, indicating the existence of a corresponding CFT, in the sense of a set of conformalprimary operators with scaling dimensions and OPE coefficients consistent with unitarity andcrossing. We have only considered the constraints from the (cid:104) φφφφ (cid:105) correlator, and numericalstudies using a multi-correlator approach will either give further constraints on the candidate CFTor disprove its existence.The motivation for our work was to see if the candidate theory approaches a simplifying limitas m → ∞ . If this were the case, it could potentially be studied using perturbative methods justlike we did for the first kink in section 3. Our results show that this is not the case, meaning that16 .0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0100105110115120125130135 (a) S , (cid:96) = 0 (b) S , (cid:96) = 2 (c) S , (cid:96) = 4 (d) X , (cid:96) = 0 (e) X , (cid:96) = 2 (f) X , (cid:96) = 4 (g) Y , (cid:96) = 0 (h) Y , (cid:96) = 2 (i) Y , (cid:96) = 4 (j) Z , (cid:96) = 0 (k) Z , (cid:96) = 2 (l) Z , (cid:96) = 4 Fig. 6:
Plots from the extremal functional method for the even representations. In the horizontal axis we plotthe scaling dimension and in the vertical the logarithm of the action of the functional on convolvedconformal blocks (denoted by F ∆ ,(cid:96) in [23]). .5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5105110115120125130135 (a) A , (cid:96) = 1 (b) A , (cid:96) = 3 (c) A , (cid:96) = 5 (d) B , (cid:96) = 1 (e) B , (cid:96) = 3 (f) B , (cid:96) = 5 Fig. 7:
Plots from the extremal functional method for the odd representations. the candidate CFT remains non-perturbative, or “strongly coupled” for all values of m . It willtherefore be challenging to corroborate the existence of this CFT.There is, however, one test that any conformal field theory must pass, be it perturbative ornot, namely consistency with the predictions from the lightcone bootstrap [24]. These papersproved the twist additivity , stating that any conformal field theory containing operators O and O , must also contain an infinite family of spinning operators [ O , O ] ,(cid:96) with twists (recallthat τ = ∆ − (cid:96) ) lim (cid:96) →∞ τ (cid:96) = τ ∞ = τ + τ . (4.7)This statement means that a given theory will contain many different accumulation points τ ∞ ,see [35] for observations in the ε expansion of the O ( N ) model predating the lightcone bootstrap,however not all these values may be visible in a given correlator. In fact, the only accumulationpoint guaranteed to exist in the direct channel of the (cid:104)O O O O (cid:105) is τ + τ , but other values arenot excluded. The conclusion is that we expect that in our candidate CFT the value τ ∞ = 2∆ φ = 1 . φ is close to the scalar unitarity bound d − , the twists ofthe double-twist operators [ φ, φ ] ,(cid:96) are also close to the spinning unitarity bound d −
2, c.f. (2.3). The interacting theory also contains subleading twist families [ O , O ] k,(cid:96) with twists approaching τ + τ + 2 k .
18n such a theory, the double-twist operators have leading twist, and will include the stress-energytensor at (cid:96) = 2.In the generic case, where ∆ φ − d − is finite, there are three possibilities: • The double-twist operators [ φ, φ ] ,(cid:96) remain the operators at leading twist for each spin, whichmeans that they acquire “large” anomalous dimensions γ (cid:96) = τ (cid:96) − τ ∞ in order to accommodatethe stress-energy tensor at (cid:96) = 2. This is the situation in 3D critical O ( N ) models at finite N , and indeed also in the perturbative fixed point of section 3 in this paper. • The stress-energy tensor belongs to an additional twist family below the double-twist operators.This case happens for instance when φ is a composite operator in weakly coupled theories.An example is correlators of gauge-invariant operators in weakly coupled N = 4 SYM, wherethe double-twist family consists of double-trace operators and the additional leading twistfamily consists of the single-trace weakly broken currents. • The stress-energy tensor and other conserved currents are isolated operators not belongingto any twist family. This happens in N = 4 in the strong coupling expansion, where thelower limit of analyticity in spin is shifted upwards [33], but is not expected for a bona fide CFT with usual Regge behavior.We now study the plots in Figs. 6 and 7 to see if they are consistent with any of the mentionedscenarios. Since the position of the kink could not be precisely determined, we do not expectthese plots to give very precise values for the operator dimensions. We can, however, look at thequalitative behaviour of the lowest dimension operators.In the S and X representations, the spectrum plots are comparatively sparse. There the lowestdimension operators at each spin are consistent with the first scenario, with large anomalousdimensions. In the singlet representation, these anomalous dimensions are negative, which isconsistent with Nachtmann’s theorem (convexity) and with the stress-energy tensor appearingat (cid:96) = 2. In the X representation, the anomalous dimensions are positive, and it seems like thefamily can be extended to spin zero to include the X operator. On the contrary, the Y , Z , A ,and B representations show a comparably dense spectrum, which is consistent with the secondscenario of a twist family below the double-twist operators. In Fig. 8 we display a cartoon thatsummarizes these observations, however, further numerical study will be needed to confirm ordisprove this picture. Note that in these theories, the anomalous dimensions never become particularly large; the largest values isattained in the 3D O (2) CFT with τ ∞ − τ T = 0 . By studying also the spectrum plots at spin 6 and 8 (not displayed), we note that the twists of the leadingoperator continues to decrease according to the behavior displayed in Fig. 8. Note however the feature at ∆ = 5 inFig. 6f, a similar feature is noted at spin 6 (but not at spin 8) and is likely to be a spurious zero. φ (a) Twist spectrum in X . φ (b) Twist spectrum in S . φ (c) Twist spectra in Y and Z . φ (d) Twist spectra in A and B . Fig. 8:
Cartoons for the hypothetical twist families in the various representations showing spin on the horizontalaxis and twist on the vertical axis. The unitarity bound (2.3) is shown in blue and the value 2∆ φ = 1 . S = T and A = J have twists exactly on the unitarity bound. While it is encouraging that the spectrum plots are not inconsistent with the constraints fromthe lightcone bootstrap, we would like to address some issues that complicate the picture. Asalready mentioned, the uncertainty in the position of the kink induces uncertainty in the spectrumplots. Moreover, the plots should not be interpreted as displaying the full spectra in the respectiverepresentations since they are only showing operators with a non-negligible contribution to the φ four-point function. The complete spectrum of the theory is more dense than our cartoonsof Fig. 8 indicate. For instance, the singlet representation is expected to contain accumulationpoints 2 τ O for all operators O in the theory and in our case this would include an accumulationpoint 2∆ Z = 1 .
2, which is below the value 2∆ φ in (4.8). A numerical bootstrap study of mixedcorrelators may reveal more operators. In [25], an ambitious attempt was made to determine theoperator spectrum in the 3D Ising CFT, using a system of mixed correlators constructed out of σ and (cid:15) , the smallest dimension Z odd and even operators. Based on [36], the extremal functionalwas then applied to a sample of points on the boundary of the allowed region (in this case anisland), and only stable operator dimensions were deemed to be candidates for primary operatorsin the spectrum. A similar computation was performed in [37] for the 3D O (2) CFT. The methodappears to give somewhat reliable predictions for the spectrum at low operator dimensions, but20isses higher-spin operators asymptoting to double-twist dimensions of operators not included inthe system of correlators studied. It would be desirable to perform a similar study of the theoryat hand.
5. Discussion
In this work we studied CFTs with global symmetry MN m,n = O ( m ) n (cid:111) S n in d = 3 dimensions,with the motivation of evaluating further the potential existence of two distinct fixed points insuch theories, as was recently suggested in [10] for the MN , and MN , theories. For theorieswith n = 2 we found evidence supporting this conclusion by considering various values of m andobserving two distinct kinks in bootstrap bounds, even for large m . For n >
2, our bootstrapbounds did not include a clear second kink in the expected region for large n , although such akink does exist in the MN , theory (see [10, Fig. 4]). Our results suggest that there exists acritical line n c ( m ) below which there are two distinct MN CFTs.The second kink we examined appears to not correspond to a known theory obtained withinthe standard Wilson–Fisher paradigm. Looking at the operator spectrum at this kink, we verifiedthat it satisfies general expectations derived from the Nachtmann theorem [28] and the lightconebootstrap [24]. If this kink is due to a corresponding full-fledged CFT, this would indicate thatthe Wilson–Fisher paradigm is incomplete, i.e. that there exist fixed points in d = 3 that cannotbe obtained from continuations of fixed points in d = 4 − ε . Recently, other bootstrap works havereported kinks that do not appear to be of Wilson–Fisher type [38], but the qualitative features ofthese kinks differ from ours and they may be of different origin. Interestingly, the picture emergingfrom our spectrum analysis at large m shows some similarities with results in large N O ( N ) bosonic tensor models [39]. It would be interesting to further investigate the consistency of thespectrum of the second kink, using a numerical application of the Lorentzian inversion formulaas has been done for the Ising and O (2) CFTs [37, 41]; see also [25]. This would require a moreprecise determination of the spectrum and estimation of OPE coefficients.With regards to the experimentally accessible MN , case, out of the five scenarios of [4, Sec.III.B.3], all of our bootstrap results favor “Scenario II”, with two fixed points and thereforesecond-order phase transitions in both groups of systems. However, we stress that the values ofthe exponent β obtained in experiments are in mild tension with the bootstrap ones for the first For instance, results of [25] clearly identify the accumulation points 2∆ σ = 1 . (cid:15) = 2 .
825 and 2∆ σ +2 = 3 . τ T = 2 and τ T + ∆ (cid:15) = 2 . O (2) CFT, not all expected operatorsin the charge 4 sector were found numerically in [37]. In these models, ∆ φ = d/
4, and the bilinear operators in some representations, including those containingconserved currents, acquire large anomalous dimensions, similar to Fig. 8a and 8b; see [40] for recent work in thesupersymmetric case. β − ν (cid:62)
0. Of course, by constructionthe numerical bootstrap results are consistent with unitarity. Given the relatively good agreementof ν between bootstrap, experimental and Monte Carlo results, we conclude that there is need fora more accurate determination of β with experiments and Monte Carlo simulations, controllingsystematic errors.It would be desirable to compute results at large m using conventional diagrammatic techniquessimilar to the large N expansion of the critical O ( N ) model. There, the leading scalar operatordimensions have been computed to orders N − (∆ φ [42]) and N − (∆ S [43] and ∆ T [44]), andresults for the next-to-leading scalar singlet at order N − also exist [45]. A difference with the O ( N ) model is that the auxiliary field is now X , which is not a singlet under the global symmetry.Additionally, it would be informative to perform a numerical bootstrap in intermediate dimen-sions 3 < d <
4, like that in [46], in order to examine the behavior of the kinks as we approach d = 4. This will allow us to make better contact with the ε expansion for the first kink, as wellas determine if the second kink persists closer to d = 4. We note here that theories defined fornon-integer values of d are expected to be non-unitary [47], but this is not expected to causeproblems when bounding scaling dimensions of low-lying operators. Mixed correlator bootstrapstudies in d = 3 and 3 < d < O ( N ) models. Acknowledgments
We would like to thank J. Gracey and A. Vichi for helpful discussions and comments on themanuscript. The project has received partial funding from the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation programme (grant agreementno. 758903). This research used resources provided by the Los Alamos National LaboratoryInstitutional Computing Program, which is supported by the U.S. Department of Energy NationalNuclear Security Administration under Contract No. 89233218CNA000001. Research presented inthis article was supported by the Laboratory Directed Research and Development program of LosAlamos National Laboratory under project number 20180709PRD1.
Appendix A. Large m results for general d In this appendix we present the results of section 3.1 for general spacetime dimension d = 2 µ . Thescalar operator dimensions are given by∆ φ = µ − n − n η m + O ( m − ) , (A.1)22 S = 2( µ −
1) + 4( µ − µ − n − − µ ) n η m + O ( m − ) , (A.2)∆ X = 2 − µ − µ − n − µ + 6 µ − − µ ) n η m + O ( m − ) , (A.3)∆ Y = 2( µ −
1) + 4( n − − µ ) n η m + O ( m − ) , (A.4)∆ Z = 2( µ −
1) + 2 (2 − µ ) n − − µ ) n η m + O ( m − ) , (A.5)where η = ( µ − µ − − µ )Γ( µ ) Γ( µ +1) ; the spinning operator dimensions by∆ S (cid:96) = (cid:96) + 2∆ φ − µ ( n − J n (cid:18) µ − (cid:96) + 1)Γ(2 µ − (cid:96) + 2 µ − (cid:19) η m + O ( m − ) , (A.6)∆ X (cid:96) = (cid:96) + 2∆ φ − µJ n (cid:18) ( µ − n −
1) + ( n − (cid:96) + 1)Γ(2 µ − (cid:96) + 2 µ − (cid:19) η m + O ( m − ) , (A.7)∆ Y (cid:96) = ∆ A (cid:96) = (cid:96) + 2∆ φ − µ − µ ( n − J n η m + O ( m − ) , (A.8)∆ Z (cid:96) = ∆ B (cid:96) = (cid:96) + 2∆ φ + 2 µ ( µ − J n η m + O ( m − ) , (A.9)where J = ( µ − (cid:96) )( µ − (cid:96) ) and we have substituted the value for a X = ( n − µ − µn ( µ − η ; andthe central charge corrections by C T C T, free = 1 − n − − µ ) µ ( µ + 1) n (cid:0) µ (2 − µ )[ π cot( πµ ) + S (2 µ − − µ + 2 µ + 4 (cid:1) η m + O ( m − ) , (A.10) C J C J, free = 1 − µ − n − µ ( µ − n η m + O ( m − ) , (A.11)where S ( x ) denotes the standard analytic continuation of the harmonic numbers away from integerarguments. For µ = 3 / µ = 2 − ε/
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