33D Ising Model: a view from the Conformal Bootstrap Island
Slava Rychkov a,ba
Institut des Hautes ´Etudes Scientifiques, Bures-sur-Yvette, France b Laboratoire de Physique de l’Ecole normale sup´erieure, ENS,Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France
Abstract
We explain how the axioms of Conformal Field Theory are used to make predictions aboutcritical exponents of continuous phase transitions in three dimensions, via a procedure calledthe conformal bootstrap. The method assumes conformal invariance of correlation functions,and imposes some relations between correlation functions of different orders. Numerical anal-ysis shows that these conditions are incompatible unless the critical exponents take particularvalues, or more precisely that they must belong to a small island in the parameter space.
Physics has many emergent laws , which follow in a non-obvious way from more fundamentalmicroscopic laws. Whenever this happens, we have two separate goals: to understand how theemergent law arises, and to explore its consequences.One example is the Gibbs distribution of equilibrium statistical mechanics: the probability forthe system in thermal equilibrium at temperature T to be found in a state n of energy E n is pro-portional to exp( − E n /T ). One may be interested in deriving this emergent law from microscopicmodels of thermalization, or in exploring the myriad of its physical consequences.This text, based on a recent talk for an audience of mathematical physicists, is about the “con-formal field theory” (CFT), a set of emergent laws governing critical phenomena in equilibriumstatistical mechanics (such as the liquid-vapor critical point or the Curie point of ferromagnets).CFT makes certain assumptions about the state of the system at a critical point. These assump-tions can be given reasonable physical explanations, but for the purposes of this talk we will viewthem as axioms.CFT is an “emergent law of second degree” with respect to the Gibbs distribution, by itselfemergent. It sidesteps the Gibbs distribution similarly to how the Gibbs distribution sidesteps athermalization model. Future work should derive the CFT axioms with mathematical rigor fromthe Gibbs distribution. Our goal here will be to explain the axioms and how they lead to concretepredictions for observable quantities through a procedure called the “conformal bootstrap”. a r X i v : . [ m a t h - ph ] J u l FT/bootstrap approach to critical phenomena is an alternative to the better-known Wilson’srenormalization group (RG) theory. The RG is more directly related to the Gibbs distributionthan CFT, although it too is not fully mathematically justified. The RG will not be treated hereexcept for a few comments.We will not give many references, which can be found in the recent review [1]. See also lecturenotes [2, 3, 4]. An excellent set of recorded lectures is [5].
We will describe the axioms of Conformal Field Theory (CFT) on R d , d (cid:62)
3. These axioms arewell established in the physics literature. We will present them in a form hopefully more accessibleto mathematicians. In particular, we will try to avoid (or at least explain) excessive physics jargon.Similar axioms, with additional bells and whistles, hold in d = 2 dimensions [6]. Suppose we are given a collection of real-valued functions T = { G i ,...,i n ( x , . . . , x n ) } , (1) defined for x p ∈ R d , x p (cid:54) = x q ( p, q = 1 , . . . , n ) , where n (cid:62) and the indices i p are non-negativeintegers. Functions (1) are called “ n -point correlators of fields A i , ..., A i n ” and are also denoted by (cid:104) A i ( x ) A i ( x ) . . . A i n ( x n ) (cid:105) . (2)The collection T is called a CFT if it satisfies certain axioms stated below. Different CFT’s arejust different collections of correlators satisfying those axioms. Note that the “field” A i is just a label, a name, and Eq. (2) is just a notation for G i ,...,i n ( x , . . . ., x n ).The statistical average operation suggested by this notation does not have a direct meaning in theCFT axioms. It will be handy in the interpretation of the axioms (section 4). Axiom 1 (Simple properties) . Correlators have the following properties:(a) They are invariant under permutation of any two fields: (cid:104) A i ( x ) A i ( y ) . . . (cid:105) = (cid:104) A i ( y ) A i ( x ) . . . (cid:105) , etc . (3) (b) Index i = 0 is associated with the “unit field”, replaced by 1 under the correlator sign: (cid:104) A ( x ) × anything (cid:105) = (cid:104) anything (cid:105) . (4) Our axioms should be viewed as a sketch of future complete axiomatics, which has not yet been written up inthe mathematics literature. A different approach to axiomatize CFTs in d (cid:62) d (cid:62) One also uses the term “Conformal Field Theory” in a meta-sense, as the study of all possible CFTs. One can also consider CFTs with fields having fermionic statistics, whose correlators change sign under permu-tations. Such CFTs are important e.g. for describing quantum critical points of many-electron systems. Here weonly consider commuting fields for simplicity. c) The 1-point (1pt) correlators are given by (cid:104) A ( x ) (cid:105) ≡ , (cid:104) A i ( x ) (cid:105) ≡ i (cid:62) . (5) (d) The 2pt correlators are given by (cid:104) A i ( x ) A j ( y ) (cid:105) = δ ij | x − y | i ( x, y ∈ R d ) . (6) where δ ij is the Kronecker symbol, and ∆ i (cid:62) d − ( i (cid:62) ) is a real number called “scalingdimension of field A i ”. For the unit field we have ∆ = 0 .(e) The set of scaling dimensions { ∆ i } is called the “spectrum”. It is a discrete set withoutaccumulation points (i.e. there are finitely many scaling dimensions below any ∆ ∗ < ∞ ). Axiom 2 (Conformal invariance) . Correlators are conformally invariant, in the sense that theysatisfy the constraint G i ,...,i n ( x , . . . ., x n ) = (cid:32) n (cid:89) p =1 λ ( x p ) ∆ ip (cid:33) G i ,...,i n ( f ( x ) , . . . ., f ( x n )) , (7) or equivalently, using notation (2) , (cid:104) A i ( x ) . . . A i n ( x n ) (cid:105) = (cid:32) n (cid:89) p =1 λ ( x p ) ∆ ip (cid:33) (cid:104) A i ( f ( x )) . . . A i n ( f ( x n )) (cid:105) , (8) where f ( x ) is an arbitrary conformal transformation of R d and λ ( x ) = (cid:12)(cid:12) ∂f∂x (cid:12)(cid:12) /d is its scale factor. Recall that conformal transformations satisfy the constraint ∂ f µ /∂x ν = λ ( x ) R µν ( x ) where R µν ( x ) ∈ SO( d ). For d (cid:62)
3, these transformations form a group SO( d + 1 , Remark 1.
Conformal transformations of R d may send points to infinity, and should be thoughtmore properly as acting on R d ∪ {∞} , the d -dimensional analogue of the Riemann sphere. Totreat the point at infinity on equal footing with the other points, one can put R d ∪ {∞} in one-to-one correspondence with the d -dimensional unit sphere S d via the stereographic projection. Thissubtlety will be glossed over here. We will state without proof a few basic consequences of the above axioms. One can check thatthe 2pt correlators given in Axiom 1(d) are consistent with Axiom 2. Note that the same scalingdimension ∆ i has to appear in all n -point correlators involving the field A i . The 3pt correlatorsare fixed by Axiom 2 up to an overall factor: (cid:104) A i ( x ) A j ( x ) A k ( x ) (cid:105) = c ijk x ∆ i +∆ j − ∆ k x ∆ i +∆ k − ∆ j x ∆ j +∆ k − ∆ i , (9)3here c ijk is totally symmetric by Axiom 1(a), and we denoted x ij = | x i − x j | . For 4pt correlatorsAxiom 2 implies the following functional form: (cid:104) A i ( x ) A j ( x ) A k ( x ) A l ( x ) (cid:105) = (cid:18) x x (cid:19) ∆ i − ∆ j (cid:18) x x (cid:19) ∆ k − ∆ l g ijkl ( u, v ) x ∆ i +∆ j x ∆ k +∆ l , (10)where g ijkl ( u, v ) is a function of conformally invariant cross-ratios: u = x x x x , v = u | ↔ = x x x x . (11)By Axiom 1(a) functions g ijkl with permuted indices are all related, e.g. permutation 1 ↔ u − ∆ i +∆ j g ijkl ( u, v ) = v − ∆ k +∆ j g kjil ( v, u ) , etc . (12) Group-theoretically, the transformation A ( x ) → λ ( x ) ∆ A ( f ( x )) (13)is an irreducible representation π ∆ of the conformal group on scalar functions A : R d → R .Eq. (7) means that the correlators G i ,...,i n belong to the invariant subspace of the tensor productrepresentation ⊗ np =1 π ∆ ip (so they can be called “invariant tensors”).We formulated Axioms 1,2,3 for the fields transforming as (13), called “scalar fields”. Theseaxioms can and should be extended to allow for fields with tensor indices. First of all, we haveto add fields ∂ α A i ( x ) which are partial derivatives (of arbitrary order) of the fields A i . Theircorrelators are defined as derivatives of the original ones: (cid:104) ∂ α A i ( x ) . . . (cid:105) := ∂ αx (cid:104) A i ( x ) . . . (cid:105) . (14)This is, in a sense, just a convenient notation. The basic fields A i ( x ) whose correlators transformas (13) are called “primaries”, while their derivatives “descendants”. Transformation rules forcorrelators of descendants can be obtained by differentiating (13).The second extension is a bit less trivial. We should generalize (13), allowing for fields withvalues in a finite-dimensional vector space V , dim V >
1, transforming under the conformal groupvia A ( x ) → λ ( x ) ∆ ρ ( R ( x )) A ( f ( x )) , (15)where ρ is an irreducible representation of SO( d ) acting in V . Such fields are called “primaryspinning fields”. One example is V = { symmetric traceless rank- l tensors } . Correlators of spinningfields then take values in the tensor product ⊗ np =1 V i p and satisfy a conformal invariance constraintsimilar to (7) but with factors of ρ i p ( R ( x p )) in the l.h.s. [Derivatives of spinning fields are thenalso added as in (14).] Adding spinning fields would complicate the notation a bit. We will neglectthem here, although practical conformal bootstrap computations always allow for their presence.4 The OPE axiom
The last “OPE axiom” will relate different correlators, and in particular correlators with different n . This is unlike the previous axioms which involved one n -point correlator at a time. Suppose we are given two collections of real numbers { λ ijk } , { s ( r ) ijk , r (cid:62) } , (16)where i, j, k run over the field indices (non-negative integers). With these numbers as coefficients,“Operator Product Expansion” (OPE) is constructed as a set of formal equalities (one for eachpair of fields A i and A j ): A i ( x ) A j ( y ) = ∞ (cid:88) k =0 λ ijk | u | ∆ i +∆ j − ∆ k × (cid:104) A k ( x ) + s (1) ijk u µ ∂ µx A k ( x ) + (cid:16) s (2) ijk u µ u ν + s (3) ijk u δ µν (cid:17) ∂ µx ∂ νx A k ( x ) + · · · (cid:105) . (17)where u = y − x . Using the OPE for the first pair of fields inside the n -point correlator (2) with n (cid:62)
2, we get a set of candidate identities among correlators: (cid:104) A i ( x ) A j ( y ) Π (cid:105) = ∞ (cid:88) k =0 λ ijk | u | ∆ i +∆ j − ∆ k [ (cid:104) A k ( x ) Π (cid:105) + s (1) ijk u µ ∂ µx (cid:104) A k ( x ) Π (cid:105) + · · · ] , (18)where we denoted i = i , i = j , x = x , x = y , and Π = Π np =3 A i p ( x p ) is the product of all otherfields in the correlator. In the l.h.s. we have an n -point correlator, while in the r.h.s. we have aninfinite series of ( n − Axiom 3 (OPE) . There exists a set of coefficients (16) , such that Eq. (18) holds as a true relationbetween correlators (the series in the r.h.s. converges absolutely to the l.h.s.) as long as | x p − x | > | u | for all p (cid:62) (see Fig. 1). gags 't y = Ketu . " " . " " " @ Ii > → I Xijk2D" Ik ? Xl Xz Xz Xp Xz X3 XI Xz X3 V. = ¥ - - II * just > =-3 . Xl Xz Xs Xg Xi Xz Xz Xy = • ooo * It > a I ; , ii. . x , Ea ' Figure 1:
The OPE expansion applies when all points x p ( p (cid:62)
3) lie further from x than x . What can be said about coefficients (16) which make this axiom work? To see this, let usapply Eq. (18) to a 3pt correlator. Because the 2pt correlators vanish for non-identical fields ( δ ij in Axiom 1(d)), the sum in the r.h.s. collapses to the single k value, and we get: (cid:104) A i ( x ) A j ( y ) A k ( x ) (cid:105) = λ ijk | u | ∆ i +∆ j − ∆ k (cid:20) | x − x | k + s (1) ijk u µ ∂ µx | x − x | k + · · · (cid:21) , (19) Except the rather trivial Axiom 1(b). (cid:104) A k ( x ) A k ( x ) (cid:105) = 1 / | x − x | k . On the other hand, we already know that the3pt correlator in the l.h.s. has form (9) by Axiom 2. Let us then expand (9) for small y and matchwith (19). From the leading term we find λ ijk = c ijk . Relative to this overall normalization, thesubleading terms on the l.h.s. are fixed, and this allows to determine s ( r ) ijk uniquely as rationalfunctions of ∆ i , ∆ j , ∆ k and d . We conclude that all coefficients (16) can be uniquely determinedby demanding that the OPE axiom works for the 3pt correlators. Furthermore, the axiom saysthat the same set of coefficients should then also work for any n -point correlators. Definition 1.
A conformal field theory (CFT) T in d (cid:62) dimensions is a collection of correlators (1) satisfying Axioms 1-3. Remark 2.
We included in our axioms the conditions ∆ i (cid:62) d − for i (cid:62) λ ijk ∈ R in Eq. (16). By this, we are restricting our discussion to a subclass of conformal fieldtheories called reflection positive (or unitary). Many statistical physics systems at criticality (suchas the Ising model or O ( N ) models) are known to be described by unitary CFTs. The spectrum ∆ i and the OPE coefficients λ ijk comprise the “dataset” of a CFT T :Data( T ) = { ∆ i , λ ijk } . (20)As discussed, Data( T ) is in one-to-one correspondence with the 2pt and 3pt correlators of T .Moreover, knowing Data( T ) we can reconstruct all n -point correlators, for an arbitrarily high n .Indeed, from Data( T ) we can construct the OPE (the coefficients s ( r ) ijk are not included in Data( T )since they are uniquely determined by ∆’s and d ). Then, we can recursively reduce any n -pointcorrelator to lower-point ones, until we get to the known 2pt and 3pt correlators. We thus see that the dataset Data( T ) encodes full information about the CFT T . Below wewill describe a program of classifying CFTs by classifying their data sets. But first let us discussthe interpretation of the CFT axioms. Notation (cid:104) A i ( x ) A i ( x ) . . . A i n ( x n ) (cid:105) for G i ,...,i n ( x , . . . , x n ) acquires a meaning in the interpreta-tion of the CFT axioms, as correlation functions of statistical systems at their critical points. CFTcalculations are then interpreted as predictions for the critical exponents of statistical physics mod-els. Although the CFT calculations based on the axioms are completely rigorous, the interpretationstep is at present non-rigorous. Hopefully it will be justified in the future. In particular we learn that λ ijk has to be symmetric, just as c ijk . Note also that by putting A k = A = 1 in(9) and by using Axiom 1(d), we get λ ij = c ij = δ ij . We should take care that the OPE is used for a pair of fields at positions x , x verifying conditions of Axiom3, so that it converges. This is the case if x is the unique position with the minimal distance from x . There aredegenerate configurations when such a pair cannot be found, because each point has two or more nearest neighborsat equal distance (e.g. the vertices of a regular polygon). It is then always possible to apply a small conformaltransformation which moves points to a non-degenerate configuration. In the new configuration the OPE convergesand we can compute the value of the correlator. We then conformal-transform back to the original configuration.This way we can compute correlators in any configuration of non-coincident points. H = − (cid:80) (cid:104) xy (cid:105) S x S y where S x = ± d = 3 describing the critical point of this model, and of anyother model in the same universality class. Just as the lattice Ising model, this CFT has a global Z invariance with all fields divided into Z -even and Z -odd. It contains a Z -odd scalar primaryfield denoted σ ( x ), whose correlators (cid:104) σ ( x ) σ ( x ) . . . σ ( x n ) (cid:105) (21)are interpreted as the 3d Ising model spin correlation functions (cid:104) S x S x . . . S x n (cid:105) (22)computed at the critical temperature T = T c , at distances | x p − x q | much larger than the latticespacing. While (cid:104) . . . (cid:105) in (21) is just a notation, in (22) it is a true average with respect to the Gibbsdistribution, in the thermodynamic limit. By Axiom 2, correlator (21) is conformally invariant,and thus in particular scale invariant, scale transformations being a part of the conformal group.This means (∆ σ is the scaling dimension of σ ): (cid:104) σ ( λx ) σ ( λx ) . . . σ ( λx n ) (cid:105) = λ − n ∆ σ (cid:104) σ ( x ) σ ( x ) . . . σ ( x n ) (cid:105) . (23)On the other hand, (22) clearly does not have such an exact scale invariance, already because it isdefined on a lattice. The precise statement of agreement at large distances is lim | x p − x q |→∞ (cid:104) S x S x . . . S x n (cid:105)(cid:104) σ ( x ) σ ( x ) . . . σ ( x n ) (cid:105) = C n , (24)where C is some constant, which is n -independent but non-universal (e.g. it would change if weadd next-to-nearest interactions to the lattice model, which does not change the universality class).Other 3d Ising CFT fields will correspond to other lattice-scale operators. E.g. we can considerthe product of two nearby spins (separated in an arbitrary direction) E x = S x S x +1 − (cid:104) S x S x +1 (cid:105) , (25)where (cid:104) S x S x +1 (cid:105) is subtracted so that (cid:104) E x (cid:105) = 0. The 3d Ising CFT contains a Z -even scalarprimary ε ( x ) whose correlators describe long-distance limits of the E x correlators, similarly to(24).More generally, we expect to have a CFT associated with every universality class of continuousphase transitions. This CFT will share global symmetry ( Z , O ( N ), etc) with the universalityclass, and its scaling dimensions will determine the critical exponents. It has not been proven yet, We have not included the notion of global symmetry in the CFT axioms, but this extension is straightforward.It just means that all fields transform in finite-dimensional irreducible representations of a compact global symmetrygroup G , forming a direct product with the conformal group. All correlators are invariant tensors of G, and theOPE respects this additional symmetry. Equivalently, one can consider a sequence of lattice models with a smaller and smaller lattice spacing a , andtake the limit a → x a fixed. A i and their scaling dimensions ∆ i also have counterparts in the RG approach tocritical phenomena [9]. Namely, they correspond to the eigenvectors and the eigenvalues of RGtransformation linearized near a fixed point describing a continuous phase transition. Fields ofscaling dimension ∆ i < d (∆ i > d ) correspond to the relevant (irrelevant) deformations of the fixedpoint. This dictionary is not needed for the actual CFT calculations, but only for interpreting theresults.We expect that the above-mentioned fields σ and ε are the only two relevant fields of the 3dIsing CFT. This follows from the experimental fact that the critical point of the 3d Ising modelis in the same universality class as the liquid-vapor critical point, which is reached by tuning twoparameters (pressure and temperature). Conformal bootstrap program attempts to classify CFTs by classifying their datasets. That thismay be possible was first suggested by Polyakov [10].We call a dataset D = { ∆ i , λ ijk } “consistent” if it is a dataset of some CFT: D = Data( T ).Ideally, we would like to have a list of all consistent data sets: { Data( T ) , Data( T ) , . . . } , (26)but it is not currently known how to generate such a list. The following question is less ambitiousbut still very interesting: Q1: Given a trial dataset D , decide if it is inconsistent . (27)It turns out that this has an algorithmic answer. This will allow progress on classification by rulingout inconsistent data sets (rather than by constructing consistent ones).The idea is straightforward: given a trial dataset D = { ∆ i , λ ijk } , we will try to construct allcorrelators, looking for some inconsistency with the axioms.The first step is to construct the 2pt and 3pt correlators. These are simply given by explicitformulas from Axiom 1(d) and (9) with c ijk = λ ijk . So far no room for inconsistency.Then we proceed to construct the 4pt correlators. For this we consider the OPE series reducingthem to the 3pt correlators. All information needed to write down these series is contained in ∆ i and λ ijk . But now we need to check a couple of things. First, do these series converge whereAxiom 3 says they should? For this, the trial OPE coefficients λ ijk should not grow too fast as a On the other hand, the OPE coefficients λ ijk do not feature prominently in the RG approach. We are not giving full details necessary to make this statement precise. One important subclass of CFTs are“local CFTs”, which roughly correspond to critical points of lattice models with finite-range interactions. It isexpected that most local CFTs are isolated. One exception are CFTs with “exactly marginal” fields of dimension∆ = d , which form finite-dimensional continuous families. A folk conjecture says that exactly marginal fields in d (cid:62) k for fixed i , j . The required growth condition can be shown to take a relatively simpleform: ∞ (cid:88) k =0 (4 ρ ) ∆ k λ ijk < ∞ ∀ ρ < . (28)Second, there are several ways to reduce a 4pt correlator to 3pt correlators via the OPE, andthey all should agree in the overlapping regions of convergence. See Fig. 2 for an example. Thiscondition is called “crossing”, and it is not automatically satisfied. Assuming that it also holds,we can define the 4pt correlators as the sum of OPE series. gags 't @ % • Xz=
Itt . Xp j . an . Tcg Xl Xz Xz Xp Xz X3 Xi Xz X3 II. - - ¥ - - LI XI Xz Xz Xg Xi Xz X3 Xy = • ooo • Bo • I , Xz Xz X4 Figure 2:
The 4pt correlator for this configuration of points can be reduced to 3pt correlators usingthe OPE (18) with ( x, y ) being one of the following pairs: ( x , x ), ( x , x ), ( x , x ), ( x , x ). We then proceed to higher n -point correlators. Similarly to n = 4, they are reduced to ( n − n (cid:62) λ ijk with a fixed j and arbitrary i, k : ( M ( j ) ) ik = λ ijk . (29)Then any such M ( j ) , viewed as an operator from i -indexed sequences to k -indexed sequences,should be bounded with respect to a certain weighted (cid:96) norm (with ∆-depending weights). Eq. (28) follows from this general condition when we apply the operator to a sequence consistingof a single nonzero element, and demand that the result have a finite norm.To summarize, consistent datasets are those which satisfy the general convergence and the4pt crossing conditions. The convergence condition is the less interesting of the two. Below wewill focus on the 4pt crossing, which will allow us to put constraints on the fields of low scalingdimension.
Here we will describe how to put 4pt crossing constraint into a more explicit form, by expanding4pt correlators in a basis of special functions called conformal blocks.Recall that conformally invariant 4pt correlators have form (10). When we compute the 4ptcorrelator in the r.h.s. of (10) using the OPE, we should get something consistent with this formula.Let us see how this happens. Applying the OPE to the first pair of fields, we get an expression of → M ( p , p → p , p ) is invariant under“crossing transformations”, when one incoming particle is moved (“crosses”) into the group of outgoing particles,while one outgoing particle crosses in the opposite direction. This is related to something called ‘radial quantization’, which we do not describe in this text. This convergencecondition for higher n -point corelators has not been discussed in detail in the literature. (cid:104) A i ( x ) A j ( x ) A k ( x ) A l ( x ) (cid:105) = (cid:88) m λ ijm x ∆ i +∆ j − ∆ m [ (cid:104) A m ( x ) A k ( x ) A l ( x ) (cid:105) + · · · ] , (30)where ... denotes terms proportional to s ( r ) ijm times derivatives acting on the 3pt correlator (cid:104) A m A k A l (cid:105) ,which is in turn given by λ mkl times an x -dependent function which can be read off from (9). Itcan be shown that by doing all derivatives and infinite sums over r , the r.h.s. of Eq. (30) takesthe form: (cid:18) x x (cid:19) ∆ i − ∆ j (cid:18) x x (cid:19) ∆ k − ∆ l x ∆ i +∆ j x ∆ k +∆ l (cid:88) m λ ijm λ mkl G ∆ m ( u, v ) . (31)The functions G ∆ m ( u, v ) appearing here are called ‘conformal block’. These functions are fixed byconformal symmetry. They depends on the exchanged scaling dimension ∆ m , and on the spacedimension d . Notably, they do not depend on the OPE coefficients λ ijk whose product appearsas a prefactor in (31).Theory of conformal blocks is huge and it’s not possible to do it justice in this text. It hasconnections to representation theory, orthogonal polynomials, and integrable quantum mechanics.There are no fully general closed form expressions of conformal blocks in terms of the classicalspecial functions. Fortunately, they admit rapidly convergent power series expansions which allowefficient numerical evaluation. This is what is used in practical applications.The conformal block is simple only for the exchanged unit field: A m = A = 1, when we have: G ( u, v ) = 1 , λ ij = δ ij , λ kl = δ kl , (32)where we also gave the OPE coefficients for this case (see footnote 5).Comparing Eq. (31) with (10) we see that they are consistent if we identify: g ijkl ( u, v ) = (cid:88) m λ ijm λ mkl G ∆ m ( u, v ) . (33)This gives a compact formula to compute the 4pt correlators in terms of the CFT data. We canalso obtain a compact expression for the 4pt crossing constraints, by substituting Eq. (33) into(12): u − ∆ i +∆ j (cid:88) m λ ijm λ mkl G ∆ m ( u, v ) = v − ∆ k +∆ j (cid:88) m λ kjm λ mil G ∆ m ( v, u ) . (34)As Eq. (12), this corresponds to the permutation x ↔ x . Constraints corresponding to otherpermutations take a rather similar form. They should also be considered, although we will notdiscuss them here explicitly.Now, we can test a trial dataset D for consistency, by checking Eq. (34) for all possible choicesof i, j, k, l , in the region of overlapping convergence. This region is not empty. E.g., let us fixpoints x , , so that x is far away from x , . Then the l.h.s. should converge within the set { x : x < x } , and the r.h.s. in { x : x < x } . These two balls have a nontrivial overlap. They also depend on the external dimension differences ∆ i − ∆ j , ∆ k − ∆ l but we will omit this from thenotation. In a full treatment involving spinning fields, the conformal blocks also depend on the spin of the fields. .3 Partially specified datasets In sections 5.1 and 5.2, we gave an answer to the consistency question (27). Unfortunately, thedescribed procedure is not by itself practically useful, since it assumes that the trial dataset D is fully specified, which includes infinitely many parameters (all scaling dimensions and OPEcoefficients). To correct for this, let us define the notion of a “partially specified trial dataset”,which is a list L of finitely many assumptions on scaling dimensions and OPE coefficients. We saythat L is consistent if there is at least one CFT T whose dataset Data( T ) satisfies the assumptions.The following is then a more practical version of question (27): Q2: Given a partially specified trial dataset L , decide if it is inconsistent . (35)Although this looks like a much harder question than (27), it turns out that this question can alsobe answered, based on Eq. (34), using numerical algorithms. This was first shown by Rattazzi,Tonni, Vichi and the author [12] and led to the rapid development of the numerical conformalbootstrap in the last 10 years. We will explain how this work on an example in the next section. Let us fix two real numbers ∆ , ∆ in the interval [ , L = L (∆ , ∆ ) about a 3d CFT: • Z global symmetry; • there is one field which is Z -odd, one which is Z -even, and they have scaling dimensions∆ and ∆ ; • all other fields have scaling dimensions ∆ i (cid:62) L (∆ σ , ∆ ε ). Our strategy will be to exclude alarge part of the (∆ , ∆ )-plane by showing that L (∆ , ∆ ) is inconsistent there. This will implythat the scaling dimensions of the 3d Ising CFT must belong to the remaining part of the plane. Consider first the 4pt crossing for (cid:104) A A A A (cid:105) . Putting i = j = k = l = 1 in (34), we obtain: u − ∆ (cid:88) m p m G ∆ m ( u, v ) = v − ∆ (cid:88) m p m G ∆ m ( v, u ) , p m = λ m (cid:62) . (36)We know that p = G = 1 (see (32)), so isolating those terms we write this as h ( u, v ) ≡ v − ∆ − u − ∆ = (cid:80) ∞ m =2 p m F ∆ m ( u, v ) , (37) F ∆ ( u, v ) := u − ∆ G ∆ ( u, v ) − v − ∆ G ∆ ( v, u ) . Note that F ∆ also depends on ∆ . The sum starts from m = 2 because λ = 0 for the Z -odd A . 11eometrically, (37) means that h , viewed as a vector in a space of two-variable functions,belongs to a convex cone C generated by vectors F ∆ and F ∆ with ∆ (cid:62)
3. We include all F ∆ with∆ (cid:62) m for m (cid:62)
3, but onlythat ∆ m (cid:62)
3. Denote by C ∗ the dual convex cone, which is the set of all linear functionals α whichare positive on all vectors generating the cone: α [ F ∆ ] (cid:62) , α [ F ∆ ] (cid:62) ∀ ∆ (cid:62) . (38)Suppose that there exists a functional α ∈ C ∗ such that α [ h ] < . (39)Then by acting with α on Eq. (37) we get a contradiction. So, this equation cannot be satisfiedfor any nonnegative p m . This is how one shows that the assumption L (∆ , ∆ ) is inconsistent: byexhibiting a functional α which satisfies (38) and (39).Numerically, one works with a finite dimensional space of functionals A (Λ) which are finitesums of partial derivatives at a particular point: α [ f ] = (cid:88) m + n (cid:54) Λ α m,n ∂ mu ∂ nv f ( u , v ) , (40)where Λ is a parameter, to be taken as large as possible to have the maximal constraining powers(within the available computer resources). One then minimizes α [ h ] over all α ∈ C ∗ ∩ A (Λ),looking for a functional satisfying (39). This is a convex optimization problem (continuous linearprogramming), which can be solved by efficient numerical algorithms. If the minimum is negative,then we ruled out L (∆ , ∆ ). If it is positive, and cannot be made negative by increasing Λ, thiswould mean that L (∆ , ∆ ) is consistent with crossing for (cid:104) A A A A (cid:105) .With this procedure, Ref. [13] showed that the constraint L (∆ , ∆ ) is inconsistent in a signif-icant portion of parameter space. Invoking an extra and so far unproven assumption, that the 3dIsing CFT lies at a singular boundary point of the consistent region (the so called “kink”), Refs.[13, 14] gave the first conformal bootstrap determination of ∆ σ , ∆ ε . Subsequent work has shownthat the kink assumption is unnecessary, provided that one includes crossing constraints for the4pt correlators (cid:104) A A A A (cid:105) , (cid:104) A A A A (cid:105) . We will now explain briefly how this was done. To increase the constraining power, a natural idea is to include crossing constraints for the other4pt correlators of fields A and A . While (cid:104) A A A A (cid:105) is completely analogous to (cid:104) A A A A (cid:105) ,one encounters a crucial difference when analyzing (cid:104) A A A A (cid:105) . Namely, its conformal blockexpansion involves products of two different OPE coefficients λ m λ m . These products are notnecessarily positive, because λ ijk may have either sign. On the other hand, positivity of thecoefficients p m = λ m played a crucial role in making the minimization problem of section 6.1convex. To overcome this obstacle, one analyzes all three correlators together, and considers thematrix P m = (cid:18) λ m λ m λ m λ m λ m λ m (cid:19) . (41)Crucially, this matrix is positive semidefinite: P m (cid:60)
0. This condition is convex, and provides agood substitute for the simple positivity in the bootstrap problems involving multiple correlators.12he resulting problem is that of continuous semidefinite programming, and it can still be attackedby efficient numerical algorithms. This was realized and carried out in Refs. [15, 16, 17] whichfound a consistent “island” near ∆ ≈ . ≈ . σ , ∆ ε ) must live somewhere in this tiny island (Fig. 3). also be realized as a Lorentzian ð þ Þ D quantum criticalpoint (Fradkin and Susskind, 1978; Henkel, 1984). Here wework in the Euclidean signature; the Lorentzian version isobtainable by Wick rotation and has the same set of CFT data.In its original formulation as a model of ferromagnetism,the 3D Ising model is described using a set of spins s i ¼ % on a cubic lattice in R with nearest neighbor interactions,with the partition function Z ¼ X f s i g exp ! − J X h ij i s i s j " : ð Þ At a critical value of the coupling J , the model becomes anontrivial CFT at long distances. Note that the lattice modelhas a manifest Z symmetry under which s i → − s i . Thissymmetry is inherited by the CFT, which contains localoperators that are either even or odd under its Z globalsymmetry.Another microscopic realization is in terms of a continuousscalar field theory in three dimensions, with action S ¼ Z d x ! ð ∂ σ Þ þ m σ þ λσ " ; ð Þ which also has a Z symmetry under which σ → − σ . Becauseboth m and λ describe relevant couplings, this theory isdescribed by a free scalar at short distances but has nontrivialbehavior at long distances. At a critical value of the dimen-sionless ratio m = λ the long-distance behavior is describedby a CFT, which is the same as for the above lattice model.From the conformal bootstrap perspective, the Ising CFThas a Z global symmetry, one relevant Z -odd scalar operator σ , and one relevant Z -even scalar operator ϵ . This is evidentfrom experimental realizations, where Z -preserving micro-scopic realizations require one tuning (e.g., tuning the temper-ature in uniaxial magnets) and Z -breaking microscopicrealizations require two tunings (e.g., tuning both temperatureand pressure in liquid-vapor transitions). Note that theassumption that the only relevant scaling dimensions are Δ σ and Δ ϵ is the same assumption that went into producing thedark blue detached region of Fig. 13.Kos et al. (2016) pursued a numerical analysis of themixed-correlator bootstrap system containing σ and ϵ to highderivative order. In addition, they studied the impact ofscanning over different possible values of the ratio λ ϵϵϵ = λ σσϵ . This scan effectively inputs the information thatthere is a single operator in the OPE occurring at the scalingdimension Δ ϵ , whereas the plot of Fig. 13 allowed for thepossibility of multiple degenerate operator contributions atthe dimension Δ ϵ . This led to the three-dimensionalallowed region shown in Fig. 14 and its projection to the f Δ σ ; Δ ϵ g plane shown in Fig. 15. In addition, for each pointin this region the magnitude of the leading OPE coefficientswere also bounded, with the result shown in Fig. 16. Theseworld-record numerical determinations are summarized inTable. II. FIG. 14.
An allowed region in the f Δ σ ; Δ ϵ ; λ ϵϵϵ = λ σσϵ g spaceobtained by Kos et al. (2016). FIG. 15.
Projection of the 3D region in Fig. 14 on the f Δ σ ; Δ ϵ g plane and its comparison with a Monte Carlo prediction for thesame quantities. From Kos et al. , 2016. FIG. 16.
Variation of λ ϵϵϵ and λ σσϵ within the allowed region inFig. 14. From Kos et al. , 2016. The existence of Z -breaking liquid-vapor experimentalrealizations, allowing one to get Z as an emergent symmetry andpredict the total number of relevant scalars, is a nice feature of theIsing model which does not have analogs for the O ð N Þ models. More precisely, the scan inputs that the outer product of OPEcoefficients ð λ σσϵ λ ϵϵϵ Þ ⊗ ð λ σσϵ λ ϵϵϵ Þ appearing in Eq. (103) atdimension Δ ϵ is a rank 1 matrix, rather than the more generic rank 2possibility which occurs if there are degenerate contributions. David Poland, Slava Rychkov, and Alessandro Vichi: The conformal bootstrap: Theory, numerical … Rev. Mod. Phys., Vol. 91, No. 1, January – March 2019 015002-32
Figure 3:
The bootstrap island to which the 3d Ising CFT must belong [17]. Also shown is a MonteCarlo result [18] for the same scaling dimensions. This plot used Λ = 43 in (40).
The scaling dimensions ∆ σ , ∆ ε determine the main critical exponents of the 3d Ising model α, β, γ, δ, η, ν . In what follows we will focus on η and ν , given by η = 2∆ σ − , (42) ν = 1 / ( d − ∆ ε ) . (43)Eq. (43) deserves a comment, because it expresses an off-critical quantity (the exponent ν describingbehavior of the correlation length close to the critical point) via a critical theory parameter ∆ ε .This is an example of how CFT can make predictions about small deviations from the criticaltheory, which arise at short distances from relevant perturbations , and at large distances from theirrelevant ones. Such predictions are done via a technique called “conformal perturbation theory,”which we have not explained. Eq. (44) below is another simple example.Another important quantity is the “correction to scaling” exponent ω . It appears in the rate ∼ /r ω at which the limit in (24) is achieved, assuming that all distances | x p − x q | ∼ r are of thesame order. It also appears in the subleading singularities of all quantities exhibiting powerlawbehavior near the critical point (e.g. the specific heat). While describing deviations from criticality, ω like ν can be expressed in terms of a purely critical parameter: ω = ∆ − d , (44)where ∆ is the scaling dimension of the leading irrelevant Z -even scalar operator. The conformalbootstrap determines ∆ (and hence ω ) by scanning the island in Fig. 3 and reconstructing thespectrum which provides a solution to the 4pt crossing [19].In Table 1 we report the values of the critical exponents ν , η , ω according to the conformalbootstrap, Monte Carlo simulations and RG calculations. We also include some experimentalmeasurements of ν and η . The conformal bootstrap predictions are the most precise, and they Since ω parametrizes subleading powers, it is harder to measure, and we are not aware of any published result. ν η ω [17, 19] 2016 Conformal bootstrap 0.629971(4) 0.036298(2) 0.82968(23)[18] 2010 Monte Carlo 0.63002(10) 0.03627(10) 0.832(6)[20] 1998 RG 0.6304(13) 0.0335(25) 0.799(11)[21] 1989 Binary fluid 0.628(8) 0.0300(15)[22] 2009 Binary fluid 0.629(3) 0.032(13)[23] 1994 Binary mixture 0.623(13) 0.039(4)[24] 2000 Liquid-vapor 0.62(3)[25] 1998 Liquid-vapor 0.042(6)[26] 1987 Uniaxial antiferromagnet 0.64(1) Table 1:
Some representative theoretical and experimental determinations of the 3d Ising criticalexponents. See [27], Section 3.2, for more references.
Conformal bootstrap calculations provide predictions for observable physical quantities from theCFT axioms. Agreement of these predictions with alternative theoretical determinations and theexperiment increase our belief in the validity of the axioms.Feynman [28] called the Gibbs distribution the “summit of statistical mechanics”, the entiresubject being either the “climb-up” to derive it, or the “slide-down” when it is applied. EchoingFeynman, we may call the CFT a summit of the theory of critical phenomena, the conformal boot-strap being the way to slide down. To climb up would be to prove the validity of the interpretationof the CFT axioms described in Section 4. Unfortunately, relatively little rigorous work has beendone in the way of climbing up. One should also not forget a second major peak in the samemountain range: the Renormalization Group.
Acknowledgements
This article is based on the talk at the mathematical physics workshop “Inhomogeneous RandomSystems” (Institut Curie, Paris, January 28, 2020). I am grateful Ellen Saada, Fran¸cois Dunlop andAlessandro Giuliani for the organization and the invitation to speak. I am also grateful to JacquesVillain for the invitation to write this article, careful reading of the draft, and many suggestions onhow to improve the presentation. SR is partly supported by the Simons Foundation grant 488655 The last 20 years, starting with Smirnov [29], have seen significant progress in showing rigorously conformalinvariance of specific 2d models. This program is still far from establishing conformal invariance of a generic criticaltheory, let alone the full scope of the CFT axioms, such as the existence of a complete set of local operators andthe OPE. We are also not aware of any relevant mathematical work in d (cid:62) References [1] D. Poland, S. Rychkov, and A. Vichi, “The Conformal Bootstrap: Theory, Numerical Techniques,and Applications,”
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