Generalized Gross-Neveu Universality Class with Non-Abelian Symmetry
GGeneralized Gross-Neveu universality class with non-abeliansymmetry ∗ J.A. Gracey,Theoretical Physics Division,Department of Mathematical Sciences,University of Liverpool,P.O. Box 147,Liverpool,L69 3BX,United Kingdom.
Abstract.
We use the large N critical point formalism to compute d -dimensional criticalexponents at several orders in 1 /N in an Ising Gross-Neveu universality class where the coreinteraction includes a Lie group generator. Specifying a particular symmetry group or takingthe abelian limit of the final exponents recovers known results but also provides expressions forany Lie group or fermion representation. LTH 1256 ∗ Invited talk presented at Algebraic structures in perturbative quantum field theory, IH´ES, Bures-sur-Yvette,France, 16th-20th November, 2020. a r X i v : . [ h e p - t h ] F e b Introduction.
One aspect of quantum field theory that has important applications to Nature is the study offixed points of the renormalization group functions. These are defined to be the non-trivial zerosof the β -function. Using the location of a fixed point one can compute the values of the renor-malization group functions there to produce renormalization scheme independent expressionsknown as critical exponents, [1, 2, 3, 4]. These quantities govern the dynamics of the phasetransitions in a material. Indeed accurate measurements of the exponents experimentally aswell as the symmetry properties of a material, can equally guide one to the underlying quantumfield theory or spin model describing the dynamics. One property is that more than one theorycan be a valid description of a phase transition. For instance, both a continuum quantum fieldtheory as well as a discrete spin model with a common symmetry can be valid tools to providenumerical exponent estimates. The equivalence of both theoretical techniques at a fixed pointis known as universality, [3, 4]. Away from a phase transition each theory will have differentproperties and be inequivalent.One theory that has risen to the fore in this context in recent years is that of the IsingGross-Neveu model, [5, 6]. This is primarily due to the belief that it underpins a particularphase transition in graphene. This material is made up of a sheet of carbon atoms arranged ina hexagonal lattice. When the two dimensional sheet is stretched it can undergo a transitionfrom an electrical conductor to what is termed a Mott-insulating phase. On the theoretical sidethe Ising Gross-Neveu model can be supplemented with Quantum Electrodynamics (QED) todescribe aspects of other phase transitions. Equally if the basic or Ising Gross-Neveu modelis endowed with extra symmetries it contributes to the understanding of transitions in othermaterials. For example, what is termed the chiral Heisenberg Gross-Neveu model is an extensionof the Gross-Neveu model to include an SU (2) symmetry. It is the effective theory for electronson a half-filled honeycomb lattice where there is a phase transition between an anti-ferromagneticinsulating phase and a semi-metallic one [7, 8, 9]. Its criticality properties were studied in [10].More recently a variation of this version of the Gross-Neveu model, called the fractionalizedGross-Neveu model, has been developed, [11]. It has a novel spectrum that differs from those ofother Gross-Neveu models and has an associated SO (3) symmetry. What is clear from this setof emerging variations of the Gross-Neveu universality class is the common theme of the coreinteraction being modified to include a non-abelian symmetry. In this respect it is completelyparallel to the extension of QED to include a non-abelian symmetry that equates to QuantumChromodynamics (QCD) or Yang-Mills theory with fermions in the fundamental representationof the Lie group responsible for colour symmetry. The only difference with the Gross-Neveu classof theories is in the Lorentz structure of the core interaction. As QCD has been studied at lengthusing a general Lie group symmetry rather than specifying SU (3) at the outset, which is thegroup that governs the strong interactions, it seems sensible to develop a programme to calculatein the Gross-Neveu model with a parallel non-abelian symmetry. Then the properties of thevarious physical applications can be deduced by specifying the appropriate group parameters.This is the main task here. We will consider a generalized Gross-Neveu universality classwith a non-abelian symmetry and calculate the critical exponents of the theory. This will beachieved by using the critical point large N formalism pioneered in [12, 13, 14] for the nonlinear O ( N ) σ model. Here N would correspond to the number of quark flavours in the analogy withQCD. The elegance of the approach is such that we can deduce the critical exponents in d spacetime dimensions as a function of the non-abelian symmetry group Casimirs. The fixedpoint associated with the formalism is the Wilson-Fisher fixed point in d -dimensions [3]. As theexponents are renormalization group invariants their (cid:15) expansion near 2 and 4 dimensions, where (cid:15) measures the difference in these values from d , will agree with the perturbative evaluation of2he same exponents in the respective quantum field theories of the universality class. Moreusefully since the dimension of interest for the materials application is three, one can determineseveral terms of the 1 /N series for each exponent. These provide reasonably accurate estimatesfor relatively low N when compared to results from other techniques. However, the benefitof taking the general non-abelian universality class approach is that estimates will be readilyavailable if a phase transition with a new symmetry is discovered.The article is organized as follows. Relevant background concerning the generalized Gross-Neveu universality class is given in Section 2 together with the basic critical point large N formalism. Subsequently in section 3 we solve the Schwinger-Dyson equations at criticalityat O (1 /N ) to produce the fermion anomalous dimension. Various calculational tools thatare necessary for this are reviewed as well. To provide the groundwork for finding the nextorder of this exponent, the anomalous dmension exponent of the bosonic field is determined at O (1 /N ) in Section 4. One of the other basic exponents in critical systems is that relating tothe correlation length behaviour and we determine it at O (1 /N ) in Section 5. Equipped withthese results, the large N conformal bootstrap formalism at criticality is applied in Section 6 todeduce the fermion anomalous dimension at O (1 /N ). We review our results in Section 7 andprovide concluding remarks in Section 8. To begin with we recall the Lagrangian of the chiral Heisenberg Gross-Neveu Yukawa theory is,[9], L cHGNY = i ¯ ψ iI ∂/ψ iI + 12 ∂ µ ˜ π a ∂ µ ˜ π a + g ˜ π a ¯ ψ iI T aIJ ψ iJ + 124 g (˜ π a ˜ π a ) . (2.1)which is renormalizable in four dimensions where the two couplings g and g are dimensionless.This is a generalization of the Lagrangian studied in [15] and is in the chiral Heisenberg Gross-Neveu model universality class. The renormalizability dimension is also termed the criticaldimension of the theory. The scalar-fermion interaction includes the group generator T a of theLie algebra and the indices take values in the ranges 1 ≤ i ≤ N , 1 ≤ I ≤ N c and 1 ≤ a ≤ N A where N is the number of flavours of massless fermions and N c and N A are the respective dimensionsof the fundamental and adjoint representations of the symmetry group. We note that in [9] thespecific group considered was SU (2). Within our ultimate critical exponents the generators willmanifest themselves through various colour Casimirs such as C F and C A defined by T a T a = C F , f acd f bcd = C A δ ab (2.2)where f abc are the structure constants. The scalar field ˜ π a plays a subtle role in the constructionof the large N expansion but in four dimensions it corresponds to a fundamental propagatingfield. The main aspect of the large N critical point formalism of [12, 13, 14] is that in theapproach to criticality at the Wilson-Fisher fixed point the dynamics are driven by the coreinteraction of the universal quantum field theory. For (2.1) this is the cubic interaction togetherwith the fermion kinetic terms. These two terms determine the canonical dimensions of bothfields by ensuring the action is dimensionless in d -dimensions. In effect the universal Lagrangianat criticality is L cHGN = i ¯ ψ∂/ψ + g ˜ π a ¯ ψT a ψ −
12 ˜ π a ˜ π a . (2.3)where the quadratic term in ˜ π a is necessary for large N renormalizability. We will omit thelabels i and I for brevity from now on. We say in effect since at criticality there is no couplingconstant in the sense it is conventionally used in perturbation theory. So the critical point3niversal Lagrangian that we will be the foundation for applying the large N critical pointformalism of [12, 13] is L univ = i ¯ ψ∂/ψ + π a ¯ ψT a ψ − g π a π a (2.4)where we have rescaled the scalar field ˜ π a to introduce π a . This Lagrangian (2.4) is renormal-izable in d -dimensions in the large N formalism [15, 16, 17, 18] where 1 /N is the dimensionlessordering parameter since the perturbative coupling constant is absent at criticality. Ensuringthat the d -dimensional Lagrangian (2.4) has a dimensionless action means that ψ has canonicaldimension ( d −
1) while that of π a is unity. For (2.4) this implies that g is dimensionless intwo dimensions after eliminating the auxiliary field π a producing L = i ¯ ψ iI ∂/ψ iI + g (cid:16) ¯ ψ iI T aIJ ψ iJ (cid:17) . (2.5)This is similar to the Ising Gross-Neveu model discussed in [1] and to see the equivalence, onetakes the abelian limit of (2.5) by replacing the group generators with the unit matrix. This iscompletely parallel to taking the abelian limit of QCD to produce QED. The dimensionality of π a at criticality plays a key role in the connection of the universal theory and the Lagrangian of(2.1). In the latter both couplings are dimensionless in four dimensions similar to the effectivecoupling of the 3-point interaction of (2.4). Therefore (2.4) would be strictly non-renormalizablein four dimensions and the quadratic term in π a would have a dimensionful coupling which wouldbe the mass. Instead the standard kinetic term and quartic π a interaction of (2.1) would berelevant. In other words we term the quartic interaction to be a spectator interaction that wouldbe active solely in four dimensions. Moreover underlying the first two terms of the universalLagrangian (2.4) there are an infinite number of Lorentz scalar operators built from combinationsof π a and its derivatives. A finite subset of these extra operators would become relevant in eveninteger dimensions and act as interim spectators in the infinite tower of renormalizable quantumfield theories that connect to the universal theory in the neighbourhood of their respective criticaldimensions.More concretely we now summarize the key aspects of the large N critical point formalsmfor (2.4). In the approach to the fixed point the propagators have the following asymptoticbehaviour in coordinate space, [20], ψ ( x ) ∼ Ax/ ( x ) α (cid:104) A (cid:48) ( x ) λ (cid:105) , π ( x ) ∼ C ( x ) γ (cid:104) C (cid:48) ( x ) λ (cid:105) (2.6)where the name of the corresponding field is used. The dimensionless quantities A and C arethe coordinate independent amplitudes that will always occur in the combination y = A C fromthe 3-point interaction. The next to leading order terms in (2.6) that involve the exponent λ arecalled the corrections to scaling. Here λ will be identified with the correlation length exponent ν through 1 /ν = 2 λ . In addition to the canonical dimension the two fields have anomalouscontributions and the respective full dimensions of ψ and π a are α = µ + η , γ = 1 − η − χ π (2.7)where we use d = 2 µ for shorthand, [12], and η and χ are the fermion field and vertex anomalousdimensions respectively. For applications in condensed matter problems the dimension of π a thatis conventionally used is η π = 4 − µ − η + χ π ) . (2.8)When λ corresponds to the correlation length exponent its canonical dimension will then betaken to be ( µ − A (cid:48) and C (cid:48) . Each of the exponents that we will compute as well as y will dependon N , µ and the Lie group Casimir invariants. The dependence on the former means that eachentity has a Taylor series in powers of 1 /N that is formally given by η ( µ ) = ∞ (cid:88) n =1 η n ( µ ) N n , y ( µ ) = ∞ (cid:88) n =1 y n ( µ ) N n (2.9)for η and y for example and we will determine the first three terms of η for (2.4).These general considerations cover the basic formalism for the technique introduced in [12,13, 14]. To determine all bar η we can apply the original method [12, 13] that was used for theIsing Gross-Neveu universality class in [21, 22, 23, 24, 25, 26]. This required solving the skeletonSchwinger-Dyson equations for the ψ and π a x ) α = a ( α )2 α (cid:90) k e ikx ( k ) µ − α (2.10)where a ( α ) ≡ Γ( µ − α )Γ( α ) . (2.11)we transform (2.6) to momentum space carry out the inversion and then apply the inverse Fouriertransform. This results in the coordinate space 2-point function asymptotic scaling forms whichare, [20], ψ − ( x ) ∼ r ( α − x/A ( x ) µ − α +1 (cid:104) − A (cid:48) s ( α − x ) λ (cid:105) π − ( x ) ∼ p ( γ ) C ( x ) µ − γ (cid:104) − C (cid:48) q ( γ )( x ) λ (cid:105) . (2.12)The presence of the function a ( α ) in the Fourier transform produces a complicated dependenceon µ and the exponents, leading to the compact functions p ( γ ) = a ( γ − µ ) a ( γ ) , r ( α ) = αp ( α )( µ − α ) q ( γ ) = a ( γ − µ + λ ) a ( γ − λ ) a ( γ − µ ) a ( γ ) , s ( α ) = α ( α − µ ) q ( α )( α − µ + λ )( α − λ ) . (2.13)While the large N conformal bootstrap formalism of [14] has its origins in these asymptoticscaling functions and was applied to the Ising Gross-Neveu model in [22, 23, 26], the extractionof an expression for η derives from the scaling behaviour of the 3-point function. We defer toa later section for the required technicalities of that formalism. -point Schwinger-Dyson equation. Equipped with the asymptotic scaling forms of the full propagators, (2.6) and (2.12), whichrepresent the behaviour at criticality, we use them to solve the Schwinger-Dyson equations. Inconventional perturbation theory one systematically renormalizes the divergent n -point Green’s5unctions in a renormalizable theory order by order in perturbation theory. This principle isrespected in the large N technique of [12, 13] except that the ordering of graphs in the n -pointfunctions is achieved by the variable 1 /N which is dimensionless across all spacetime dimensionsunlike the perturbative coupling constant. For (2.4) the first few terms in the respective 2-pointfunctions of the fields are given in Figure 1 where the dotted line represents the fermion and thewiggly line denotes the π a field. The two loop graphs are the O (1 /N ) corrections to the oneloop ones. The counting of powers of N arise from closed fermion loops giving a factor of N and the π a field. The expansion of the amplitude variable begins at O (1 /N ) and this translatesinto each π a line in Figure 1 carrying a power of 1 /N . One key point worth noting concernsthe lack of dressing of lines with self-energy corrections. Contributions from such graphs arealready accounted for in the inclusion of a non-zero anomalous dimension in the power of theasymptotic scaling forms. π − + + Π ψ − + + Σ Figure 1: O (1 /N ) corrections to the skeleton Schwinger-Dyson 2-point functions.At leading order the two equations of Figure 1 equate to0 = r ( α −
1) + C F y + O (cid:18) N (cid:19) p ( γ ) + T F N y + O (cid:18) N (cid:19) (3.1)where we have included the respective group theory factors which derive from the properties of T a and Tr (cid:16) T a T b (cid:17) = T F δ ab . (3.2)In this coordinate space representation the one loop graphs require no evaluation. This is becauseone integrates over the coordinate of the internal vertices. As the one loop graphs have externalvertices the corresponding terms of (3.1) are the products of the propagators. In this leadingorder instance any integration has been effected in the derivation of the scaling forms for the full O (1 /N ) it containstwo unknowns. Using (2.9) and the 1 /N expansion of r ( α −
1) and p ( γ ) these are η and y andmoreover at this order they occur linearly. Thus eliminating y between the equations of (3.1)produces η = − µ − C F µ Γ(1 − µ )Γ( µ − ( µ ) T F . (3.3)At next order the situation is not as straightforward due to the additional graphs of Figure 1being divergent which necessitates the introduction of renormalization constants. The formalismfor this was provided in [16, 17] and requires the introduction of a regularization which is achievedthrough the shift, [12, 13], χ π → χ π + ∆ (3.4)6here ∆ is a small parameter. In effect it equates to an analytic regularization of the propagatorsand we emphasize that the spacetime dimension d does not play any role in the regularizationin contrast to coupling constant perturbation theory. Consequently the extension of (3.1) to thenext order has to account for this and so the algebraic representation of Figure 1 becomes0 = r ( α −
1) + C F yZ V ( x ) χ π +∆ + C F [2 C F − C A ] y Z V Σ ( x ) χ π +2∆ + O (cid:18) N (cid:19) p ( γ ) + T F N yZ V ( x ) χ π +∆ + T F [2 C F − C A ] N y Z V Π ( x ) χ π +2∆ + O (cid:18) N (cid:19) (3.5)in coordinate space where Z V is the vertex renormalization constant. It has the Laurent expan-sion Z V = 1 + ∞ (cid:88) l =1 l (cid:88) n =1 m ln ∆ n (3.6)where the residues are m ln = ∞ (cid:88) i =1 m ln,i N i . (3.7)after expanding in powers of 1 /N . We follow [16, 17] and restrict to the MS scheme. In (3.5) the x dependence arises from the dimensionality of the integrals in the regularized Lagrangian. Asit stands the various factors prevent the limit to criticality from being taken smoothly. Moreoverthe factors associated with the one loop graphs of Figure 1 will give contributions at O (1 /N )from the expansion of ( x ) χ π . This will produce problematic logarithms but these are connectedwith the simple poles of the values of the two loop graphs denoted by Σ and Π . In particularthey have the formal structureΣ = K ∆ + Σ (cid:48) , Π = L ∆ + Π (cid:48) (3.8)where Σ (cid:48) and Π (cid:48) are finite. They were computed previously in [20] where the explicit d -dependent values are available. We note our trace conventions at this stage are the same as [20]and we use 2 × γ -matrices. To adjust for higher dimensional γ -matrix representations onesimply redefines N using N = 12 d γ N (3.9)where d γ is the dimension of the γ -matrix representation. x yz αγ β ≡ a ( α ) a ( β − a ( γ − β − γ − x y µ − α µ − γ + 1 µ − β + 1 Figure 2: Uniqueness rule for scalar-fermion vertex for arbitrary exponents α , β and γ .The key integration tool for evaluating the graphs in [20] is shown in Figure 2 and is termeduniqueness or conformal integration for the scalar-Yukawa interaction. There are several waysto establish the relation provided in Figure 2. One is to use Feynman parameters. In that7erivation the final integration is over a hypergeometric function and it cannot proceed unlessthe uniqueness condition of α + β + γ = 2 µ + 1 (3.10)is fulfilled. Setting this allows the final integration to be completed which produces the factoron the right side of Figure 2. A more elegant alternative is to apply a conformal transformationto the integral which is x µ → x µ x , y µ → y µ y , z µ → z µ z (3.11)which implies the mapping( x/ − y/ ) → − y/ ( x/ − y/ ) x/x y = − x/ ( x/ − y/ ) y/x y (3.12)for instance. The consequence is that when applied to strings of contracted γ -matrices thetransformation does not alter the initial string of γ -matrices. In the application to the left handside of the equation of Figure 2 the exponent of the scalar becomes (2 µ + 1 − α − β − γ ). Settingthis to zero allows the z -integration to proceed resulting in the expression on the right hand sideafter undoing the initial conformal transformation.With the availability of the counterterm from the vertex renormalization constant the di-vergences are removed minimally. However ln( x ) terms remain through two contributions ineach Schwinger-Dyson equation. One is from the power of ( x ) in the O (1 /N ) correction.The other arises from the factor ( x ) χ π that is present in the one loop graph. Expanding thisin powers of 1 /N the O (1 /N ) term involves ln( x ). To be able to take the x → χ π has to be suitably chosen. Doing so to ensure there are noln( x ) terms in each Schwinger-Dyson equation at O (1 /N ) requires χ π = (2 C F − C A ) µ µ − C F η (3.13)from the explicit values for the residues which satisfy L = − K and implies the same valueresults for both equations. This finally renders the algebraic representation of Figure 1 finite aswell as ensuring that it is scale free. Since the two equations have two unknown variables η and y that appear linearly, eliminating the latter leads to η = (cid:20) (2 µ − C F ( µ −
1) Ψ( µ ) − µC A µ −
1) Ψ( µ ) + (4 µ − µ − C F µ ( µ − − µC A µ − (cid:21) η C F (3.14)where we use the shorthandΨ( µ ) = ψ (2 µ − − ψ (1) + ψ (2 − µ ) − ψ ( µ ) (3.15)which involves the Euler ψ function defined by ψ ( z ) = d ln Γ( z ) /dz . π a critical exponent at O (1 /N ) . Having established the fermion critical exponent at O (1 /N ) the next stage in the large N formalism is to determine the same quantity for the boson field. In this instance from thedefinition (2.7) this requires the vertex anomalous dimension at O (1 /N ). While χ π followedas a corollary to ensuring the 2-point function was finite in the approach to criticality, in order toproceed to the next order to find χ π by the same method is too intractable. Indeed evaluatingthe analogous exponent in other models has not been achieved that way. Instead a more direct8igure 3: O (1 /N ) corrections to vertex function.approach suffices which is to examine the scaling behaviour of the 3-point vertex in the criticallimit. In other words the O (1 /N ) graphs illustrated in Figures 3 and 4 are computed.In practical terms the diagrams are more straightforward to evaluate in momentum spacethan in coordinate space. This is primarily due to simplifications in the application of theuniqueness rule of [20]. However one can connect the values of graphs in both the coordinateand momentum space evaluations through the Fourier transform (2.10). Underlying this oneneeds to use the momentum space forms of the asymptotic scaling functions which are ψ ( p ) ∼ ˜ Ap/ ( p ) µ − α +1 , π ( p ) ∼ ˜ C ( p ) µ − γ (4.1)where we have new momentum independent amplitudes ˜ A and ˜ C with the associated variable˜ y = ˜ A ˜ C . The explicit value for the latter can be related to the known expansion of y eithervia the Fourier transform relation of (2.6) and (4.1) or by repeating the exercise of the previoussection by setting up the formalism in momentum space at the outset. Both approaches lead tothe same expression for ˜ y to O (1 /N ) as well as η as a check. In relation to this momentumspace 2-point function approach the asymptotic scaling forms of the 2-point functions are ψ − ( p ) ∼ p/ ˜ A ( p ) α − µ , π − ( p ) ∼ C ( p ) γ − µ . (4.2)It is the inverse Fourier transform of these that produce the leading terms of (2.12). + + ++ + + Figure 4: O (1 /N ) corrections to vertex function.9ith (4.1) it is straightforward to evaluate the graph of Figure 3 and determine an expressionfor χ π from the scaling behaviour. It is equivalent to (3.13). The key point is that the sameprocedure of [16, 17] can now be applied to determine χ π . This requires in part the evaluationof the O (1 /N ) corrections illustrated in Figure 4 where the fermions can be routed around theclosed loop in both directions. The values of the diagrams in the absence of any group theoryconsiderations were given in [21]. With the presence of the group generator in the vertex of(2.4), obtaining the associated group factor of each graph is straightforward in most cases using(2.2) for example and the definition of the Lie algebra which in our conventions is[ T a , T b ] = if abc T c . (4.3)However for the graphs where there is a closed fermion loop with four π a fields attached a highergroup Casimir is present. In particular the group factor associated with each graph will containthe fully symmetric rank 4 tensor d abcdF = 16 Tr (cid:16) T a T ( b T c T d ) (cid:17) (4.4)among others. To treat these so called light-by-light graphs we have used the color.h routinethat accompanies the symbolic manipulation language Form , [27, 28]. The package allows oneto manipulate group theory factors associated with Feynman graphs and write them in terms ofCasimirs. It is based on the comprehensive analysis provided in [29]. However, rather than use color.h to determine the group factor solely for these light-by-light graphs we have applied itto all the graphs treated throughout this article for consistency. We note that the rank 3 fullysymmetric colour tensor d abc given by d abc = 12 Tr (cid:16) T a T ( b T c ) (cid:17) (4.5)also occurs in graphs that contain d abcdF . In the final expressions for the exponents, however,these are absent as a result of a cancellation between graphs where the fermions are routedaround the closed loop in different directions. In addition to the O (1 /N ) corrections of Figure4 there are contributions to χ π from the graph of Figure 3. These arise from the correction tothe variable ˜ y which is ˜ y as well as the parts from the 1 /N expansion of the exponents of thepropagators in (4.1). In addition one has to include the vertex renormalization constant Z V inthe O (1 /N ) graph as this cancels the subgraph divergences in the first three graphs of the firstrow in Figure 4. This cancellation is necessary in order to ensure the approach to criticality issmooth. Assemblying all these components allows us to deduce the vertex critical exponent atthe next order which is χ π = (cid:34) µ (2 µ − µ − C F − µ (7 µ − µ − C F C A − µ (2 µ − µ + 1)12( µ − C A − µ (2 µ − µ − d abcdF d abcdF N A T F + µ ( C A − C F )( C A µ − µ − C F )4( µ − Ψ( µ ) − µ ( C A T F N A − d abcdF d abcdF )8( µ − N A T F Θ( µ ) (cid:35) η C F (4.6)where Θ( µ ) = ψ (cid:48) ( µ ) − ψ (cid:48) (1) . (4.7)and the contributions from the light-by-light graphs is evident.10 Correlation length exponent at O (1 /N ) . Having established the dimensions of the two fields at O (1 /N ) using the leading term of theasymptotic forms of the propagators in the approach to criticality, it is possible to study thecorrections to scaling. These are contained in both (2.6) and (2.12) corresponding to the termsinvolving the coordinate independent additional dimensionless parameters A (cid:48) and C (cid:48) . The extraexponent λ can be regarded as any exponent but to access the correlation length exponent ν weset λ = 1 / (2 ν ) which has the canonical dimension of ( µ − /N correctionsin d -dimensions to this exponent requires a consistency equation that extends (3.1) and (3.5).To find λ we substitute the various asymptotic scaling functions into the Schwinger-Dysonequations for the 2-point function which produces the representation0 = r ( α − (cid:104) − A (cid:48) s ( α − x ) λ (cid:105) + C F yZ V ( x ) χ π +∆ (cid:104) (cid:0) A (cid:48) + C (cid:48) (cid:1) ( x ) λ (cid:105) + C F [2 C F − C A ] y ( x ) χ π +2∆ (cid:104) Σ + (cid:0) Σ A A (cid:48) + Σ C C (cid:48) (cid:1) ( x ) λ (cid:105) + O (cid:18) N (cid:19) (5.1)and 0 = p ( γ ) (cid:104) − C (cid:48) q ( γ )( x ) λ (cid:105) + T F N yZ V ( x ) χ π +∆ (cid:104) A (cid:48) ( x ) λ (cid:105) − T F [2 C F − C A ] N y ( x ) χ π +2∆ (cid:104) Π + (cid:0) Π A A (cid:48) + Π C C (cid:48) (cid:1) ( x ) λ (cid:105) + O (cid:18) N (cid:19) (5.2)where we have omitted the factor of Z V in the respective O (1 /N ) and O (1 /N ) correction terms.The counterterms from these only come into effect at the next order.Unlike the computation of η we have included the two loop graphs of Figure 1 where thecorrection to scaling is included. These are denoted by Σ (cid:48) φ and Π (cid:48) φ where φ indicates that theinsertion is on either a ψ or π a line according to whether it is A or C respectively. The reasonwhy these all have to be included in principle resides in the leading order N dependence of the2-point scaling functions. For the two key combinations that appear in the correction to scalingSchwinger-Dyson equation we note that this dependence is, [22, 24], r ( α − s ( α −
1) = O (1) , p ( γ ) q ( γ ) = O (cid:18) N (cid:19) . (5.3)In fact the constant of proportionality of the latter is the combination ( λ − η − χ π ). As η and χ π are both available this leaves λ as the unknown we seek. These terms will form part ofthe consistency equation that determines λ to O (1 /N ) and emerges from decoupling the ( x ) λ terms in (5.1) and (5.2) which follows on dimensional grounds. The resulting two equations are0 = − r ( α − s ( α − A (cid:48) + C F yZ V ( x ) χ π +∆ (cid:2) A (cid:48) + C (cid:48) (cid:3) + [2 C F − C A ] y ( x ) χ π +2∆ (cid:2) Σ A A (cid:48) + Σ C C (cid:48) (cid:3) + O (cid:18) N (cid:19) . (5.4)and 0 = − p ( γ ) q ( γ ) C (cid:48) + T F N yZ V ( x ) χ π +∆ A (cid:48) − C F [2 C F − C A ] N y ( x ) χ π +2∆ (cid:2) Π A A (cid:48) + Π C C (cid:48) (cid:3) + O (cid:18) N (cid:19) . (5.5)Alternatively the relevant terms that produce an expression for λ with respect to the large N expansion due to (5.3) can be written as a matrix M where M = (cid:32) − r ( α − s ( α − C F yT F N y − p ( γ ) q ( γ ) − C F [2 C F − C A ] N y Π (cid:48) C (cid:33) . (5.6)11xamining the (2 ,
2) element both terms are the same order in 1 /N as y is O (1 /N ). Settingdet M = 0 produces the consistency equation for λ which can be solved to give λ = − (2 µ − η . (5.7) + ++ ++ ++ Figure 5: Higher order graphs for corrections to the π a Schwinger-Dyson 2-point function neededfor λ .To proceed to the next stage and find λ the higher order 1 /N correction graphs haveto be added in to the two Schwinger-Dyson equations that produced the O (1 /N ) consistencyequations. However the ones we neglected to determine (5.7), since their N dependence in thedeterminant is an order lower that the contribution from Π (cid:48) C , now have to be included. Theseare Σ (cid:48) A , Σ (cid:48) C and Π (cid:48) A while the graph corresponding to Π (cid:48) C has to be expanded to the nextorder in 1 /N since there is N dependence in the propagator exponents. To ease the extractionof the expansion of the consistency equation determinant we formally setΠ (cid:48) C = Π (cid:48) C + Π (cid:48) C N + O (cid:18) N (cid:19) (5.8)to clarify this. The main work however resides in including the final contributions to find λ which are illustrated in Figure 5. While the individual d -dependent values have been recordedin [25] for instance, we have had to append the respective group theory factors. Again we haveused the Form color.h routine due to the presence of the light-by-light diagrams. Repeatingthe exercise of setting the determinant of the consistency equations to zero at the next order in1 /N produces the expression λ = (cid:34)(cid:34) µ (3 µ − µ + 2) C A C F T F + 4 µ d abcdF d abcdF C F T F N A (cid:35) µ − µ − η (cid:34) µ (2 µ − C A + 2 µ (2 µ − d abcdF d abcdF T F N A (cid:35) [Ψ ( µ ) + Φ( µ )]( µ − µ − (cid:104) − (2 µ − ( µ + 1)( µ − µ − C F + µ (2 µ − µ − µ − C F C A + 124 µ ( µ − µ − µ + 20) C A − µ (3 µ − µ − d abcdF d abcdF T F N A (cid:35) × Ψ( µ )( µ − ( µ − + (cid:20) − µ (2 µ + 1)( µ − C F + 34 µ (2 µ + 5)( µ − C F C A − µ ( µ − C A + 3 µ (5 µ − d abcdF d abcdF T F N A (cid:35) Θ( µ )( µ − µ −
2) + 3 µ (2 µ − µ − C F C A + (2 µ − (2 µ − µ − µ + 1)2 µ ( µ − C F − µ (8 µ − µ + 85 µ − µ + 20)48( µ − ( µ − C A − µ (4 µ − µ + 26 µ − µ + 7)2( µ − ( µ − d abcdF d abcdF T F N A (cid:35) η C F . (5.9)We have introduced the additional shorthand notationΦ( µ ) = ψ (cid:48) (2 µ − − ψ (cid:48) (2 − µ ) − ψ (cid:48) ( µ ) + ψ (cid:48) (1) . (5.10)Essential in determining this was the values of η and χ π . + + ++ Figure 6: Primitive graphs which determine η in the large N conformal bootstrap method. N conformal bootstrap. The final part of our study follows a different tack by applying the large N conformal bootstrapprogramme developed in [14], based on the early insights of [30, 31, 32, 33, 34]. In generalterms the focus is initially directed towards the 3-point vertex of (2.4) and its behaviour in the13ritical region. The fields will still obey the asymptotic scaling forms of (2.12) but in treating theGreen’s functions in the bootstrap formalism not only are there no self-energy corrections on thepropagators but there are no vertex corrections. In effect the primitive graphs are the buildingblocks and are illustrated in Figure 6. In that figure the dotted vertices do not correspond tothe vertex of the Lagrangian (2.4). Instead they denote the presence of a Polyakov conformaltriangle [30] which includes all vertex corrections at criticality. It is defined in Figure 7 for thegeneral Yukawa type interaction that includes the one of (2.4). The external exponents α i aregeneral and are determined from the underlying theory. For example α = α = α and α = γ for (2.4). The values of the internal indices a i are the solution to the simultaneous equations a + a + α = 2 µ + 1 a + a + α = 2 µ + 1 a + a + α = 2 µ + 1 . (6.1)They ensure that the internal vertices of the triangle graph in Figure 7 are all unique unlikethe vertex on the left side of the equation for (2.4). The calculational benefit of regarding thefull vertex correction as a conformal triangle is that applying a conformal transformation, (3.11)and (3.12), the graphs of Figure 6 are reduced to 2-point ones which are easier to evaluate. α α α = f ( α i , a i ) α α α a a a Figure 7: Polyakov conformal triangle for a scalar Yukawa interaction.The graphs of Figure 6, however, are the lowest order contributions to the full vertex functionthat we will denote by V (¯ y, α, γ ; δ, δ (cid:48) ) where the last two arguments are regularizing parameters.These are required in the derivation of one of the two consistency equations defining the large N bootstrap formalsm, [14, 30, 31, 32]. The first equation represents Figure 6 and is1 = V (¯ y, α, γ ; 0 ,
0) (6.2)where ¯ y is similar to the early amplitude combination y but includes the normalization of thePolyakov conformal triangle of Figure 7. Indeed (6.2) is responsible for determining the termsin the 1 /N expansion of ¯ y once the first few orders of η have been reproduced in this formalism.This is because the explicit expressions are necessary to extract η from the third order term ofthe second bootstrap equation which is T F N r ( α − C F p ( γ ) = (cid:104) χ π ∂∂δ (cid:48) V (¯ y, α, γ ; δ, δ (cid:48) ) (cid:105)(cid:104) χ π ∂∂δ V (¯ y, α, γ ; δ, δ (cid:48) ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ = δ (cid:48) =0 . (6.3)We note briefly that the regularizations δ and δ (cid:48) that appear here arise because of singularitiesin the 2-point Schwinger-Dyson equations when all the vertices are replaced by conformal tri-angles. In other words it was recognised in the original work of [33] that in the absence of anyregularization the 2-point functions with dressed vertices would be finite overall. However eachof the contributing diagrams were individually divergent. To accommodate this, and similar tothe introduction of ∆ earlier, the vertex anomalous dimension has to be continued in a parallel14ay to (3.4). This is illustrated in Figure 8 for the leading order contribution to V (¯ y, α, γ ; δ, δ (cid:48) )where we have set χ π = 2 ˜∆ π (6.4)for shorthand. This figure indicates the values of the exponents of the internal lines of thePolyakov triangle. Moreover the appearance of both δ and δ (cid:48) on the external legs of the graphon the left hand side indicate the addition of the regularizations to the exponents of the respectivefields. This is reflected internally in the conformal triangles in the right hand graph as each ofthe vertices have to be unique even when there is a regularization. α + ˜∆ π − δ (cid:48) γ α + ˜∆ π γ + ˜∆ π − δ (cid:48) α + ˜∆ π + δ (cid:48) α + ˜∆ π γ + ˜∆ π α αγ + ˜∆ π + δα + ˜∆ π − δ α + ˜∆ π − δ x y =2 δ δ (cid:48) Figure 8: Regularized one loop contribution to the vertex bootstrap equations.Given the form of (6.3) we can rederive η from knowledge of the value of the graph ofFigure 8. This was determined in [26], using the technique given in [14], for the Ising Gross-Neveu model as a function of the exponents of that theory and that result can be translated to(2.4). If we expand V (¯ y, α, γ ; δ, δ (cid:48) ) in large N in the same notation as (2.9) then to the orderthat will eventually be necessary to evaluate η we recall, [26], V = − Q ˜∆ π ( ˜∆ π − δ )( ˜∆ π − δ (cid:48) ) exp[ F ( δ, δ (cid:48) , ˜∆ π )] (6.5)with Q = − a ( α − a ( γ )( α − Γ( µ ) (6.6)and F ( δ, δ (cid:48) , ˜∆ π ) = (cid:20) B γ − B α − − B − α − (cid:21) ˜∆ π − [ B γ − B ] δ + (cid:20) B − B α − − α − (cid:21) δ (cid:48) + (cid:20) C α − − α − (cid:21) δδ (cid:48) + (cid:20) C γ + C − C α − + 2( α − (cid:21) ˜∆ π δ + 12 (cid:20) α − − C α − − C (cid:21) δ (cid:48) + (cid:20) C − C γ − C α − + 2( α − (cid:21) ˜∆ π δ (cid:48) −
12 [ C γ + C ] δ (cid:20) C α − − C γ − C − α − (cid:21) ˜∆ π + O (cid:16) ˜∆ π , δ , δ (cid:48) (cid:17) (6.7)where the order symbol indicates that terms cubic in any combination of the parameters areneglected. The functions B z and C z are defined by, [14], B z = ψ ( µ − z ) + ψ ( z ) , B = ψ (1) + ψ ( µ ) C z = ψ (cid:48) ( z ) − ψ (cid:48) ( µ − z ) , C = ψ (cid:48) ( µ ) − ψ (cid:48) (1) (6.8)in terms of the Euler ψ function. With (6.7) and setting V = V /N in (6.2) and (6.3) it isstraightforward to expand both bootstrap equations to O (1 /N ) and verify the earlier expressionsfor η and η .Having established the formalism reproduces available results the extension to the next orderto find η requires several steps. The first is to compute the next term in the 1 /N expansion of(6.7) as that will contribute to the O (1 /N ) part of (6.3). However contributions from all thegraphs of Figure 6 bar the one loop one have to be determined and included as they correspondto V . The relevant parts of these graphs were calculated in [26]. By this we mean that acomputational shortcut was used akin to that used in [14]. From examining the terms in theformal Taylor expansion of (6.3) in powers of 1 /N the part involving V occurs in the combination (cid:20) ∂∂δ (cid:48) V ( y, α, γ ; δ, δ (cid:48) ) − ∂∂δ V ( y, α, γ ; δ, δ (cid:48) ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) δ = δ (cid:48) =0 . (6.9)In [26] the contribution to (6.9) from each of the higher order correction graphs in Figure 6 weredetermined. Indeed in most cases only the value for the difference in derivatives could be found.Appending the group theory values from the color.h routine allows us to finally extract η . Wefind η = (cid:34) (2 µ − µ − µ + 16 µ − µ ( µ − C F − µ (43 µ − µ + 6)8 µ ( µ − C F C A − µ (4 µ − µ − µ − µ ( µ − C A − µ (4 µ − µ − µ + 10)4 µ ( µ − d abcdF d abcdF T F N A + (cid:20)
12 (11 µ − µ − − µ (19 µ − µ − C F C A − µ (2 µ − µ − C A − µ (2 µ − µ + 4) d abcdF d abcdF T F N A (cid:35) Ψ( µ ) µ ( µ − + 3(2 C F − µC F + µC A ) µ − Ψ ( µ )+ (2 C F − µC F + µC A ) µ − Φ( µ )+ (cid:34) (2 µ − µ + 1) C F − µ (5 µ − C F C A − µ ( µ − C A + µ ( µ + 8) d abcdF d abcdF T F N A (cid:35) × (cid:20) Θ( µ ) + 1( µ − (cid:21) µ − − (cid:34) C A − d abcdF d abcdF N A T F (cid:35) (cid:20) Θ( µ ) + 1( µ − (cid:21) [2Ψ( µ ) + Ξ( µ )] µ µ − (cid:35) η C F . (6.10)Again the pieces arising from the light-by-light graphs can be clearly identified. In addition tothe Euler ψ function and its derivatives a new function arises which is related to the function I ( µ ) defined in [14]. It corresponds to the derivative of a two loop self-energy graph where aregularizing exponent of one of the propagators is differentiated. In [35] I ( µ ) was expressed as16n F hypergeometric function where the differentiation manifests itself as derivatives of theparameter arguments of the hypergeometric function. Here we have set I ( µ ) = − µ −
1) + Ξ( µ ) (6.11)so that Ξ( µ ) has an (cid:15) expansion involving multiple zeta values, [14, 35, 36]. For instance,Ξ(1 − (cid:15) ) = 23 ζ (cid:15) + ζ (cid:15) + 133 ζ (cid:15) + O ( (cid:15) ) . (6.12)In strictly three dimensions, [14], I ( ) = 2 ln 2 + 3 ψ (cid:48)(cid:48) ( )2 π (6.13)and the expansion around three dimensions is known up to ten terms [37]. We focus in this section on general aspects of the critical exponents of (2.4) that we havedetermined. One of the main reasons for considering a generalized universality class was the factthat known results for specific models could be extracted as well as be of use where a different Liegroup underlies the physics problem. To assist with that we have collected electronic expressionsin an attached data file. One aspect of the results that needs to be stated is that we have checkedthat the d -dimensional exponents are in agreement with several models where the results weredetermined directly. For instance, the original Ising Gross-Neveu model of [1] corresponds tothe abelian limit of the symmetry group by specifying C F = 1 , T F = 1 , d abcdF d abcdF = 1 , C A = 0 (7.1)while the Mott insulating phase [8, 9] that corresponds to taking the symmetry group to be SU (2) takes the values C F = 34 , T F = 12 , d abcdF d abcdF = 564 , C A = 2 . (7.2)For the more recent application of (2.4) to the fractionalized Gross-Neveu model discussed in[11] the respective values are C F = 2 , T F = 2 , d abcdF d abcdF = 203 , C A = 2 . (7.3)For each of these cases the (cid:15) expansion of the exponents near four dimensions with d = 4 − (cid:15) are in full agreement with known three and four loop perturbative results, [11, 15, 38, 39, 40, 41,42, 43]. In the case of the Ising Gross-Neveu model exponents these also are in accord with twodimensional perturbation theory, [44, 45, 46, 47, 48, 49, 50]. Moreover taking the limits for thethree cases, all the large N exponents agree with previous work, [19, 20, 21, 22, 23, 24, 25, 43, 51].One advantage of the arbitrary group approach in d -dimensions means that the structure ofthe exponents can be studied in various representations. For example if we restrict the fermionsto be in the adjoint representation A whence C F = C A and T F = C A . In that case we find η adj = (cid:34) (13 µ − µ + 2)4 µ [ µ − + (3 µ − µ −
1] Ψ( µ ) (cid:35) (cid:16) η adj (cid:17) adj π = µ µ − η adj χ adj π = (cid:34)(cid:34) µ [ µ − d abcdA d abcdA C A N A − µ µ − (cid:35) Θ( µ ) + µ (3 µ − µ − Ψ( µ ) − µ ( µ + 1)( µ − µ − − µ (2 µ − µ − d abcdA d abcdA C A N A (cid:35) (cid:16) η adj (cid:17) λ adj = (cid:34)(cid:34) µ (3 µ − µ + 2) + 4 µ d abcdA d abcdA C A N A (cid:35) µ − µ − η adj − (cid:34) µ (2 µ −
3) + 2 µ (2 µ − d abcdA d abcdA C A N A (cid:35) [Ψ ( µ ) + Φ( µ )][ µ − µ − − (cid:20)
124 ( µ − µ − µ + 549 µ + 4 µ − µ + 96)+ µ (3 µ − µ − d abcdA d abcdA C A N A (cid:35) Ψ( µ )[ µ − [ µ − + (cid:34) µ (5 µ − d abcdA d abcdA C A N A − µ ( µ − µ − (cid:35) Θ( µ )[ µ − µ − µ − µ + 4770 µ − µ + 3951 µ + 724 µ − µ + 768 µ − µ [ µ − [ µ − − µ (4 µ − µ + 26 µ − µ + 7)2[ µ − [ µ − d abcdA d abcdA C A N A (cid:35) (cid:16) η adj (cid:17) η adj = (cid:34) (3 µ − µ − (cid:104) Φ( µ ) + 3Ψ ( µ ) (cid:105) − µ (2 µ − µ − µ − d abcdA d abcdA C A N A − (8 µ − µ − µ − µ + 2448 µ − µ + 480 µ − µ [ µ − − (cid:20)
112 ( µ − µ − µ + 267 µ − µ + 18)+ 12 µ (2 µ + 1)( µ − d abcdA d abcdA C A N A (cid:35) Ψ( µ ) µ [ µ − + (cid:34) µ ( µ + 8) d abcdA d abcdA C A N A −
196 ( µ + 8 µ − µ + 24) (cid:35) Θ( µ )[ µ − − µ µ − (cid:34) − d abcdA d abcdA C A N A (cid:35) Θ( µ )Ψ( µ ) − µ Ξ( µ )16[ µ − (cid:34) − d abcdA d abcdA C A N A (cid:35) − µ µ − (cid:34) − d abcdA d abcdA C A N A (cid:35) Ξ( µ )Θ( µ ) (cid:35) (cid:16) η adj (cid:17) (7.4)where d abcdA is the adjoint version of the fully symmetric rank 4 Casimir. These exponentssimplify substantially in three dimensions since λ | d =3 = 1 − π N + (cid:34) d abcdA d abcdA C A N A + 5248 π − (cid:35) π N η | d =3 = 83 π N + 121627 π N + (cid:104) [9072 ζ − π ln 2][ C A N A − d abcdA d abcdA ]18 [25920 π − d abcdA d abcdA + [151072 − π ] C A N A (cid:105) π C A N A N . (7.5)The group valued coefficient of the terms involving ζ and π ln 2, which derive from I ( ), hasan interesting combination of Casimirs. Indeed there might be instances of this factor beingzero for certain Lie groups. However, we have computed the value of ( C A N A − d abcdA d abcdA ) forall the classical and exceptional Lie groups and found that it is always non-zero. As the Ising Gross-Neveu universality class is central to a number of phase transitions in variousmaterials, we have examined a generalized version of the underlying quantum field theory thatincorporates the respective condensed matter systems. The key aspect is that the core interactionis endowed with a non-abelian symmetry that has a parallel in gauge theories. There the gaugeinteraction of QED is extended from an abelian to a non-abelian one to produce QCD by theinclusion of the generators of a Lie group thereby endowing QED with a colour symmetry. Thesimilar extension of the Ising Gross-Neveu model is simpler in some respects. One obviousone is the absence of gauge symmetry. A benefit, however, is that considering (2.4) at theoutset means results for specific phase transitions can be quickly deduced by specifying theLie group. Indeed if a phase transition were discovered in a material that was in the sameuniversality class as the Ising Gross-Neveu model but possessed a new symmetry other thanthe specific examples we have noted here, then information on the exponents can readily bededuced from our results. Throughout we have focussed on the application of the critical pointlarge N technique developed in [12, 13, 14] to determine d -dimensional critical exponents. Theadvantage of this is that results are available for the renormalization group functions of the fourdimensional quantum field theories in the same universality class too. By the same token thelarge N exponents contain a wealth of information on the structure of the same functions. Forinstance, coefficients in the anomalous dimension beyond the first few known loop orders canbe accessed at successive orders in 1 /N . This is particularly useful in that our O (1 /N ) and O (1 /N ) exponents can reveal where the new colour group Casimirs, such as d abcdF d abcdF , appear.In indicating the parallel of the QED to QCD generalization, examining the large N O (1 /N )exponents in this universality class, albeit with a simpler vertex structure, does provide usefulinsight into what to expect in the calculation of critical exponents in QCD at O (1 /N ). Wehave to qualify this comment by noting that while there are similarities, in the QCD large N critical exponent computation for ν for instance, there will be more graphs to consider thanthose of Figure 5. This is because in (2.4) Feynman diagrams with subgraphs involving three π a lines connecting to a fermion loop are zero after taking the γ -matrix trace. In QCD this wouldnot be the case due to each vertex adding an extra γ -matrix to the trace. While such graphsremain to be computed the associated group theory factor that would result should not involveany Casimir higher than d abcdF d abcdF . Acknowledgements.
This work was fully supported by a DFG Mercator Fellowship and inpart with the STFC Consolidated ST/T000988/1. The graphs were drawn with the
Axodraw package [52]. Computations were carried out in part using the symbolic manipulation language
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