JHEP09(2018)103 algebra 3 Springer W July 11, 2018 : February 25, 2018 September 14, 2018 September 18, 2018 : spectrum is fixed : : ls” of = 4, we study the Revised N N e-Yang edge singularity. Received entification with previous , China Accepted Published byproducts of our numerical symmetry, we find a series of 2 symmetry. . Setting Z 3 S ces, ⊗ → ∞ N S Published for SISSA by https://doi.org/10.1007/JHEP09(2018)103 N ) at Ising ǫ ∆ , [email protected] , Ising σ b . 3 1712.00985 and Ning Su The Authors. a c Conformal Field Theory, Renormalization Group

We study scalar conformal field theories whose large , [email protected] Seoul 133-791, Korea CAS Key Laboratory of TheoreticalInstitute Physics, of Theoretical Physics, ChineseZhong Academy Guan of Cun Scien East Street # 55,E-mail: P.O. Box 2735, Beijing 100190 Department of Physics, Hanyang University, b a saturate the numerical bootstrap bounds of CFTs with Open Access Article funded by SCOAP Abstract: Junchen Rong Keywords: ArXiv ePrint: Scalar CFTs and their large N limits kinks whose locations approach (∆ by the operator dimensionsUsing the of numerical either bootstrap the to study Ising CFTs model with or the Le cubic anisotropic fixed point withbootstrap three work, spin components. we As known discover CFTs another remains series a of mystery. kinks whose We also id show that “minimal mode JHEP09(2018)103 ] 3 4 8 8 ]. 1 3 8 5 11 12 15 18 3 19 14 15 , 4 ] and 71 -expansion ]. In higher ǫ 6 s provided the theory, can be used ]. There has been a 4 7 φ neither the expansion (see [ lly based on certain per- r example, the numerical N k of [ dimensions ed matter systems to extract l field theories. It has played dimensions inimal models” [ ǫ ks on the conformal bootstrap algebra ǫ 2 formal bootstrap program [ 3 − − ns and critical phenomena. A large W ] or the large 3 theory in 4 theory in 6 ]. For the perturbative regions, the bootstrap 4 3 symmetry – 1 – φ φ 74 2 – Z 72 Ising model ⊗ or bigger symmetry N symmetry breaking, which includes the Ising model [ → -expansion [ N S ǫ 2 S Z ]. symmetry ]. The simplest example among them, the 2 Lee-Yang singularity 70 , N – 1 S 8 → works very well. For the three dimensional Ising model, it ha N The numerical bootstrap is applicable even in regions where The critical exponents calculated in field theories are usua A.1 3-loop renormalizationA.2 of generic 3-loop renormalization of generic 3.2 Other bootstrap3.3 results: unidentified Other kinks bootstrap results: “minimal models” of 2.1 “Restricted Potts2.2 model” The spectrum2.3 of (de)coupled CFTs Potts model 3.1 Bootstraping CFTs with most precise critical exponents so far [ nor large and related topics is [ result was also shown to agree with the field theory result. Fo turbation methods, suchreferences as therein). the As a non-perturbativehas method, proven the to con be usefulan in important studying role two dimensional in conforma thedimensions, classification significant of progress two dimensional wasrevival “m of made this in program the since then. seminal An wor incomplete list of wor to study phase transitions with Scalar field theories are useful innumber studying of phase transitio these models have beenthe applied critical to exponents different [ condens 1 Introduction A Renormalization of scalar field theory B Bootstrap with 4 Discussion 3 Bootstraping CFTs with Contents 1 Introduction 2 Renormalization of scalar theories JHEP09(2018)103 ] 2 ]. N . Z 76 78 g , (1.2) (1.1) global 75 d point 2 , we look Z ngularity. 2 . The name ⊗ dimensions. ]. The Borel coupling N ǫ N e critical Ising S ), on the other 71 S − 1.1 -state Potts model ith N . In accord with this 2 ) points whose large ) vector model [ i ]. In section N exist a series of kinks, φ N i 79 ( econd model that we will φ f order one. The specific as used as an intermediate ( O d by the scaling dimensions ensions, quartic interactions 2 ough symmetry group is therefore 8 g teraction in 4 ter numerical bootstrap study. en that the scaling dimensions ed point of ( , it also preserves an extra m limit of the Potts model [ (3) invariant Heinsberg model, k + N ]. This model was also recently tical φ l O = 4, we are able to observe the j S φ 79 φ k i N φ φ j φ ijk i d φ 6 g ]. symmetry is broken and one gets the model klm + ) can be neglected. The 2 77 d is invariant under the action of i Z φ 1 dimensional representation of the symmetric ] with three component spins. Interestingly, the – 2 – 1.1 µ ijm ijk ∂ 84 − d d i – in the critical Ising model. This confirms the large 1 8 φ , the g N ǫ µ k 81 ∂ φ + = j 1 2 do not agree with the i calculation based on the scalar theories [ φ φ n dimensions, given by i -expansion. Setting terms in ( S = -expansion series for anomalous dimensions and comparing ǫ µ φ N ǫ ǫ 4 ∂ 2 L i φ symmetry of the original Potts model. Suppose we turn on ijk approach a point given by the scaling dimension of the spin φ − d µ 1 N 3! ∂ and ∆ N S 2 1 φ = L limit whose operator spectrum is fixed by the Lee-Yang edge si limit of the ) is known to have two extra fixed points other than the free fixe -expansion series for scaling dimensions of operators in th N N ǫ 1.1 theory in 6 transform in the ]. Its Lagrangian is given by 3 i φ ) invariant fixed point where symmetry is enhanced [ theory is known to have a non-unitary fixed point at imaginary 80 φ and the thermal operator 3 N ( φ . The totally symmetric tensor σ O N S The model ( We then employ the numerical bootstrap method to study CFTs w We will study scalar field theories admitting conformal fixed behaviour predicted by the fact, this symmetry under which all theslighter scalars bigger change than their the signs. Its the a trilinear interaction “restricted Potts model” is due to the fact that besides scaling dimensions of ∆ hand, has a large and the at their operator spectrum to setTaking up the the large background for the la We would like to first consider a scalar theory with quartic in revisited in [ agrees perfectly with theresummation large of It can also be used toof study scalars the are Potts model. irrelevant. The In close to six dim which describes the continuumconsider limit is of a the Potts model. The s The scalars is known to undergo a first order phase transition for large en with the corresponding series inof the all Ising the model, it operators canof that be operators we se have in studied the approach critical a Ising limit model. fixe The non-unitary fix The model was referred tostep as to a study “restricted the Potts model”, continuum and limit w of the Potts model [ model also agrees with the bootstrap result [ group models that we will study are closely related to the continuu behaviour is controlled by another CFT with central charge o bootstrap for the scaling dimensions of operators in the cri whose locations at large operator symmetry. We observe that in three dimensions, there indeed famous cubic anisotropic fixed point [ N JHEP09(2018)103 and (2.2) (2.1) (2.4) (2.3) even), as , which is ...N symmetry = 3 as an 2 A 3 Z S T. S denotes N . The scaling = 1 2 ⊕ . The quadratic P α 2 hypertetrahedron A . ⊕ algebra saturate the and 1) . Take . , S with 3 1 3 n P nts α i W ⊕ → e , , , to identify them with any he O(n) group can be de- = (0 and table n ootstrap with 3 1 can be constructed explicitly ic tensor representation, and 6= 0 6= 0 6= 0 (they are clearly ection ⊗ = 1 e , 2 2 2 ijk N , e N d S . g n = 0 ′ ! 1. The O(n) invariant fixed point consists of all SO(2) rotations that , g , g , g 2 T 2 1 g , 3 − α k ⊕ − S = = 0 6= 0 6= 0 e , n N α j 3 1 1 1 1 e 2 g g g g ⊕ √ α i = ) are summarised in appendix e A − n 1.1 α

: : ⊕ X – 3 – 1 2 1 dimensional space. From a group theory point S = P P = Ising model 2 − → , the totally symmetric tensor is defined as ijk N n → α i d ]. In principle, it is easy to extend the result to three , e e ]. As a byproduct of our numerical bootstrap study, we ), the invariant tensor ⊗ 86 ! n 85 free theory : 2 1 1.1 − , ]. We will however only focus on the two loop results. 3 2 80 √ critical O(n) point :

= 1 e ]. It is possible to define a set of “vielbeins” is a subgroup of O(n), with -dimensional representation is reducible, 79 2 N 1 through a recursion relation. These vielbeins tell us how a Z − ⊗ vertices can be embedded in N S ...N N The beta functions of this model have, in total, four fixed poi = 1 numerical bootstrap bound. These results are presented in s keep this triangle invariant. Using in two dimensions, we also show that the “minimal models” of The details of the two loop calculation of ( form an equilateral triangle. The symmetric group according to [ with operators fall into various irreps of the symmetry group based on the generalloops formula using in the [ result of [ 2 Renormalization of scalar2.1 theories “Restricted Potts model” also discover a seriesCFTs of with new Lagrangian kinks. descriptions. By We are, doing the however, numerical not b able of view, the Remember that the productcomposed of into two three vector irreducible representations representations of followin t For the restricted Potts model ( i as opposed to the prediction in [ example, the three vielbeins the O(n) singlet representation, A denotes the antisymmetr dimensions of the operators we have studied are given in tabl since is also present. We will focus on the two extra new fixed points JHEP09(2018)103 ing (2.5) . . 2 1 P P t t . It is interesting ′ T ] on cubic anisotropic and 88 ) n , nd try to better understand Ising ǫ 87 of low lying operators in the ∆ ], and certain critical exponents l, , because of the existence of the − 91 Ising ǫ Ising ǫ Ising σ N = Ising σ – S D ∆ ∆ ∆ k ( ∆ →∞ ′ 89 →∞ n Ising σ Ising ǫ Ising ǫ Ising ǫ ′ − × × , = n Ising σ Ising ǫ Ising ǫ × × ∆ ∆ ∆ 2 ∆ ∆ 2 j ∆ ∆ D 2 ∆ ∆ 2 84 – = ) ) ) ) ) ) ) ) i 81 ) ) ) ) if otherwise. A.5 A.6 A.9 A.9 A.7 A.8 A.5 A.6 A.10 A.10 A.7 A.8 – 4 – ∆ ( ( ( ( ( ( ∆ ( ( ( ( ( ( , , behaviour can already be partially inferred from 1 0 2 2  Z − + + + + + Z − + + + + + N = ijkl ′ ]. It preserves a symmetry group which is the generalized ′ Q S, 1st S, 2st n T S S, 1st S, 2st n T S 92 n ] and the much earlier work of [ n representation of O(n) branches into ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈ 80 ∈ T 4 4 2 2 2 4 4 2 2 2 φ φ φ φ Operator φ φ theory was studied, whereby the model was obtained by replac φ φ φ φ Operator φ φ 4 φ , the ) with . Scaling dimensions of low-lying operators at the fixed poin . Scaling dimensions of low-lying operators at the fixed poin ijk 1.1 d in ( limit. Table 2 Table 1 ], another N klm limit can be expressed in terms of the Ising model spectrum. d 80 N ijm In [ d are known up to six loops [ systems. We will explainthe this large point in the present section, a This model has a long history of being studied [ the to observe that forlarge both fixed points, the scaling dimensions 2.2 The spectrum of (de)coupledWe CFTs should mention that the large T’ denotes the symmetric traceless representation. For invariant tensor combining the result of [ JHEP09(2018)103 , N m G S (2.8) (2.9) (2.6) (2.7) (2.10) .  ) j 2 lculations of O 2 ∂ ( i 1 acts independently O , , . ,..., i ” stands for wreath 3 enumerates the CFT ≀ D N o invariant under O . i 6= 0 6= 0 = 0 G as an example, one can ) − alised. The summation , i 2 i 2 2 2 2 2 X O up O j 2 ∆ µ + 2 = 0 ly for them to have definite on. For operators with non- ∂ O N 2 , g , g , g 2 1 ( ) ariant under the action of g . Suppose the composite CFT j 1 i decoupled copies of this CFT 1 √ e index and -fold product of CFTs. Suppose N 2∆ O O 6= 0 = 0 6= 0 = 1 1 2 = S N N 1 1 1 1 ≀ O + ∂ 3 ∆ g g g g ( . j 2 j 2 G O are scalars, we can also construct the − D O . The symbol “ O j 2 2 µ i 1 N − , ∂ O O , then O i 2 i 1 ) G 1 ), it also has four fixed points ,.... i G j 1 O O 3 ∆ 6= + 2 ⊗ O µ i X and 1.1 i µ 1 ∂ ,O N X ∂ 1 2 − N ( – 5 – S O 2∆ 2 . N group interchanges them. Let’s first consider only − 1 ,O ∆  = 1 1 √ 2 free theory : j j N . Like ( O 6= 6= N S N = i i limit (see section 5.1.2), and therefore the two models → ∞ X X N 2 S √ 2 Z ≀ N N N N O ⊗ G − − critical O(N) point : 1 1 N , 2 2 i 1 S invariance. If N N O ensures that the composite operator is made of operators fro cubic anisotropic point : √ √ N j i ) = S X = = 6= ,N i N -fold product of Ising Models : 1 =1 =0 (2 ,l ,l ] that certain numbers that appear in the renormalization ca √ N S =0 =1 80 invariant operators of the following form = n n ] ] 1 2 2 N O S O O 1 1 pairs ensures O O j [ .... [ -fold product then has the following operators which are als 6= N i It is straightforward to work out the spectrum of the easily make copies. Picking two operators from the same copy, take where space-time indices are suppressed for simplicity. Th following operators The coefficient in frontzero of spin, the operators the is space-timespin. due indices to The need normalizati condition to be arranged proper two different copies ofover the CFTs, so that it would not be renorm It was shown in [ symmetric group approach the same limit at large permutations, both models have the same large The has the following conformal primary operators which are inv on each copy of the CFTs, while the operators which are invariant under the whole group a certain CFT preserves the symmetry group preserve the symmetry group product, which can be viewed as a shorthand notation. The gro JHEP09(2018)103 . . l 2 P + O n (2.12) (2.11) 2∆ 23 x +2 . . 1 2 i O N conformal come from S 2∆ 14 al primaries j +∆ x 1 l O = functions. It is j OOOO h 6= l X and 1 = N i i at the fixed point 2 + O 1 1 erators with spin-0 and N  (after projecting out the ppearing anywhere in the re given by the four-point O i ors priate derivative structure + ǫ “generalized free fields”, as ee massive scalar with AdS 2∆ 23 Ising ǫ O . x ged so as to ensure that they 1 o-point functions. The leading ∆ 2∆ 24 O ′ k at the 4-pt function consisting x Ising ǫ Ising ǫ × 1 2∆ 14 . O ing double trace operators made of i x 2∆ 13 in the singlet representation of + x i l ) ∆ = ∆ ∆ = 2 ∆ = ∆ O 4 = j x OOOO 2∆ 24 ( 6= l h k x X 1 O = O i ) , for double trace operators in the AdS/CFT j 3 2 2∆ 13 ǫ – 6 – x 1 i x ( N n,l ǫ ] comes from a different copy. Its contribution to k , ]. The sub-leading behaviour receives contributions j , 2 i + i ′ l + O 6= ǫ ǫ i ) O 95 i O O i i 1 2 O , P x O P 2∆ 34 ( P 93 2∆ 34 j x N x 1 N 2 and 1 N )[ 1 O 1 O O in the second line makes sure that √ − ) √ k 2 1 1 OOOO l 2∆ 12 i 2∆ 12 h x N O ) Ising ǫ x l ( x 2 4 i = = l ∆ √ x k O = ( k − = h   k j X O 6= D 1 ) = j N 6= X ( i 3 = X x 2 i,j,k,l i j invariant). These operators will be important for our later − ( 1 2 2 N Ising ǫ 1 O 1 1 = N ) N N expansion clearly factorises into disconnected two-point  ∆ + S 2 i ]. The scaling dimensions of these operators are simply ∆ = ∆ ]. One can also follow it to construct “double trace” conform − x 1 = = = N ( 96 95 = . The leading operator in the n-channel is simply O – ) 2 2 1 93 L x 2 ( which is m i Specialising to the Ising model, the first three S-channel op hO ǫ “S-channel operators” is short for operators transforming 1 i lowest scaling dimensions are the four-point function therefore reducesterm to in a the product of tw the same CFT copy, while See table bootstrap study. mass from both a disconnectedfunction piece of and the a composite connected CFT, piece, as which denoted a by equivalent to the boundary four-point function given by a fr with higher spin and twist.literature, Even a though similar we procedure are shouldoperators not exist with aware for of construct non-zero it spins. a of It identical is scalar also operators, interesting to loo context [ They have the same scaling dimension as the S-channel operat P We have borrowed the notation [ The condition The derivatives acting onare the conformal operators primaries. areis carefully The arran exactly procedure the of samestudied choosing as in an constructing [ appro conformal primaries for JHEP09(2018)103 ) ). 1 N . ], ( 1.1  O 82 (2.13) (2.16) (2.15) (2.14) (2.17) 1 = 2, 1 N of ( −  2 D P O + and ]. Their critical -fold product of ) vector models. 1 l coupling, which 88 N P N Ising [ α Ising α turning on the double − t change its mass since 1 , . = ) corresponds to the change ]  ǫ Ising ) singlet operator is given by α ). The IR fixed point can be 97 1 ∆ N N e fixed point , ppear in the spectrum. Explicit 2.17 −  ators” and so on. The only mod- 88 Ising ǫ approaches the O and D , . ∆ 2 ]. Their four-point functions are ex- = 82 P → − , 2 71 Ising ǫ S ). The spectrum of S-channel operators  is found to be accompanied by a “double D i ∈ ∆ ( ǫ 2 1 approaches the IR cubic anisotropic fixed N ), the effective Lagrangian becomes [ φ , g − × 2.11 i 1  Ising ǫ  X 2.5 P D ∆ O 1 – 7 – N N 1 − ) point, its dimension is given by ∆ = → +  √ and ∆ N D O ]. The replacement ( -fold product of Ising models and also fall into the copies of the Ising model action plus certain Ising ǫ Ising , + N α N ∆ ) vector model [ 102 Ising limit as in ( – ν − N Ising σ Ising N 98 1 g ∆ . What’s more, operators like = = 1 . This type of flow at large N was studied in the early days of → j 1 ǫ φ g i ǫ ∆ , ν j an operator with 6= ∆). The exact same phenomenon happens for O(  i 1 1 N − P  D O ∆( + − , the renormalization is clearly dominated by the Ising mode ] = 1, while at the critical O( i N = φ Ising corrections. i 2 η φ L 1 N i = At the cubic anisotropic fixed point of ( The IR fixed points fit into the class studied by Victor Emery in P 2 η AdS The action of the model becomes At the free theory limit, the scaling dimension of the first O( self average as in the critical O( ∆[ corrections. It can be shown that this is also true for both th the AdS/CFT correspondence [ of boundary conditionsM for the scalar field in AdS, and does no should be exactly the same as the pected to factorise in the large- exponents are related to Ising critical exponents by [ At large agreeing exactly with table categories of “single trace operators”,ification “double that trace one oper needs to make is the replacement, Translated into operator dimensions, this means trace” operator with the scaling dimension 2 plus reached from the UVtrace fixed deformation point of the decoupled Ising model by Notice that in table Ising models, which lives in the UV, while explains why the scaling dimensions ofcomparison the of Ising operators their a operator spectrum shows that point. JHEP09(2018)103 ) N for S 1.2 (3.2) (3.1) -state (2.19) (2.18) A.2 N ven by the . i ]. The three loop theory. ) 3 4 φ x 105 ( (see appendix l , φ that appear in the OPE 3 = 1 limit of the behaviour of the scalar ) tarity on the other hand 3 104 at are invariant under x N roduct expansion (OPE) is ( ays of computing four-point N y CFT. It can also be shown people have been using ( k able sed on crossing symmetry and ment φ lem [ ) . 2 . , “double trace operators” and so x ( Lee-Yang φ  j φ ∆ 1 ] that the ) N 1 →∞ 3 Lee-Yang φ − O, studied in the previous section can be  n Lee-Yang φ Lee-Yang φ x ( 103 ∆ 1 i ∆ ∆ D ∆ O O P φ X − h + symmetry ) ) ) ∼ D = 2 j – 8 – i Z to be real. φ ) → 4 A.12 A.12 A.12 or bigger symmetry 3 × ∆ ( ( ( ⊗ x O -state Potts models. The theory has a non-unitary is given by i ( Lee-Yang 2 limit, it is clear that the scaling dimensions of opera- l N g φ N N O φ N 1 S ) S = O 3 Lee-Yang φ λ x S S g ( n → ∞ ∆ k ∈ ∈ ∈ φ 3 2 N ) 2 φ Operator φ φ x ( j Lee-Yang singularity . It was pointed out in [ φ ) N 1 → x ( -expansion result shows that the single trace spectrum is gi i . Scaling dimensions for continuum N-state Potts from ǫ φ h ), the continuum limit of Table 3 1.2 By assuming certain conditions on the spectrum of operators unitarity. Crossing symmetry meansfunctions that should the lead following to two equivalent w results observed in numerical bootstrap. Conformal bootstrap is ba In this section, we show that the fixed point 3.1 Bootstraping CFTs with 3 Bootstraping CFTs with Before closing thismodel ( section, we briefly mention the large spectrum of the Lee-Yang edge singularity, with the replace that the coupling constant at large tors are fixed by the spectrum of the Lee-Yang edge singularit By the same argument as in the previous section, operators th performed. This isrequires all true the for OPE any coefficients conformal field theories. Uni The lines connecting the operators denote how the operator p 2.3 Potts model should fall into the categorieson. of “single The trace explicit operators” Potts model gives the percolation model. Based on this fact, to calculate the criticalrenormalization exponents for of operator the dimensionsmore percolation details). is prob By summarised taking in the t fixed point at generic JHEP09(2018)103 e. re m: ) for both Ising ǫ . This confirms ∆ ns chosen to be 1 , sion of Ising model 0.540 first n-channel scalar Ising σ dication of the existence 0.535 ave the details of how this 0.530 , increases, the location of the kink φ have scaling dimensions greater than N ) plane and the result is presented in j region). Yellow, red, green and blue curves 0.525 n φ φ ) should approach (∆ ∆ × ϕ , n i φ Δ ∆ φ , 0.520 φ – 9 – XX . The conditions that we have assumed for the B -channel has scaling dimension greater than or equal -channel has scaling dimension greater than or equal S 0.515 n , one can then check whether such an assumption is ), as denoted by the black cross in figure 2 φφO symmetry (small ∆ has scaling dimension ∆ Ising ǫ λ 0.510 2 i ∆ Z φ , ⊗ , a clear kink can be observed in the numerical bootstrap curv . The region above the curves is excluded, which means there a Ising σ N , it is clear that (∆ } 0.505 . We can introduce one extra condition on the assumed spectru S 2 N ). The bounds are obtained at Λ = 19. 2 50 P ,

100 respectively. The black cross denotes the scaling dimen Ising ǫ 1.3 1.2 1.1 1.0 1.5 1.4

and

, n 49 Δ ∆ 1. 1 and . , 10 , 1 6 , P } ∪ { + 0 Ising σ , n n 25 = 4 . Numerical bootstrap bound on the scaling dimensions of the . The result is obtained by setting Λ = 19, with the range of spi N 1 ,... to ∆ The first spin-0 operator in the the external operator the first spin-0 operator in the to ∆ all the other operators that appear in or equal to the unitarity bound. 1 For large enough From table • • • • ∈ { and testing the positivity of operators (∆ consistent with unitarity andmethod crossing was symmetry. implemented in We appendix will le The appearance ofof kinks a in conformal numerical field bootstrap theory. is More a interestingly, strong as in approaches the point (∆ fixed points no unitary CFTs with the assumed spectrum. are for the prediction from the previous section. l spectrum are: Figure 1 operators in CFTs with figure We have scanned a certain region of the (∆ JHEP09(2018)103 . is 1 ≈ P 2 ], as Z = 4 is Ising ǫ ⊗ 107 N 4 l [ S = 100, since itself, it is not N ) with 1 1.1 0.540 first n-channel scalar We should emphasis , while preserving 2 2 P 0.535 r not, since there is no clear ). -channel gap to saturate the 5179(2). Just like for the 3D er increased, the change of the ) plane, it not clear which one . 2 he case. n n ]. Our non-perturbative result ariant Heisenberg model using bove condition is not imposed. error bars are the estimation of ∆ 85 = 0 0.530 , 495(6)) as our prediction for the . = 3. From figure φ φ 1 , N region). The dashed line is calculated at 0.525 φ -channel operator has scaling dimension -channel operator has dimension ∆ S S ϕ 5179(2) ) with . Δ is located around the kink. At 1 0.520 2.5 P – 10 – (3) invariant Heisenberg model shows that their ]. O ) = (0 S curve has no significant change after introducing this ]. The two groups clearly have the same order as 108 0.515 ∆ 79 , [ 100 , the leading φ , the leading S 1 3 2 2 P P symmetry (small ∆ Z ]. An analysis of the six loop perturbative calculation in 0.510 2 ⊗ Z 106 3 ⊗ S 4 S = 0.505 2 Z , this assumption would clearly exclude point

≀ 1.3 1.2 1.1 1.0 1.5 1.4

n . The functional is obtained by setting the N Δ = 48. This means that the “restricted Potts model” ( 3 3 ], we can treat (∆ 5874. While for S 3 = 4 curve. One can study the corresponding extremal functiona . computed at Λ = 27 (which is shown in figure 2 1 73 = 4 case deserves some special attention. The symmetry group n N × ≈ N . Numerical bootstrap bound on the scaling dimensions of the ) should agree with each other to high precision [ S Ising ǫ ∆ 2 = 3! ∆ The Clearly the functional is discontinuous at around ∆ , This difference was noticed recently in [ φ 2 − × 4126. We have checked that the . the two fixed points are very close to each other in the (∆ isomorphic to of them sits closer to the original bootstrap curve when the a the cubic anisotropic model and in the from numerical bootstrap, however, shows this might not be t 4! clear whether there is akink CFT in saturating the the bootstrap boundshown o in figure bound on ∆ equivalent to the cubic anisotropic model ( (∆ Figure 2 condition. This shows that the fixed point Ising model [ the Monte Carlo method [ scaling dimensions of the correspondingthe operators. operators’ The dimensions red in the three dimensional O(3) inv operators in CFTs with Notice that for the fixed point 1 At large enough Λ = 19, whilebounds the is solid no line long visible is by calculated naked at eyes. Λ = 27. If Λ is furth D JHEP09(2018)103 ]. Unlike explicitly. n 109 ) is presented ondition, while n was close to the ∆ remal functional is , φ 0.540 φ 0.520 l charge and setting the , we can again observe ere ∆ N s assumption need to be d not survive when a gap 0.519 gian descriptions for them. 0.535 isotropic fixed point using unctional is caused by the nvariant CFTs [ urating the unitarity bound. an be found on the bootstrap ntains the energy momentum s fact is tested by adding the checks for them to actually be 0.518 to the region with much higher are the first and second operators scaling dimension ∆ 0.530 s. odel obtained using the Monte Carlo 0.517 error bars are the scaling dimensions of 0.516 0.525 ϕ 0.515 Δ 1.60 1.55 1.50 1.45 1.40 – 11 – computed at Λ = 27, while the extremal functional 0.520 n -channel has scaling dimension greater than or equal S . Surprisingly, for large enough 0.515 4 0.510

1.0 0.8 0.6 0.4 1.6 1.4 1.2 S Δ = 10 curve as an example, the allowed region for (∆ N ]. We have checked that suppose one increase Λ at which the ext . The solid line corresponds to the result without the above c 05, . 5 . The extremal functional obtained by minimizing the centra 106 to 3 the first spin-2 operator in the • . This is presented in figure φ -channel gap to saturate the bound on ∆ unitarity bound. It is straight∆ forward to extend the result following condition on the assumed spectrum Instead, we will show that theseCFTs. kinks Any pass full-fledged some consistency conformaltensor field in theory its necessarily spectrum. co If There the should kinks be a weis spin-2 observe introduced operator correspond for sat to the actual spin-2 CFTs, operators they in shoul the S-channel. Thi the CFTs in the previous section, we are not able to find Lagran confirmed by measuring theeither critical Monte Carlo exponents simulation, especially of by the measuring the cubic an calculated, the change of dots is no long visible byhere naked we eye have assumedcubic that anisotropic the fixed discontinuity point in rather the than extremal an f unknown CFT. Thi 3.2 Other bootstrapThe results: study unidentified in kinks the previous section was focused on the region wh Taking the Figure 3 corresponding operators in theMethod O(3) [ invariant Heisenberg m itself is calculated atrespectively Λ appearing = in 23. the extremal The functional. darker yellow The and red blue dots in figure n some kinks in the numericalcurve bootstrap obtained curve. by bounding Similar the kinks S-channel c operators in O(N) i JHEP09(2018)103 symmetry. 2 n-2 gap, while Z ⊗ N £¤¦ 0 0£ S -channel is introduced. symmetry, simply by S N t the energy momentum S algebra rst n-channel scalar operator rst n-channel scalar operator 0.90 3 ¤§¦ £ 0 y, when the gap for the spin- W rator in the 0.85 ¤§0 £ 0 apply to CFTs with ϕ ϕ B Δ Δ region). Yellow, red, green and blue curves are for – 12 – φ ¤¥¦ £ 0 0.80 ¥0 ¤ £ 0 0.75 symmetry (large ∆ symmetry. The solid line corresponds to bounds without a spi 2 2 0.825 ¢ Z

Z

3.15 3.10 3.45 3.40 3.35 3.30 3.25 3.20 2.5 2.0 ¡3 3

n n ⊗ Δ Δ ⊗ N 10 S S 100 respectively. The curves are obtained at Λ = 23. , 10 . Numerical bootstrap bound on the scaling dimension of the fi . Numerical bootstrap bound on the scaling dimension of the fi , 6 , = 4 3.3 Other bootstrap results: “minimal models” of for the dashed line,2 the operator above is condition imposed, istensor is included. the present curve Clearl in moves the downward, spectrum. showing tha The crossing equations we derived in appendix the dashed shows the result when a small gap for the spin-2 ope N The method can also be easily generalized to study CFTs with Figure 5 in CFTs with Figure 4 in CFTs with changing the assumptions on the spectrum to be JHEP09(2018)103 , 3 3 ′ n W W algebra. algebra 3 3 . , the k n W W φ L ]. It contains ijk d ons of their 111 symmetry in two ∼ , while the second φ 3 j X S φ symmetry, therefore, econd n-channel scalar 3 × X al models” here means 0.4 i Z φ -2 operators X educible representations of dchikov in [ . We found that “minimal 6 , one gets the symmetric group X , 0.3 φ X group (which is not invariant under ) for CFTs with ′ n N have scaling dimensions greater than or ∆ S ϕ , j φ φ Δ X × 0.2 i ], saturate the unitarity bound. The – 13 – φ 110 -channel has scaling dimension ∆ which satisfy non-trivial commutation relations with n n X W would appear in its own OPE, i 0.1 has scaling dimension ∆ φ -channel has scaling dimension greater than or equal to ∆ i n φ ] that all these models have a global symmetry. The crosses correspond to minimal models with 110 3 S 0.0

is an invariant tensor of the

1.0 0.5 0.0 2.0 1.5 n ' Δ algebra, as classified in [ ijm 3 d . The central charges of these models and the scaling dimensi 2 W Z . Numerical bootstrap bound on the scaling dimension of the s ), the scalar operator ⊗ 2 3 Z spin-0 operator in the equal to the unitarity bound. all other operators that appear in the external operator the first spin-0 operator in the Z We have studied the allowed region of (∆ and among themselves. Like for the Virasoro algebra, “minim ⊗ • • • . It was shown in [ = 3 n N 3 S space-time dimensions. This result is presented in figure Notice since the fusion rules ofW the models consist of a finite number of irr L is an extension of the Virasoro algebra introduced by Zamolo algebra contains spin-3 operators taking into account complex conjugationS of complex scalars The first cross to the left is the 3-state Potts model. models” of the Virasoro algebra as a subalgebra. Besides the usual spin Figure 6 operator in CFTs with JHEP09(2018)103 , p N G ≤ (3.3) (3.5) (3.4) (3.6) ⊗ ′ , m N S ry group 3) would be , +  − m + 1) p 12 p 1, ] that minimal − 3( -algebras. 4(2 in the numerical = 4 special case, − 2 algebra also share calculation like in 1 p 2 112 W = )) N ′ 3 P etry group )) ≤  tc. A naive guess is N ′ m ints approaching the W ′ m ! n − + + m 1 3 1 1 ssible to replace the large ularity have the symmetry so be interesting. It is also f the ( n espectively in the large m vered in [ p (

]. For the um more carefully. The best lt to other ) effect receives contributions p isolating this fixed point using ootstrap bound for CFTs with − Φ  − ) ′ 114 /N ) proper large- ∆ , ′ n (1 corresponds to operators with n × − O 6 113 + n , , n = 2 76 ′ n + 1 , φ + 1)( 74 + 1)( p ∆ p  5 2 – 14 – 1) + (( = 3(( and ∆  − ′ n 12 p ∆ ) corrections to the operator dimensions and compare , ( p + 1) 3) /N 1 p − ( + 1) − (1 ] for a review). 1 p p p O  71 12 3( 2( CFTs that approach a CFT with symmetry group G. We leave = = 2 = are positive integers whose ranges are N p   p -expansion, a proper resummation is necessary. Finally, it C ǫ ! ! ′ m and 1 2 1 1 , we have shown that one can observe the fixed point ′ ′ n m

n , n 3.1 , while the CFTs that approach the Ising model have the symmet ′ Φ

 1 Φ ∆  ⊗ × ) vector model (see [ 4. The horizontal and vertical axis in figure . It is therefore natural to consider scalar models with symm ∆ N 2 N S m, n, m ≥ Z = 2 p In section ⊗ symmetry. It is interesting to observe that minimal models o φ limit by other CFTs such as the XY-model, Heisenberg model, e 2 N ∆ this for future investigation. as a candidate for large S the following. The CFTsgroup that approach the Lee-Yang edge sing N 4 Discussion We have shown(de)coupled that Ising there model existlimit. and two the series It Lee-Yang of would edge be conformal singularity interesting fixed r to po understand whether it is po the same feature. It would be interesting to extend this resu interesting to investigate the possibilitythe of O( performing a from all orders in the respectively. They satisfy models of the Virasoro algebraZ also saturate the numerical b them with our numerical bootstrap result. Since the bootstrap curve. Itway would to be do this interesting is to probablymixed study by correlator first bootstrap, its studying along spectr the the possibility lines of of [ desirable to try to extract the a further comparison with experiment or Monte Carlo would al irreducible representations are given by where which saturates the numerical bootstrap bound. It was disco and JHEP09(2018)103 – ), 1.1 116 (A.1) (A.2) rators , 96   3 2 3 2 , g g 7 7 94 spectrum of angian ( B A , N + + 93 2 2 2 2 g g 1 supported by ITP- 1 , one can define the g g . adislav Vaganov and 6 6 dimensions 2 ijk B A i ijk ijk d 1 ijk ijk ǫ i ch clearly contains oper- d d limit of decoupled CFTs. + + δ d d n in AdS [ 2 5 71 72 3 2 single trace operators with 2 − of “single trace operators”, , the large sor g T T T T T g N upported by HPC Cluster of e the necessary and sufficient it should be more similar to in this case. 2 1 2 1 g g 2.2 = = = = = 5 5 4 6 9 9 3 B A i i i i i 2 3 4 8 7 i i i i + + 2 2 7 6 ki i i i i 3 (de)coupled CFTs in the context of 3 d 1 1 d d d d 3 g g 4 5 9 9 i i i i i 4 4 1 3 3 6 8 N i i i i theory in 4 B ji A 1 1 4 4 i i i i d 4 ] 2 + + d d d d i 6 8 8 φ 1 2 2 2 factorization, any CFT with an Einstein-like 2 i i i 5 5 5 ii g g 105 i i ], we can calculate the scaling dimensions of op- 3 3 d 2 2 ki N i i – 15 – d B A 86 d d 4 as [ 7 7 i i i + + 3 3 3 } i i 2 ji 2 ]. However, since the CFT operators do not saturate 1 1 g g i i d 72 . 1 1 2 d d i 4 g g 6 6 ,T 120 1 i i 2 2 5 5 ii , 71 B A d ki ki d d ,T + + 119 4 4 5 i i 2 2 1 1 3 3 ], besides large g g ,T ji ji 1 1 3 d d 93 A B 2 2 , the operators that can be interpreted as “single trace” ope i i ,T   1 1 2 1 1 ii ii C C T 2.2 d d { + + 1 2 ǫg ǫg up to two loop order. For a scalar field theory given by the Lagr − − ǫ − = = 1 2 β β Before we close, let’s think about the large ]. As conjectured in [ the AdS/CFT correspondence.(de)coupled CFTs As naturally explained breaks“double down trace in operators” into section and the soconditions categories on. for It a is not CFT yet to clear what have ar a weakly coupled dual descriptio erators in 4 we get the beta function A Renormalization of scalar fieldA.1 theory 3-loop renormalization of generic Yi Pang forCAS. helpful The discussion numerical computations and inSKLTP/ITPCAS. comments. this work are The partially s work of NS is the unitarity bound, higher spin symmetry is clearly broken Acknowledgments We would like to thank Youjin Deng, Zhijin Li, Ziyang Meng, Vl are simply the S-channelators operators with of arbitrary the spin.Vasiliev’s component higher CFT, If spin theory whi the [ dual theory indeed exists, As shown in section local bulk dual descriptionspin must higher also than have 2. a This large is gap clearly for not all the case for the large 118 Using the general formula summarised in [ following constants Suppose a group preserves a totally symmetric invariant ten with the coefficient given by table JHEP09(2018)103 , 5 2 ijk
T T 2 d 2 2 3 (A.3) T π T
6 , +88 −
+128 ) 2 2 2 .... 3 n 2 T T g 3  2 + T
π 2 2 + 2 n T 3 3 +92 5 T T 3 2 T +512 1 + 1)) 2 T g 2 g 3) ( 96 n T 1 3 , 2 2 g − − 3 T ) 4 T T +30 2)( 2 3 2 3 5 n T 1 the above formulas. Using T 2 π T g T − 71 − 4( π + n nT 384 192 − ( + 2 function. 2 +96 2 2 32 − − + 6) 2 2 2 T 2 g β 2 +512 ( 2 T T 2 − T n T ) 2 2 1 3 T 2 + 1 5 3 T 2 g n g T T 3 T 2 2 7 π 3 3 g π 3 T 2 nT − + 1)( nT , T +56 1 288 T 2 +64 ) 256 n 2 g + 8 2 2 4 2 T 32 2 328 2 71 − 2 5 T T
− 168 g π + 2) + T 5 T − 3 3 − 5 T 3 + (6 − 3 − 2 T T 2 2))( n 3 T 3
nT 2 ( 2 π 2 T + 2 256 T nT 2 2 2 π +64 T − T 2 2 nT – 16 – π 3 240 g 4 +256 2 + 2 π π n 128 T 5 2) π T 64 − 2 1 , , 3 5 3 + T5 2 (( ], it is easy to calculate the constants that appear 3 − +164 +84 2 g − 2 2 T − T nT +64 2 T 2 +256 ( 2 2 3 3 T 3 2 3 3 − 256 2 g 2 79 3 T 2 71 2 n +768 +320 T T T 48 T ) 2 ( 86 3 T T nT nT 5 2 2 T 3 + 6 2 − 1 2 nT − T 1 + 1) + 1) T ) 4 − g T in [ 2 2 2 64 π 2 g also preserves a totally symmetric invariant tensor 2 3 3 176 nT nT (  2 1 π n +32 n n 120 184 π 3 3 T π ) 8 2 π T − g T 2 4 2 3 − 2 2 n n . Coefficients that appear in π N nT − − 2 5 128 3 ijk − g π 2 T + 2 T 2 1)( 2)( 2 2 T S 128 +4 d 2 2 2 384 1 3 T − 2 3 + 1) 2 1024 2 g nT T 2 192 1 +256 T − − 2 − T − nT nT 2 +96 T + 2 T 3 g n 2 2 T +32 + 3 2 2 20 1 − 3 2 n n 3 3 64 T 24 72 +88 g T ( ( 2 T 112 2 nT 3 T − − 3 2 2 2)( nT + 2) + 2 T − Table 4 − − 1 π 2 nT 3 2 T 2 − 3 n T 20 3 1 3 3 g 2 3 = = nT 2 n 2 3 3 π T − 2 2 T T ( 16 T π 3 π − nT π nT 2 3 T 2 2 2 π + 6) + n 2 nT 3 2 T T g n n + 2 ( nT ( T + 2) + T n 4 128 +2 2 88 112 3 ( 18 10 2 +384 +32 2 π 20 n T 2 − 76 +128 +96 2 +320 2 2 − − π ( 1 − − g 2 2 2 3 nT 2 2 − 2 2 T 2 2 2 − nT g 2 2 2 2 T ( 2 3 T nT g 16 ( 256 4 2 − 2 5 T T 4 2 2 nT n nT +2 2 2 T nT nT T 2 2 2 π π π 3 nT n n 2 = = = = π π π 16 5 44 9 5 32 ′ ), they are φ n S − − − − 256 − 0 128 64 16 192 0 nT +16 − 16 T γ ∈ ∈ ∈ 2 2 The anomalous dimensions are given by 2 φ φ A.1 φ 4 5 6 7 1 2 3 6 7 2 3 4 5 1 γ γ γ B B B B C A A B B B A A A A A in ( they fall into thethe type explicit of construction models of that can be calculated using Since the symmetric group JHEP09(2018)103 5 3 n (A.6) (A.7) (A.5) (A.4) n ). The ... 2.3 + + 3200  6 ... + 23565 n ... 4 ... + n + + , 379  37176 2
2 .... ǫ ǫ − − 8724 ) to get the spectrum. 32

7 + n n − − . 2 12424 A.3 ǫ 5
19 1109
− n  n − + 1) + 1741 3 use ( n n n 2 + 22 3 + 84478 ... 3 ... 2 n + 8) + 2058 + n + 1) + 11 1616 + 6 n 45 + 8) + 3) 2 + 3) + 29( ..., 5 ǫ n + 34622 n 2 − n n − 2 + 2614 ǫ n
2 5 + − 6( 2 2
2 n 293 82423 2 n 2 5)( n n 2 5)( − 3 − ǫ ... n ǫ + 1) 9 − 3 2 − + 88 2 2 −
..., − , n + 318 n 7 + 88 n 3 n n 4 5) ( n + 31 n 38823 9 n 2 + 8) + n 9( − + 7 ǫ + 672 n 3 + 8) + 3) + 1) − 2 − 2 n = 0, we find the four fixed points in ( 19 3 3 n n ǫ − – 17 – 108 ( n n n n 162( 2 + 18 n + 1)  9 n 2( 162( ( 3 + 18 β 3 2 3 2( n + 8 − + 8) + 44) 2 ( n 6 + 2) 182 = 2 4 108( n
+ 45287 n 162( n − + 1) + n n n 1 + 208 + 8) 4 + 8) 6 − − ( 4( β ǫ + n 4 n n + 1) 4 n + 1 n (
+ n ǫ 5 n + + + 2 n + ǫ 2(
6 ǫ ǫ ǫ 19 2 2 2 ǫ − 19 2) + 8)  + 2)(13 2)( 2 + 11 + 8) 1 15310  2 − − − n + 9 − n ǫ n 1 + 5 + 1) − n − n − ( − n n 6 5 ǫ ( ǫ n n − 5 ), solving − 5) ( ( ( + = 1 = 1 = 1 − − 2 − 2 ) + 8 1 2 − 2 2 + 9 n + 3) A.2 = = = 2 + 4) 2 P φ P φ + 8 + 7) ǫ N n n n n ( n + 8 n n 6 5 3 ( n ∆ ∆ n n 3 φ O 71 72  T 3 ( ( 2 n 3( 2( 2 T T  ∆ 162( − − − − − + = 2 + = 2 = 2 = 2 = 2 = 2 + = 2 ′ ) ) S S S T ) n n n N N ∈ ∈ ∈ ∈ ∈ ∈ ( N ( ∈ 1 2 2 2 2 2 ( 1 2 2 2 2 P φ P φ φ O φ O , we have P φ P φ φ O φ ∆ ∆ ∆ ∆ ∆ ∆ ∆ For the quadratic operator in the S-channel, we have free fixed point is not renormalised.For For other points, we can Plugging them into ( For the quadratic operator in the n-channel, we have For the quadratic operator in the T’-channel, we have JHEP09(2018)103 ]. ... 121 (A.9) (A.8) , (A.11) (A.10) ... + ]: 2 2 3 +  104 6 n n ...... 2 n ǫ + +
.... 2  17709 ǫ + 6 + 9239
n − + 3583  + 110744 4 7 n 17 n , but rather focused + 1741 n n dimensions n N − 3 ǫ 5 12424 + 8947 388 2938 2 n singularity. The authors n − was studied by [ − − + 3) 262778 − + 864 ǫ n 8 5 n 2 ... 2 n − n here for comparison [ ted using the eigenvalue of the n ], where they have also studied + 273 2 − + 5)( 19 4 + 7) n 2  n ..., + 13127 − 13402 + 23259 122 ǫ n 3 ( , + 548 2
= 6 n + + 21242 3 − + 102 ..., n 6 2 2 3 1920 n D ǫ  105 + n + 8) n 2 162( .... − + 3) 57 theory in 6 n 2 19 + 278375 3 162 5 ǫ n + + 1139 − 3 108 3 n + 3430 2 + 8) + 7) 7 n − n φ + 17917 ǫ + 112 3 5)( + n 2 n ǫ n − 4 n 3 ( 5 n ǫ 17 2 3 27 2 − n 2 – 18 – theory in  3 ǫ − n +7572 3 2 602 − − − + 855 3 − ǫ 2  174366 φ 2 n − + 3) 1 81( 1) n − 4 ǫ + 11) ( + 8) n = 4 = 1 = 2 − -state Potts model with generic − 2 − n 4
ǫ ǫ n n N n ǫ n 6 5 27(  ( ′ + 165 + 8) Ising ǫ Ising σ Ising ǫ + 11) ( 1 − − + 8) + 11 3 + 3) n ∆ ∆ ∆ + 167 2 n 2 + 7) − 2 n n 5 n 6 n n 5 n ǫ 5 n 6 2 n 17 − − 3( + 71463 2( − − 5) ( 2 5) ( 2 19 5 + 3) 2 2 2 n n n − − n − n n − 2 3 ( n n 3( 3 ( 4 27 ( P , the final result turns out to be = 4 = 4 + 1   81( 19740 − P 162( + − 1st 2nd + S, = 4 = 4 + S, ∈ = 2 + = 2 + ∈ 1 4 . For 1 4 P φ ′ ′ i j 1st P φ 2nd T T ∆ S, ∂β ∂λ ∆ ∈ ∈ S, ∈ 1 2 2 2 2 4 P φ P φ ∈ P φ 2 4 ∆ ∆ P φ ∆ ∆ A.2 3-loop renormalization of generic The four loop result wasthe obtained more renormalization recently of in [ thedid Potts not model present the and result the for Lee-Yang the edge while for the point Three loop renormalization of generic The scaling dimension of thematrix quartic operator can be calcula It is useful to record the renormalization of the Ising model JHEP09(2018)103 ,  (3) (B.1) (B.2) (A.14) (A.12) (A.13) nζ 4 332009 n 4 + 2. n − n φ 2265408 3 (3) n ζ − nvenience, we will = ∆ , 2 66665 ) ], one can easily get 2 4 n φ ǫ − ( . (3) + 36755 105 ) ζ O 4 4 (3) + 300903 (3) (3) ǫ ζ + ζ ( n ingularity for comparison, 4 2 nζ kl + 1614816 3 O n (3) + 68025 n ǫ δ n ζ ij + 3  2 , δ 2 ǫ 3 38880 n ǫ 2 ǫ

(3) + 1123920 829440 = 1 4  3 , ζ 8375 2 ), one can also define the following + 28512 − α l n 59049 + 1) 7) 72 . 653 e n 5 2 T , since it is the conformal descendant n  + 206 + 1467072 129600 α k − − 2.2 2 n − n ( 3 e , 6561 3 φ , its dimension is ∆ ǫ = n 3 36755 n − α j n 2 n  − 3 (3) + 450911 e 4 7) φ (3) 71 + α i ζ n 7) 171 T e nζ 243 − ∼ ) into the formulas in [ 81(3 ijkl 210179 64 − − = n + 857 (3) α φ Q + 2 + 1187136  ζ X n – 19 – 2 5 − 8375 187238 A.4  27 n 3 (3) + 8375 2 n T + ǫ = ζ n − − n + 1 defined in ( 27(3 43 160 5 (3)
2 = (3) + 1097253 27(3 2887488 n n ζ ǫ 247  ζ 3 2 (3) + 163514 (3) ijkl (3) ijk − 1) 729 T 671 ζ ζ 86 n ζ = − d + Q 3 2 4 3 − 2 = 319602 − n n n − 2 n ǫ 15552 n 2 klm n ǫ − 7) 81 d T 43 9 − symmetry 2( 250 10 − 2  (3) 15552 ijm − + 1687392 N +466560 +580608 3 n − − ζ 207360 224126 d  ǫ + 544 , besides ǫ 3 n + S − − 5 5 α i 2 n 9(3 7) ǫ e n 3 7) 7) 11) 7) = 6 = 2 ǫ 4) − 3 φ − − − − result here. Plugging ( 1) 125 φ n − 1 n n ∆ n n − − ∆ 316224 1831190 n N 3(3 n − − 8( 2(5 3(3 2 ( 1 limit to study the percolation problem. For the reader’s co 243(3 243(3 − − + + → = 2 = 6 + = 4 N φ S S ∈ ∈ . As fixed by the equation of motion ∆ 3 2 φ φ φ ∆ ∆ Using the “vielbeins” It is not necessary to present the dimension of ∆ of They satisfy B Bootstrap with invariant tensor carrying four indices on the We also note heresetting the renormalization for the Lee-Yang edge s one gets note the generic JHEP09(2018)103 ]. T ) is 125 (B.6) (B.8) (B.5) (B.3) (B.4) (B.7) , u, v ( T, the O ! 124 ,l ) entation of ⊕ O ∆ A u, v g , ( ⊕ O . S ,l irreducible repre- klm i O ) d 4 → ∆ x g ( , due to the existence n ijm ′ l 2 O d ant tensors φ λ T ⊗ 2 ) I 3 nd the spectrum. An ana- o each CFT, on the other ⊕ x X O∈ ( eld theories, and four-point ), n + 1) ormal field theories, which is k n .

group. It can be checked that φ ( n 3 I ns was calculated in [ . ) ) I n O 2 ij ( N ijkl . 1)( x δ ′ S ( P 2 j . T − , I φ ) I ) n I ⊕ X ( ( 1 ijkl + 1) n x φ P ( − n 3 l i 2 ⊕ 2∆ 34 v n φ ” of the = = dim kl , k x h 1 1)( I A δ 1 φ v global symmetry can be written as jk ij ) ) = − ⊕ δ klm δ I I 2∆ 12 ( i N ( il ijkl d – 20 – nmkl n S 1 n x ) δ ( S P P 4 ) x − I ijm → = . ( = ) ( ijkl d l I . jl jl n i ( P 2 ijmn φ δ δ ) } ) jkl 4 P ⊗ − ik ik 3 d x δ δ x n ( + 1) ( l 1 1 2 2 ikl ,A k φ d n + 3 ] for the reason behind the spin choice. Also ) φ ′ − + , 3 ) n 2 x kl 1)( jk jk ,T ( 123 δ x δ δ ) group is further decomposed into n ( k + − ij il il j n φ δ δ δ ( ) φ n , n 1 ) 2 1 1 2 2 n ( O -dimensional representations can be decomposed as + 1 x 1 n ( x . See [ = = = = are two vectors carrying indices in the n-dimensional repres ( j i N φ ) ) ) i 2 ′ ∈ { φ ) S n A v h 1 T (1) ( ( ijkl ijkl ijkl I ( ijkl x P P P ( P i ” of stands for the dimension of the representation and φ I h I i 1 denotes operators with even(odd) spin and transforms in the v where ± . One can also define the following linear independent invari I A four-point function in CFTs with ijk , the tensor d N where dim these projectors satisfy the following relations lytical expression for the conformal block in even dimensio the conformal block which encodesuniversal all for the any kinematics CFTs.hand, of is conf The widely dynamical believed to information be specific encoded t in the OPE coefficients a sentation “ transforms in the irreducible representation “ The product of two Operator product expansions arefunctions should convergent not for depend conformal on fi how the OPE is preformed, so Compared with the product rule for the rotational group of S Suppose and Here representation of the JHEP09(2018)103 , , } } (B.9) + + (B.12) (B.10) (B.14) (B.11) (B.13) ′ ′ ,T ,T + + to study con- , n , n n. , + + 1 1 , } . − 3.1 − 1               ∈ { ∈ { 1 n − ,A 2 such that F I 2 2 0 − 0 F F − H H F + ′ 1 n − α ,T               + 1) 2 ]. . , + n ) ) esentation to be, ( , n ) = ) = 127 tional n . ) and + ) v, u v, u um in section 1 ( ( = u, v u, v ]. For the approximation of the ... ,l ,l ′ ( ( ... ∆ ∆ T ∈ { ) O O ) 9 10 ,l ,l 126 − G G + , , I O O φ φ φ φ n A must have at least one scalar operator 7 8 ( ( dim , , ∆ ∆ ∆ ∆ ∆ ∆ φ 5 6 ~ ~ v v V V u u ’s being positive. Therefore we conclude , , , 2 O 3 4 − + − , , λ with ) + ) φ φ 1) ∆ ∆ − = 1 = 2 , u u 2 u, v u, v , defined by l l – 21 – n ( ( ,l ( ( ( ,l ,l , ∆ n ∆ ∆ , , ) = 0 H        G G = 2 2 φ φ ) ) 2 ,I A ∆ ∆ symmetry. If such a functional can be found, then there u, v n n − − − symmetry and ∆ +2 v v ( and , 2 ] and references therein. The numerical computations in 2 n 1) 1) n 2 F O . For readers interested in the implementation of numerical D D , 2 ,l − F − − − Z        dim ,l − 2 = = Z n 2 2 ) n n − n ∆ O 126 n + + I n ⊗ n H ,l ,l ( 2( 2( ⊗ ( F ( D l l ∆ ∆ n 0 0 F ∆ ∆ F N ~ H V − N n, ≥ ≥ ≥ ≥ F S H S               = 2 φφO n λ I ) = ) = for ∆ for ∆ for ∆ for ∆ ∈ dim X ) to be satisfied with all the O u, v u, v , , , , , ( ( I are short for 0 0 0 0 , ) B.9 O O X ) + ,l ,l ′ + ≥ ≥ ≥ ≥ H O O T = 1 (1 ( ) ) = 1 ) ) ) ∆ ∆ ) ) ) ) ) 0 ,l , ~ ~ S V V I I + + − 0 ( ( 0 , ∆ ∆ and n ,l , A (1 ( ~ ~ ( 0 V V ∆ ∆ ( ( ~ dim V ~ F V ~ V ( ( α α Before proceeding, let’s recall the dimensions of each repr ( α α α The logic for numerical bootstrap is to look for a linear func is no way for ( this work were performedconformal using blocks, we the partially SDPB used package the [ code from JuliBoot [ formal field theories with that a unitary CFT with whose dimension is less than ∆ bootstrap, we refer them to [ From this equality we get the following crossing equations where Here This realises the conditions imposed on the operator spectr JHEP09(2018)103 ]. , (B.17) (B.16) (B.15) (B.18) ommons , SPIRE symmetry, IN     [ 3 Zh. Eksp. S F H , F − − ,     World Scientific } , − , , ) = (1973) 161 ,A ese projectors, we can for bootstrapping O(2) + ]. klm 76 u, v = 0, and d ( , n ) -theories ′ ) O + 4 redited. ,l T − 1 ijm φ ( ijkl . O A d SPIRE ]. ( P dimensions ∆ 9 8 . We need to search for a linear IN ∈ { ~ 3.3 V k I φ 99 = . 3 , SPIRE kl ijk δ Annals Phys. d IN     . , ij 0 ][ δ ∼ with H (1974) 9] [ 1 2 j 0 2 ≥ F φ ) − − , ) 39 . 0 , × + jl jl     φ n i ]. δ δ ( group (which is not invariant under SO(2)), the – 22 – ∆ φ ik ik Critical properties of ~ Tensor representations of conformal algebra and 3 V ) = 0 δ δ ( ) = S 1 1 2 2 α SPIRE u, v + − u, v ( , IN ( [ O Critical exponents in = 0, one can check that kl jk jk O ) ,l δ ′ δ δ ,l ) cond-mat/0012164 + ), which permits any use, distribution and reproduction in O [ Critical phenomena and renormalization group theory I T il il ij O n ( ]. ∆ δ δ δ ( ∆ ~ V 2 2 2 1 1 1 ~ ) plus one extra condition Sov. Phys. JETP V = = = 2 φφO , SPIRE ) ) ) (1972) 240 λ B.13 n S IN A I     ( ( ( ijkl ijkl ijkl ∈ (2002) 549 28 ]. However, when studying conformal field theories with CC-BY 4.0 P P 0 P Nonhamiltonian approach to conformal quantum field theory group, dim F X O H 3 75 (1974) 23 [ This article is distributed under the terms of the Creative C     I S 368 X 66 ) = satisfying ( α is an invariant tensor of the u, v would appear in its own OPE, ( i O ) φ ,l ijm + = 2, hence Teor. Fiz. Phys. Rev. Lett. conformally covariant operator product expansion Phys. Rept. Singapore, (2001) [ O d (1 ∆ n ~ V [4] S. Ferrara, A.F. Grillo and R. Gatto, [5] A.M. Polyakov, [1] A. Pelissetto and E. Vicari, [2] H. Kleinert and V. Schulte-Frohlinde, [3] K.G. Wilson and M.E. Fisher, functional These are exactlyinvariant the CFTs in same [ crossing equations that were used since Open Access. any medium, provided the original author(s) and source areReferences c with which are the samederive the projectors following as crossing equations for the SO(2) group. Using th Attribution License ( For scalars This is the numerical bootstrap program used in section JHEP09(2018)103 , , ] ) (2012) 091 , m ( ]. 12 , O ]. superconformal × , ) (2013) 161602 SPIRE n eal projective space ]. ( ]. , SPIRE = 2 JHEP IN [ O ]. , IN N 111 ][ ]. arXiv:1502.04124 , ]. [ ]. SPIRE SPIRE The analytic bootstrap and Deconstructing conformal The IN SPIRE IN ]. IN ][ s, ][ strap ]. ]. SPIRE ][ SPIRE uffin, SPIRE (1984) 333 rone, IN ]. 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