OOn conformal perturbation theory
Andrea Amoretti ∗ and Nicodemo Magnoli
2, 3, † Physique Th´eorique et Math´ematique Universit´eLibre de Bruxelles, C.P. 231, 1050 Brussels, Belgium Dipartimento di Fisica, Universita di Genova,Via Dodecaneso 33, 16146, Genova, Italy INFN, sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy (Dated: May 25, 2018)
Abstract
Statistical systems near a classical critical point have been intensively studied both from the-oretical and experimental points of view. In particular, correlation functions are of relevance incomparing theoretical models with the experimental data of real systems. In order to computephysical quantities near a critical point one needs to know the model at the critical (conformal)point. In this line, recent progresses in the knowledge of conformal field theories, through theconformal bootstrap, give the hope to get some interesting results also outside of the critical point.In this note we will review and clarify how, starting from the knowledge of the critical correlators,one can calculate in a safe way their behavior outside the critical point. The approach illustratedrequires the model to be just scale invariant at the critical point. We will clarify the method byapplying it to different kind of perturbations of the 2 D Ising model. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] M a y ONTENTS
I. Introduction 2II. Conformal perturbation theory: a review 4A. First order expansion 5B. Higher order expansion 7III. Examples: 2D Ising model applications 8A. Thermally perturbed two point functions 9B. Trapped Ising model 11C. Magnetically perturbed three-point functions 13IV. Conclusions 14Acknowledgments 16A. Mellin transform and asymptotic properties of integrals 161. Integrals for the trapped Ising model 18References 18
I. INTRODUCTION
Recently there has been a renewed interest in conformal field theories (CFT) in higherdimensions. Some new insight has been obtained through the conformal bootstrap method[1–5]. One of the most important results has been the computation of the critical exponentsand the Wilson coefficients of the 3 D Ising model [2, 3]. The knowledge of the system at thecritical point gives us the possibility to analyze its behavior away from criticality, considering e.g. the perturbations due to the presence of an external magnetic field or an energy operator.The knowledge of the correlators in the vicinity of the critical point is extremely relevant tocompare theoretical predictions with experimental data, to study, in the 2 D case, the relationbetween perturbed CFTs and integrable field theories [6] or the flow between different CFTs[7]. From the theoretical point of view, to obtain the behavior of observables outside acritical point is not an easy task, since usually the standard approach based on the Gell-2ann and Low theorem [8] is not useful in this situation. This is due to the fact that, inthe case of a conformal field theory perturbed by a relevant operator, the corrections to theconformal correlation functions are typically plagued by infrared divergences. This problemhas been firstly pointed out by Wilson in its seminal paper [9]. In order to understand theissue and its possible solution it is useful to consider the theory of the free massive boson.It is well known that the euclidean propagator of this theory can be expressed in terms of amodified Bessel function depending on the product of the mass m and the distance r (timesa power of r ). The short distance ( mr <<
1) behavior of the Bessel function contains termsproportional to
Log ( mr ), which cannot be obtained by performing a small mass perturbationat any finite order. Additionally, naive perturbative mass expansion is affected by infrareddivergences, being the massless “free” theory scale invariant. It is clear that the correlationfunction by itself has no finite perturbative expansion. As pointed out in [9], one canovercome the problem by expressing the short distance behavior of the correlators in termsof the Operator Product Expansion (OPE). In this way, the non perturbative contributionof the correlators, which is encoded in the vacuum expectation values (VEV) of the localoperators in the OPE, separates from the one that can be computed perturbatively, namelythe Wilson coefficients. Eventually, if the expectation values of the operator are known, theproblem reduces to perturbatively compute the Wilson coefficients.Even though techniques to perturbatively compute the short distance IR finite behaviorof the Wilson coefficients have been developed a long time ago [10], it seems that the methodis not well known in the recent literature. In this paper we will review the approach firstlydescribed in [10], which can be applied to compute the behavior of the correlators of modelsof arbitrary space dimensions in the vicinity of a scale invariant critical point. In order torender the method as clear as possible, we will apply it to several example of perturbationsof the 2D Ising model which have not yet been considered in the literature . Even thoughthe purpose is pedagogical, each example has a physical interest by itself, which will bediscussed in details in the conclusions.The paper is organized as follows. In Section 2 we describe the general method toperturbatively compute the Wilson coefficients, clarifying some aspects not discussed in[10]. In section 3 we apply the method to three examples related to the 2 D Ising model. See e.g. [11–13] for other examples of perturbations of the 2D Ising model and [14] for an application tothe Lee-Yang model. E inthe broken phase, where the spin operator σ has a non-trivial expectation value. Specifically,we will compare the perturbative result for the correlator (cid:104) σ E (cid:105) with the analytical expressionobtained by Hecht [20]. This provide us with an example in which conformal perturbationtheory gives the correct results even in a phase where the underlying symmetry of the modelis somehow broken. Secondly we apply the method to the Ising model perturbed by themagnetic field in the presence of a trap. This example might be of relevance to comparetheoretical predictions with data from experimental setups in which the presence of a trap isneeded to confine the system in a limited region, such as Bose Einstein condensates and coldatoms [21–24]. Finally we will show that the method can be used to consistently computeperturbations to the three point functions as well. In this case we will show that the three-point correlation function satisfies an associativity condition outside the critical point thatcan give non trivial relations between the Wilson coefficients and its derivatives. The lastSection of the paper is devoted to conclusions and final remarks.
II. CONFORMAL PERTURBATION THEORY: A REVIEW
Let us review the technique derived in [10], to reconstruct the short distance behavior ofthe correlators of operators in a D dimensional Euclidean quantum field theory at a fixedpoint of the renormalization group, described by the action S e , perturbed by one ore morerelevant operators O i with couplings m i , so that the perturbation to S e takes the form:∆ S = − (cid:90) d D x (cid:88) i m i O i ( x ) . (2.1)Since the operators O i are supposed to be relevant, their dimensions x i have to satisfy theinequality: 0 < x i < D . (2.2)In what follows, we will be especially interested in statistical physics applications. In thisspirit, the unperturbed theory described by S e is a conformal field theory, and the methodwhich we are going to outline describe a consistent way to go outside the critical point.In general, the short distance behavior of the correlators of a perturbed field theory isdescribed by the operator product expansion (OPE): (cid:104) Φ a ( r ) ... Φ a n ( r n ) X (cid:105) m ∼ C ca ...a n ( r − r n , ... , r n − − r n ; m ) (cid:104) Φ c ( r n ) X ( R ) (cid:105) m , (2.3)4here Φ i ( r i ) describes a complete set of composite operators of dimension x i , X ( R ) couldbe either the identity operator ( X ( R ) = ) or a multi-local operator defined on | R | > | r | , ... , | r n | , and the index m indicates that the correlators are computed at fixed sources m i .If the unperturbed theory ( m i = 0) is a conformal field theory, the convergence of theOPE at m i = 0 is well understood [25]. However, the convergence of OPE in a generalperturbed field theory is far from being proved. By the way, for the sake of the methodto be valid, it is sufficient to assume the asymptotic weak convergence of the OPE, namelythat the truncated OPE expansion∆ ( N ) a ...a n ( X ( R ) , m ) ≡ (cid:104) (cid:32) Φ a ( r ) ... Φ a n ( r n ) − x c ≤ N (cid:88) c C ca ...a n ( { r i } ; m ) Φ c ( r n ) (cid:33) X ( R ) (cid:105) m (2.4)satisfies the condition lim N →∞ lim R →∞ ∆ ( N ) a ...a n ( X ( R ) , m ) ∼ m i . The dependence on the coordinates { r i } of theWilson coefficients C ca ...a n has to be understood as in (2.3). The condition (2.5) is equivalentto the assumption that arbitrary order derivatives of ∆ ( N ) a ...a n with respect to the couplings m i asymptotically converge in the m i → N →∞ lim m → lim R →∞ ∂ n m ...∂ n k m k ∆ ( N ) a ...a n ( X ( R ) , m ) = 0 (2.6)for every k and n i . A. First order expansion
Having described the basic hypothesis on which we rely, we are now ready to describe themethod. For the sake of simplicity, let us firstly concentrate on the first order expansion.The goal will be to express the first derivative of the Wilson coefficients C ca ...a n with respectto a certain coupling m i in terms of quantities of the unperturbed theory.From the weak convergence hypothesis (2.6), we know that:lim N →∞ lim m → lim R →∞ ∂ m i ∆ ( N ) a ...a n ( X ( R ) , m ) = 0 . (2.7)5n order to expand the previous expression in a useful way, we notice that, due to the basicaction principle [26, 27], the following equality holds: ∂ m i (cid:104) X ( R ) (cid:105) m = (cid:90) d D x (cid:104) : O i ( x ) : X ( R ) (cid:105) , with : O i ( x ) : ≡ O i − (cid:104)O i (cid:105) . (2.8)Keeping in mind the definition of ∆ ( N ) a ...a n ( X ( R ) , m ) (2.4), and using (2.8), equation (2.7)becomes:lim N →∞ lim R →∞ (cid:34)(cid:90) | ¯ r | < | R | d D ¯ r (cid:104) : O i (¯ r ) : (cid:32) Φ a ( r ) ... Φ a n ( r n ) − x c ≤ N (cid:88) c C ca ...a n ( { r i } )Φ c ( r n ) (cid:33) X ( R ) (cid:105)− x c ≤ N (cid:88) c ∂ m i C ca ...a n ( { r i } ) (cid:104) Φ c ( r n ) X ( R ) (cid:105) (cid:35) = 0 , (2.9)where the Wilson coefficients C ca ...a n ( { r i } ) and the expectation values (cid:104) ... (cid:105) without the index m are the ones evaluated at m i = 0. The relation (2.9) establishes a set of constraints whichinvolve the first derivative of the Wilson coefficient C ca ...a n with respect to m i and correlatorsof the unperturbed field theory. Eventually, if one knows the correlators and the Wilsoncoefficients of the model at the critical point, the system (2.9) can be solved in terms of ∂ m i C ca ...a n obtaining the first order m i expansion of the Wilson coefficients.To clarify further equation (2.9), it is worth to note that it can be further simplified ifone considers X ( R ) = , and noticing that, due to dimensional consideration,lim | R | ... | R k |→∞ (cid:90) | r | < | R | d D r ... (cid:90) | r k | < | R k | d D r k (cid:104) : O i ( r ) : ... : O k ( r k ) : Φ c ( r n ) (cid:105) = 0 (2.10)if the dimension x c of Φ c is less the the sum of the mass dimensions y i ≡ D − x i of eachoperator O i , namely if x c − k (cid:88) j =1 y j > . (2.11)This is due to the absence of a physical scale in the unperturbed theory S e . With this inmind, the sum in the second term in (2.9) reduces to all the Φ c with x c < y i , while the onlynon trivial contribution to the sum in the third term is the one with Φ c ( r n ) = . Finally,one obtains: ∂ m i C a ...a n ( { r i } ) =lim R →∞ (cid:90) | ¯ r | < | R | d D ¯ r (cid:104) : O i (¯ r ) : (cid:32) Φ a ( r ) ... Φ a n ( r n ) − x c ≤ y i (cid:88) c C ca ...a n ( { r i } )Φ c ( r n ) (cid:33) (cid:105) . (2.12)6t this point one might wonder if the previous results might be plagued by ultravioletdivergences. It is worth to stress that in the CPT of a generic CFT there are no issuesconcerning ultraviolet divergences, as pointed out in [10]. On the contrary, the free bosoncase considered in the introduction does not belong to this category and one has to be carefulin defining the perturbing composite operator φ (see [10] for a similar discussion in the 2DIsing model and the example IIIA in this article). In this situation there appear logarithmicterms in the short distance expansion. B. Higher order expansion
In a similar way, one can consider the expansion of the Wilson coefficients up to higherorders in the derivatives with respect to the couplings m i . By expanding the relationlim N →∞ lim R →∞ lim m → ∂ m ...∂ m k ∆ ( N ) ab ( X R , m ) , (2.13)and repeating the same steps done in the previous Section, one obtains:lim | R |→∞ (cid:110) (cid:90) | r | < | R | d D r ... (cid:90) | r k | < | R | d D r k (cid:104) (cid:104) : O i k : ... : O i : (cid:16) Φ a ( r ) ... Φ a n ( r n ) − x b ≤ ¯ x (cid:88) b C ba ...a n ( { r i } )Φ b ( r n ) (cid:17) X ( R ) (cid:105) (cid:105) − x b ≤ ¯ x (cid:88) b ∂ i C ba ...a n ( { r i } ) (cid:90) | r | < | R | d d r ... (cid:90) | r k | < | R | d D r k (cid:104) : O i k : ... : O i : Φ b ( r n ) X ( R ) (cid:105) ... − x b ≤ ¯ x (cid:88) b ∂ i ...∂ i k − C ba ...a n ( { r i } ) (cid:90) | r k | < | R | (cid:104) : O i k : Φ b ( r n ) X ( R ) (cid:105)− x b ≤ ¯ x (cid:88) b ∂ i ...∂ i k C ba ...a n ( { r i } ) (cid:104) Φ b ( r n ) X ( R ) (cid:105) (cid:111) , (2.14)where ¯ x = (cid:80) k y i k − x X ( R ) . For generic X ( R ) the previous relations provide a set of con-straints which relate the derivatives of the Wilson coefficients with respect to the couplingto correlators of the unperturbed theory. This system can be consistently solved in termsof the derivatives of the Wilson coefficients if one knows the properties of the model at thecritical point. 7nce again, we can simplify the previous relation considering X ( R ) = , finally obtainingan equality for the derivatives of C a ...a n , namely: ∂ i ...∂ i k C a ...a n ( { r i } ) =lim | R |→∞ (cid:110) (cid:90) | r | < | R | d D r ... (cid:90) | r k | < | R | d D r k (cid:104) (cid:104) : O i k : ... : O i : (cid:16) Φ a ( r ) ... Φ a n ( r n ) − x b ≤ ¯ x (cid:88) b C ba ...a n ( { r i } )Φ b ( r n ) (cid:17) (cid:105) (cid:105) − x b ≤ ¯ x (cid:88) b ∂ i C ba ...a n ( { r i } ) (cid:90) | r | < | R | d d r ... (cid:90) | r k | < | R | d D r k (cid:104) : O i k : ... : O i : Φ b ( r n ) (cid:105) ... − x b ≤ ¯ x (cid:88) b ∂ i ...∂ i k − C ba ...a n ( { r i } ) (cid:90) | r k | < | R | (cid:104) : O i k : Φ b ( r n ) (cid:105) (cid:111) . (2.15)As a final comment, we note that having obtained the relations (2.9) and (2.14) one hasto prove that the expansion is IR finite, namely that all the non perturbative contributionsof the correlators are encoded in the expectations values of the operators in the OPE (2.3).The proof has been outlined in [10] and, since it goes beyond the purposes of the presentpaper, we refer to [10] for further details. III. EXAMPLES: 2D ISING MODEL APPLICATIONS
In this Section we will describe some applications of the conformal perturbation theorytechniques previously described to the 2D Ising model. As known (see e.g. [28]), at thecritical point, T = T c , the Ising model is described by the continuous unitary conformal fieldtheory M (3 / , σ and E , withdimension x = 0 , / σ ][ σ ] = [ ] + [ E ] , [ E ][ E ] = , [ σ ][ E ] = [ σ ] . (3.1)The previous relations implies that correlation functions involving an odd number of σ sidentically vanishes.In order to set the conventions and the normalization, we list below some of the correlationfunctions and Wilson coefficients of this theory which will be useful for our purposes in the8ollowing sections: C EE ( z ) = 1 | z | , C σσ ( z ) = 1 | z | , C E σσ ( z ) = | z | , C σσ E ( z ) = 12 | z | , (3.2)and (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) = | z | | z || z | , (3.3) (cid:104) σ ( z ) σ ( z ) E ( z ) E ( z ) (cid:105) = | z ( z + z ) − z z | | z z z z || z | | z | , (3.4) (cid:104) σ ( z ) σ ( z ) σ ( z ) σ ( z ) (cid:105) = | (1 − x ) z z | − (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) √ − x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) − √ − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (3.5)where z is a complex coordinate in the 2 D plane, z ij ≡ z i − z j and x = z z z z . A. Thermally perturbed two point functions
As a first example, we consider a case in which we move away from the critical point byperturbing the Ising model with the energy operator, namely: S = S Ising + λ (cid:90) E ( z ) d z . (3.6)Let us analyze the two point function (cid:104)E ( z ) σ ( z ) (cid:105) λ . Assuming the the operator productexpansion is valid outside the critical point we obtain: (cid:104)E ( z ) σ ( z ) (cid:105) λ = C E σ ( z ; λ ) + C EE σ ( z ; λ ) (cid:104)E (cid:105) λ + C σ E σ ( z ; λ ) (cid:104) σ (cid:105) λ . (3.7)The previous correlator vanishes identically at λ = 0 since it contains just one insertion of σ .Moreover, by using the relation (2.15), one can easily prove that C E σ ( z ; λ ) and C EE σ ( z ; λ )vanishes identically at all order in perturbation theory, since the correlation functions thatone has to compute always contain an odd number of insertion of σ . Eventually, we find that,outside the critical point, the correlator is proportional to expectation value of σ , namely (cid:104)E ( z ) σ ( z ) (cid:105) λ = C σ E σ ( z ; λ ) (cid:104) σ (cid:105) λ , (3.8)and is non-zero only in the magnetically ordered phase. Expanding (3.8) up to the firstorder in the coupling λ we find: (cid:104)E ( z ) σ ( z ) (cid:105) λ = ( C σ E σ ( z ) + λ∂ λ C σ E σ ( z ) + ... ) (cid:104) σ (cid:105) λ , (3.9)9here the dots stand for higher order corrections. Regarding the expectation value of σ , bydimensional analysis we obtain: (cid:104) σ (cid:105) λ = A σ λ , (3.10)where A σ is a non-universal constant. To compute C σ E σ ( z ) we rely on the operator productexpansion (2.3) and on the orthogonality of the 2-pt correlation functions at the criticalpoint, namely: C σ E σ ( z ) = lim | z |→∞ (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105)(cid:104) σ ( z ) σ ( z ) (cid:105) = lim | z |→∞ | z | | z || z | = 12 | z | . (3.11)In order to evaluate the second term of the parenthesis in (3.9) we use the relation (2.9)derived in the previous Section. Specifically, by setting X = σ we obtain: ∂ λ C σ E σ ( z ) lim | z |→∞ (cid:104) σ ( z ) σ ( z ) (cid:105) = lim | z |→∞ (cid:90) | z | < | z | d z [ (cid:104) σ ( z ) σ ( z ) E ( z ) E ( z ) (cid:105)− C σσ E ( z ) (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) − C σ σ E ( z ) (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) ] (3.12)where σ ≡ L − ¯ L − σ and L n are the Virasoro operators.The last term in (3.12) does not contribute to the integral. Regarding the other twoterms, after dividing for (cid:104) σ ( z ) σ ( z ) (cid:105) , performing the | z | → ∞ limit and setting z = 0 and z = z we obtain: ∂ λ C σ E σ ( z ) = (cid:90) d z z ¯ z + ¯ z z | z || z || z − z | . (3.13)The integral has an UV divergence when z approaches z . We regularise it by setting acutoff at | z | = | z | + (cid:15) . The final result is: ∂ λ C σ E σ ( z ) = 2 π (cid:18) log 2 | z | (cid:15) − (cid:19) . (3.14)To get rid of the UV regulator one has to consider the connected part of the correlator bysubtracting to (3.9) the quantity (cid:104)E ( z ) (cid:105) λ (cid:104) σ ( z ) (cid:105) λ . In order to do this, we need to computethe expectation value (cid:104)E ( z ) (cid:105) λ . This has been done in [10] by noting that the action principlehas to be valid, namely: ∂ λ (cid:104)E (0) (cid:105) λ = (cid:90) d z (cid:104)E ( z ) E (0) (cid:105) λ = − πλ log 2 πλe γ E (cid:15) , (3.15) This can be done due to conformal invariance at the critical point. (cid:104)E ( z ) σ ( z ) (cid:105) λ − (cid:104)E ( z ) (cid:105) λ (cid:104) σ ( z ) (cid:105) λ = A σ λ (cid:18) | z | + 2 πλ (log 4 πλ | z | − γ E ) + ... (cid:19) . (3.16)Note that the computation (3.15) is non perturbative, since, as explained in [10], relies onthe exact (cid:104)E E (cid:105) λ , which is known analytically outside the critical point for the special caseof the 2D Ising model. As a consequence it is important to verify, as we have done, that theconnected Green’s function (3.16) does not depend on the UV regulator (cid:15) . The expansion(3.16) is in perfect agreement with the exact result of the correlator found in [20]: (cid:104)E ( z ) σ ( z ) (cid:105) λ − (cid:104)E ( z ) (cid:105) λ (cid:104) σ ( z ) (cid:105) λ = 2 πλA σ λ (cid:90) ∞ πλ | z | ds s − e − s . (3.17) B. Trapped Ising model
As a second example, inspired by the analysis of [29], we consider the 2D Ising modelwith a trap perturbation, namely: S = S CF T + (cid:90) d z U ( z ) σ ( z ) , with U ( z ) = v p | z | p ≡ ρ | z | p . (3.18)In the previous expression p ( ≥
2) is a generic exponent of the trap potential while ρ isthe characteristic trap parameter. In what follows we will consider the effect of a small ρ perturbation, namely the limit of large trap, on the two point functions of the Ising model.As explained in [29], this might be of relevance in experimental studies of trapped criticalsystems such as cold atoms and Bose-Einstein condensates.As noted in [29, 30], by using renormalization group arguments, one can deduce thescaling behavior of the expectation value of the spin and the energy operator in the centerof the trap: (cid:104) σ (0) (cid:105) ρ = B σ ρ θ , (cid:104)E (0) (cid:105) ρ = B E ρ θ , (3.19)where the exponent θ = p is the characteristic trap exponent, while B σ and B E are nonuniversal constants.The perturbation that we are considering breaks translational invariance but preservesrotational symmetry. Hence, the fusion rule of the perturbed model remains the same as theunperturbed one and, if we fix one operator in the center of the trap, it is possible to rely11n OPE and on the perturbative techniques described in the previous sections to computethe two point functions outside the critical point. The previous argument yields: (cid:104) σ ( z ) σ (0) (cid:105) ρ = C σσ ( z ) + C E σσ ( z ) B E ρ θ + ∂ ρ C σσσ ( z ) B σ ρ θ +1 + ... , (3.20) (cid:104)E ( z ) E (0) (cid:105) ρ = C EE ( z ) + ∂ ρ C σ EE ( z ) B σ ρ θ +1 + ... , (3.21) (cid:104) σ ( z ) E (0) (cid:105) ρ = C σσ E ( z ) B σ ρ θ + ρ ∂ ρ C σ E ( z ) + ∂ ρ C E σ E ( z ) B E ρ θ +1 ... . (3.22)The last terms in (3.20), (3.21) and (3.22) can be computed by using the conformal pertur-bation theory techniques described in the previous sections. Specifically, one has to evaluatethe following integrals: − ∂ ρ C σσσ ( z ) lim | z |→∞ (cid:104) σ ( z ) σ (0) (cid:105) = lim | z |→∞ (cid:90) | z | < | z | d z | z | p (cid:104) (cid:104) σ ( z ) σ ( z ) σ ( z ) σ (0) (cid:105)− C σσ ( z ) (cid:104) σ ( z ) σ ( z ) (cid:105) − C E σσ ( z ) (cid:104) σ ( z ) σ ( z ) E (0) (cid:105) (cid:105) , (3.23) − ∂ ρ C σ EE ( z ) lim | z |→∞ (cid:104) σ ( z ) σ (0) (cid:105) =lim | z |→∞ (cid:90) | z | < | z | d z | z | p (cid:104) (cid:104) σ ( z ) σ ( z ) E ( z ) E (0) (cid:105) − C EE ( z ) (cid:104) σ ( z ) σ ( z ) (cid:105) (cid:105) , (3.24) − ∂ ρ C σ E ( z ) = (cid:90) d z | z | p (cid:104) (cid:104) σ ( z ) σ ( z ) E (0) (cid:105) − C σσ E ( z ) (cid:104) σ ( z ) σ (0) (cid:105) (cid:105) , (3.25) − ∂ ρ C E σ E ( z ) lim | z |→∞ (cid:104)E ( z ) E (0) (cid:105) =lim | z |→∞ (cid:90) | z | < | z | d z | z | p (cid:104) (cid:104) σ ( z ) σ ( z ) E ( z ) E (0) (cid:105) − C σσ E ( z ) (cid:104) σ ( z ) E ( z ) σ (0) (cid:105)− C σ σ E ( z ) (cid:104) σ ( z ) E ( z ) σ (0) (cid:105) (cid:105) (3.26)A detailed discussion on how to treat the last expressions can be found in Appendix A. Herewe outline the results in the case of harmonic trap ( p = 2), which is the relevant one intypical experimental setups [29]: ∂ ρ C σ EE ( z , p = 2) = | z | π (3 + 2 γ E − ∂ ρ C σ E ( z , p = 2) = −| z | π cot (cid:0) π (cid:1) Γ (cid:0) (cid:1) (cid:0) (cid:1) (3.28) ∂ ρ C E σ E ( z , p = 2) = −| z | (cid:0) − (cid:1) Γ (cid:0) (cid:1) √ π (3.29) ∂ ρ C σσσ ( z , p = 2) = −| z | . n (cid:54) = 2 Z (see e.g. [31]): F [ | z | n ]( | q | ) = 2 n n Γ (cid:0) n (cid:1) | q | − n − Γ (cid:0) − n (cid:1) . (3.31)while F [ | z | ]( q, ¯ q ) = − π∂ q ∂ ¯ q δ ( q ) δ (¯ q ) , F [ | z | − ]( | q | ) = − log | q | µ − γ E , (3.32)where µ is a renormalization parameter. The last equivalence in (3.32) has been obtainedby expanding (3.31) around the singular point n = − (cid:104) σ ( q ) σ ( − q ) (cid:105) ρ = 1 | q | / (cid:0) − . B E ρ θ | q | − . B σ ρ θ/ + 0 . | q | (cid:1) (3.33) (cid:104)E ( q ) E ( − q ) (cid:105) ρ = − log | q | µ − γ E − . B σ ρ θ +1 ∂ q ∂ ¯ q δ ( q ) δ (¯ q ) (3.34) (cid:104) σ ( q ) E ( − q ) (cid:105) ρ = − . q / B E ρ θ +1 + B σ ρ θ/ q − . q / ρ (3.35) C. Magnetically perturbed three-point functions
As we have pointed out in the first Section, CPT can be also applied to 3-point correla-tors . In what follows we will analyze the 2D Ising model perturbed by a magnetic field h ,namely S = S Ising + h (cid:90) σ ( z ) d z . (3.36)We will apply the conformal perturbation theory techniques described in Section II to thestudy of 3-point correlation functions outside the critical point. Specifically we will analyze (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) h . The assumption that the Operator Product Expansion is valid outsidethe critical point implies: (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) h = C σσ E ( z , z , z ; h ) + C σσσ E ( z , z , z ; h ) (cid:104) σ (cid:105) h + C E σσ E ( z , z , z ; h ) (cid:104)E (cid:105) h , (3.37)The leading correction to the correlation function comes from the term proportional to (cid:104)E (cid:105) h ,namely: (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) h = C σσ E ( z , z , z ) + C E σσ E ( z , z , z ) (cid:104)E (cid:105) h + ..... (3.38) See [32] for an example using the Potts model.
13n order to evaluate the previous expression we need the quantities C σσ E ( z , z , z ) and C E σσ E ( z , z , z ), as well as the expectation value (cid:104)E (cid:105) h . Dimensional analysis tell us that thelatter one is given by: (cid:104)E (cid:105) h = A E h , (3.39)where A E is a non-universal constant. Moreover, by definition, C σσ E ( z , z , z ) is given by: C σσ E ( z , z , z ) = (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) = | z | | z || z | . (3.40)where we have used (3.4). Finally, in order to compute C E σσ E ( z , z , z ) we rely, as in theprevious subsection, on operator product expansion (2.3) together with the orthogonalityof the 2-pt functions at the critical point. Eventually, in the limit | z | → ∞ , and using theequations (3.2) and (3.4), we obtain: C E σσ E ( z , z , z ) = lim | z |→∞ (cid:104) σ ( z ) σ ( z ) E ( z ) E ( z ) (cid:105)(cid:104)E ( z ) E ( z ) (cid:105) = lim | z |→∞ | z ( z + z ) − z z | | z z z z || z | , (3.41)Performing the | z | → ∞ limit we get: C E σσ E ( z , z , z ) = 14 | z − z | | z z || z | . (3.42)Putting all together, we finally obtain: (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) h = 12 | z | | z z | (cid:18) A E h / | z + z | | z | + ... (cid:19) , (3.43)where the dots stands for higher order correction in the magnetic field h . Finally we notethat in the limit | z | (cid:29) | z | , | z | (cid:29) | z | the following operator product expansion relationholds: (cid:104) σ ( z ) σ ( z ) E ( z ) (cid:105) h = (cid:104) σ ( z ) σ ( z ) (cid:105) h (cid:104)E ( z ) (cid:105) h = A E h | z | . (3.44) IV. CONCLUSIONS
In this note we have shown that in order to get infrared finite correlation functions neara critical point one has to modify the traditional approach to perturbation theory. We haveillustrated the correct approach by giving various non trivial examples of its application.Specifically, we have applied the method to compute the corrections due to several kindof relevant perturbations of the 2D Ising model not yet considered in the literature. In14he special 2D case the computation of the VEVs can be performed relying only on theknowledge of the CFT model using the truncated conformal space (TCS) approach [18], andthis simplify the discussion. However it is worth mentioning that the method illustrated inSection 2 is valid in any space-time dimension and it has already been applied to the 3 D Ising model [15–17], even though, at present, in the 3D case the computations of VEVs reliesonly on numerical simulations.As a first example, we have considered the perturbation due to the energy operator. Inparticular we have proven that, in this case, the conformal perturbation theory result forthe correlator (cid:104) σ E (cid:105) totally agrees with the exact correlator in the broken phase obtainedin [20]. This might be seen as a first step in the direction of applying the method tohigher dimensional models which exhibit a spontaneous symmetry breaking, as the 3 DO ( N ) model, where recent conformal bootstrap computations of the Wilson coefficient atthe critical point [33] may render the method described here applicable. In this case, thedynamics of Goldstone bosons might be relevant, and recently it has been pointed out in [34]that they might be the responsible for CPT to not reproduce correctly logarithmic termsin the current-current correlator. The method that we have illustrated just relies on thevalidity of the OPE outside the critical point. Since, as observed in [40], the OPE shouldalso be valid in case of spontaneous symmetry breaking, it is worth to analyze if it is possible,by applying this method, to reproduce the correct result for the current-current correlatorin the O ( N ) model. As a second example, we have considered the perturbation due to the presence of a trap.This example might be of relevance both from the experimental and theoretical point ofview. From the experimental side, the present analysis might be useful for comparing theresults with experimental data on trapped systems such as cold atoms [21–24]. From thetheoretical point of view, it is worth to mention that the trap case has some similarities withconsidering the system at finite temperature. In fact, the temperature acts as a box in theeuclidean time direction, so that the VEV will be modified, but the Wilson coefficients willremain unchanged. In this direction the method could be applied to compute the responsefunctions near a quantum critical point at zero or finite temperature [34].With the third example, we have proven that the method can be extremely powerful Another interesting application of the present method is the analysis of the transport properties near thecritical point in the presence of spontaneous symmetry breaking of translations. In this case the resultscan be compared with the analogous holographic computation (see e.g. [35–39]). (cid:104) σσ E (cid:105) due to the presence of an ex-ternal magnetic field. As a possible future direction, the present analysis combined withrecent conformal bootstrap results, might be useful in order to get more insight in higherdimensional CFTs outside the critical point.The present method can be also applied in cases where the critical model is perturbedwith more than one relevant operator . As an example, it would be interesting to applyit to the Ising model perturbed with both the magnetic field and reduced temperature inorder to analyze the interplay of scales due to the coexistence of two kind of perturbations.Finally, results from CPT can be useful in connection with sum rules as in the QCD case[43], or in the quantum critical case [34, 44]. ACKNOWLEDGMENTS
A special thanks goes to Michele Caselle, Slava Rychkov and William Witczak-Krempafor providing comments on a preliminary version of the present paper. We would like alsoto thank Daniele Musso for useful conversations. Nicodemo Magnoli thanks the support ofINFN Scientific Initiative SFT: Statistical Field Theory, Low-Dimensional Systems, Inte-grable Models and Applications.
Appendix A: Mellin transform and asymptotic properties of integrals
The structure of the integrals that we have to calculate is of the following form: I ( m ) = (cid:90) d z Θ( m | z | ) g ( z ) , (A1)where Θ( m | z | ) = e − m | z | is an infrared cutoff function which can be eventually set to 0 at theend of the calculation. In order to obtain the asymptotic ( m ∼
0) expansion of the integral,we will make use of the Mellin transform of I ( m ). Assuming that the leading behaviors of I ( m ) are m a when m → m − b when m → ∞ , one can define the Mellin transform ˜ I ( s )on the strip − a < Re ( s ) < b in the complex s plane as:˜ I ( s ) = (cid:90) ∞ dmm m s I ( m ) . (A2) See e.g. [41, 42] for other methods to treat CFTs perturbed by relevant operators.
16t is well known, see ([45]), that the poles of the Mellin transform are in one to one corre-spondence with the asymptotic expansion of the original function at m = 0. In fact one canwrite: I ( m ) = (cid:88) i Res ( ˜ I ( s )) m − s (cid:12)(cid:12)(cid:12) s = − a i , (A3)where a ≡ a < a < ... are the powers of m in the asymptotic expansion of I ( m ) at m ∼ s = 0.The Mellin transform previously described is particularly useful for our purposes if onenotes that, by making use of the convolution theorem, it is possible to express the integral(A1) as: ˜ I ( s ) = Γ( s )˜ g (1 − s ) , (A4)where ˜ g (1 − s ) = (cid:90) d z | z | − s g ( z ) , (A5)is essentially the Mellin transform of g up to angular coefficients. Eventually, if one knowshow to compute the Mellin transform of g , the integrals necessary to evaluate the derivativecorrections of the Wilson coefficients are easily done. In our case, all the integrals have thefollowing form: I ( m ; x ) = (cid:90) d ze − m | z | | z | α | z − x | γ | z − | β (A6)The Mellin transform of this integral has been computed in [46], and is given by:˜ I ( s ; x ) = Γ( s ) D ( α − s/ , β, γ, x ) , (A7)where the integral D ( a, b, c, x ) = (cid:90) d | z | a | z − x | c | z − | b (A8)can be expressed as: D ( a, b, c, x ) = S ( a ) S ( c ) S ( a + c ) | I x | + S ( b ) S ( a + b + c ) S ( a + c ) | I ∞ | . (A9)In the last expression, S ( a ) ≡ sin ( πa ) and I x ≡ x a + c Γ( a + 1)Γ( c + 1)Γ( a + c + 2) F ( − b, a + 1 , a + c + 2 , x ) , (A10) I ∞ ≡ Γ( − a − b − c − b + 1)Γ( − a − c ) F ( − a − b − c − , − c, − a − c, x ) , (A11)where F is the Hypergeometric Gauss function.17 . Integrals for the trapped Ising model In this subsection we sketch the form of the integrals relevant to compute the pertur-bations of the trapped Ising model analyzed in Section (III B). These integrals can beevaluated by using the Mellin transform technique previously described. In this respect, itis sufficient to consider just the first terms in the integrals (3.23)-(3.26). Eventually, consid-ering a generic value of p , and after evaluating the | z | → ∂ ρ C σ EE ( z ) = −| z | p (cid:90) d w | w | p − | w − | | w − | − , (A12) ∂ ρ C σ E ( z ) = − | z | p + (cid:90) d w | w | p − | w − | , (A13) ∂ ρ C E σ E ( z ) = − | z | p + (cid:90) d w | w | p − | w − | − | w + 1 | , (A14) ∂ ρ C σσσ ( z ) = − | z | p + (cid:90) d w | w | | w − | − ( p − s +4) | w + 1 | − ( p − s +3) . (A15)As one can easily see, the previous integrals are of the form (A6), and can be evaluated byusing the Mellin technique transform described previously. [1] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, “Solvingthe 3D Ising Model with the Conformal Bootstrap,” Phys. Rev. , vol. D86, p. 025022, 2012.[2] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, “Solvingthe 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise CriticalExponents,”
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