Conformality of 1/N corrections in SYK-like models
CConformality of 1 /N corrections in SYK-like models Stéphane Dartois ∗1 , Harold Erbin †2 , and Swapnamay Mondal ‡3 Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405 Orsay Cedex, France, Eu . Lpt , Département de physique de l’
Ens , École normale supérieure,
Upmc
Univ. Paris 06,
Cnrs , Psl
Research University, 75005 Paris, France Sorbonne Universités,
Upmc
Univ. Paris 06, École normale supérieure,
Cnrs , Lpt , 75005 Paris, France Sorbonne Universités,
Upmc
Univ Paris 06,
Umr
Lpthe , F-75005, Paris, France Cnrs , Umr
Lpthe , F-75005, Paris, France
Abstract
The Sachdev–Ye–Kitaev is a quantum mechanical model of N Majorana fermions whichdisplays a number of appealing features – solvability in the strong coupling regime, near-conformal invariance and maximal chaos – which make it a suitable model for black holes inthe context of the AdS/CFT holography. In this paper we show for the colored SYK modeland several of its tensor model cousins that the next-to-leading order in the N expansionpreserves the conformal invariance of the 2-point function in the strong coupling regime, up tothe contribution of the Goldstone bosons leading to the spontaneous breaking of the symmetryand which are already seen in the leading order 4-point function. We also comment on thecomposite field approach for computing correlation functions in colored tensor models. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] J un ontents A.1 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.2 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References 23
In a series of seminal conferences [1–3] Kitaev brought attention to the – now so called – Sachdev–Ye–Kitaev (SYK) model which displays a set of appealing features in the context of holography –,for which a detailed account has been given in [4]. This model – a simplification of a previous oneby Sachdev and Ye [5] – corresponds to a quantum mechanical system of N Majorana fermions(possibly organized in different families [6]) with an interaction of order q with Gaussian randomcouplings.The first key property is that it is solvable at large coupling (or equivalently large time orinfrared regime) in the large N limit. This is very precious since systems that are tractable inthe large coupling regime are very scarce. Moreover in this infrared limit the system displaysan approximate conformal symmetry. Conformal invariance in one dimension is equivalent toreparametrization invariance, and thus is infinite-dimensional which leads to many simplifications:in particular the system at zero and finite temperature are easily related in this regime. Thissymmetry is spontaneously broken and leads to Goldstone bosons, their dynamics being describedby the Schwarzian action. The latter are responsible for the last property of the model: theLyapunov exponent, which measures the chaos in the system, reaches the maximal bound proposedin [7] and thus the system is maximally chaotic. All together these properties point towards a (near)AdS / CFT interpretation of the model (see [8–11] for references on near AdS ). In gravitationaltheories the maximally chaotic objects are black holes: hence one can expect that the bulk dual ofthe SYK model contains black holes, and the fact that one can access the strong coupling regimeoffers an inestimable window on the quantum properties of black holes.Another interesting property is its equivalence with random tensor fields theories in the large N limit, as was pointed in [12] (some selected references on tensor models include [13–24]). The2urau–Witten model [12] is the simplest colored tensor model and consists in a set of q = D + 1real fermionic tensor fields with D indices of size N transforming in the fundamental of O( N ) ⊗ D ,the invariance group being O( N ) D ( D +1) / (up to a discrete factor). Two other models of interestare the case of q complex fermionic tensor fields with a U( N ) invariance and the so-called multi-orientable model which is given by a complex fermionic field with D = 3 indices [25] (there is no q because one is considering an uncolored tensor model [18]). The bosonic 0-dimensional versions ofthese models have been studied in [13, 14, 26–29]. The main simplification in these models occurbecause the randomness is moved in the fields and there is a single (fixed) coupling constant. Whileit is necessary to average over the random couplings by performing the Gaussian integration overthem (quenching), implying that one describes a thermodynamical ensemble, the tensor modelsfeature a unique fixed coupling constant and represent a genuine quantum system [12]. Moreoverthe combinatorics and renormalization properties have been largely studied and one can make useof all the tools already developed.The disordered and tensor SYK models have been extended in several directions: higher di-mensions and lattices [30–38], N = 1 , N = 4), non-quenched disorder [42–44]. Various properties have been studied in the last year:spectrum and thermodynamical properties [45–55], correlation functions [45, 56–61], dynamics ofthe Goldstone bosons [4, 60, 62, 63], relation with matrix models (for both the disordered andtensor versions) [46, 51, 64–68], transport properties [30, 69, 70], renormalization and phases [52].Experimental realizations have been proposed in [71–73].It was shown in [4, 45] that the next-to-leading order (NLO) correction in the coupling constantbreaks explicitly the conformal invariance in the leading order (LO) in the N expansion. Theproblem we address in this paper is the reversed one, i.e. is the conformal symmetry explicitlybroken in the NLO in N for the LO in the coupling constant? We consider this question in themodels mentioned above: the colored SYK model with disorder, and the real, complex and multi-orientable SYK tensor models. We find that in the first three models the NLO 2-point functionis compatible with conformal symmetry and thus should scale in the same way as the LO 2-pointfunction. This means that in the infrared the dimension of the fermions is not modified by thefirst subleading correction in the large N expansion. This finding may have some implications forthe construction of the bulk dual of SYK which has started in [59] (see also [4, 9, 56] and [74,75] for other proposals). Our method consists in analysing the transformation properties of theNLO 2-point function from the Schwinger–Dyson equation (the Feynman graphs contributing atthis order have been studied in [58], see also [28]): this is sufficient to reach our conclusions exceptfor the multi-orientable tensor model. In the latter case the conclusion depends on the explicitform of the NLO 2-point function and the full analysis is outside the scope of this paper. It isimportant to note that in all this paper the divergent contribution to the LO 4-point function dueto the spontaneous breaking of the conformal symmetry is implicitly excluded (as is implied byany statement in previous works about the conformality of some object) [4]: this contribution canbe taken into account only by looking at the NLO in the coupling which regularizes the divergence.A fruitful approach for computing the correlation functions and determining the structure ofthe graph appearing at some order in N is to write the action in terms of composite fields – to beidentified with the 2-point function and self-energy – instead of the fundamental fermions [4, 56,57, 62]. We briefly discuss in appendix A how such an approach can be undertaken for the coloredtensor models.The structure of the paper as follows. In sections 2 to 4 we study successively the SYK model,the (real and complex) colored tensor models and the multi-orientable tensor models. The resultsare discussed in section 5. Appendix A describes how to perform a composite field analysis for thereal colored tensor model. This study is restricted to the colored SYK model (already discussed in [6, 44, 58]) because the combinatoricsof graphs involving (anti)symmetric tensor is notoriously difficult and was one of the reason for the lack of progressin tensor models, until Gurau solved this problem by introducing colors [13]. SYK model with disorder
In this section we consider a specific case of the colored SYK model introduced in [6]. This hasthe main advantage of simplifying the study of the combinatorics (that has been done in [58]) andmakes it easier to compare with the Gurau–Witten colored tensor model later described. However,the model we study here keep all the interesting features of the usual SYK model at leading order.The colored SYK model we consider is a model of qN real massless fermions ψ ci where c ∈{ . . . q } , i ∈ { . . . N } , with q being the color index. This model is defined through the followingEuclidean space partition function Z SYK
N,λ = Z d λ e − Nq − λ P N { ik } qk =1 λ i ...iq λ i ...iq Z D Y c =0 D ψ c e − R d t L [ ψ,λ ] (2.1)where L [ ψ, λ ] = 12 D X c =0 N X i c =1 ψ ci c ∂ t ψ ci c + i q/ q ! N X { i k } qk =1 λ i ...i q q Y c =1 ψ ci c . (2.2)No particular assumption is made on the symmetry of the random couplings λ i ...i q and it isconvenient to define g = λ . (2.3)The reason is that there is no need for antisymmetry on the indices here since no color appearstwice in the interaction term. Moreover this simplifies further the combinatorics as this prohibitsmelonic graphs from contributing to subleading amplitudes in 1 /N as well.The free scalar two-point function G f ( t , t ) is defined, after an arbitrary choice of color c (which is kept implicit in the notation), by G f ( t , t ) = 1 N D X i T ψ c i ( t ) ψ c i ( t ) E = 12 sign( t − t ) , (2.4)whose Fourier transform writes G f ( ω ) = − iω . (2.5)We also have that, D T ψ ci c ( t ) ψ c i c ( t ) E = δ cc δ i c i c G f ( t , t ) . (2.6)The exact disorder averaged two-point function G e ( t , t ) is defined by the following relations G e ( t , t ) = 1 N D X i T ψ c i ( t ) ψ c i ( t ) E , (2.7) h T ψ ci c ( t ) ψ c i c ( t ) i = δ cc δ i c i c G e ( t , t ) . (2.8)The Feynman graphs of this model are made up of the following building blocks:• The vertices are q + 1 valent.• The edges are of two types. The fermionic edges carry a color label c ∈ { . . . q } . Thedisorder edges carry a 0 label. Edges are labelled in such a way that no two adjacent edgeshave the same label (color or disorder).• The faces are cycles made alternatively of edges labelled 0 and c , for some color label.4he free energy of the colored SYK has a 1 /N expansion of the form F SYK
N,λ = log Z SYK
N,λ = X ‘ m ≥ N − ‘ m F [ ‘ m ] ( λ ) , (2.9)where ‘ m ( G ) is a characteristic number of the Feynman graph G . ‘ m ( G ) is the number of multi-colored cycles of the graph G \ that is obtained from G by contracting all edges labelled 0. Fromthese considerations we get that the exact two-point function also admits a 1 /N expansion G e ( t , t ) = X ‘ m ≥ N − ‘ m G [ ‘ m ] ( t , t ) . (2.10) The leading order of the SYK model has been described in several works [4, 45], and the coloredSYK model has been described in [6], therefore we only give a very brief account and the readermay refer to the excellent presentations mentioned above for more details. The leading ordertwo-point function ( ‘ m = 0), G [0] satisfies the following equations G [0] ( t , t ) = G f ( t , t ) + g Z d t d t G f ( t , t )Σ [0] ( t, t ) G [0] ( t , t ) , (2.11)where Σ [0] is the leading order self-energy. The above relation is easily obtained from the usualrelation between the two-point function and the self-energy G e ( t , t ) = (cid:0) G f ( t , t ) − − Σ( t , t ) (cid:1) − , (2.12)where the inverse here means that the two variables functions G f ( t , t ) , Σ( t , t ) are seen asmatrices for the convolution product. The graphs appearing at leading order are the melonic graphs, also called melon graphs (see [12, 15] for a description of these graphs). This implies thatΣ [0] ( t, t ) = G [0] ( t, t ) q − . (2.13)Therefore we have G [0] ( t , t ) = G f ( t , t ) + g Z d t d t G f ( t , t ) G [0] ( t, t ) q − G [0] ( t , t ) . (2.14)In Fourier space this equation rewrites − i ω G [0] ( ω ) = 1 + g Σ [0] ( ω ) G [0] ( ω ) . (2.15)Consequently, in the infrared limit G [0] → ¯ G [0] , the left hand side drops and one has0 = 1 + g ¯Σ [0] ( ω ) ¯ G [0] ( ω ) , (2.16)where ¯ G [0] stands for the infrared limit of G [0] and ¯Σ [0] stands for the infrared limit of Σ [0] . Inrest of the paper, any barred quantity would denote the infrared or large coupling limit of thecorresponding unbarred quantity. In position space, we have g Z d t ¯ G [0] ( t , t ) q − ¯ G [0] ( t, t ) = − δ ( t − t ) . (2.17)As a side comment, let us note that in the full paper, the bar over any quantity such as forinstance here ¯ G [0] , denotes the large coupling limit of the corresponding quantity. One recovers the same results if one considers the large coupling g limit. .3 The Next-to-Leading Order In this subsection we want to investigate the Next-to-Leading Order (NLO) of the colored SYKmodel.We want to study the possible corrections to the scaling dimension of the two-point function inthe conformal sector. To do so we use the results of [58]. The Next-to-Leading Order is given bythe graphs with ‘ m = 1. That is to say that their contracted graphs have one multicolored cycle.From this work it is possible to write the self-energy at NLO. It writes graphically asΣ NLO := Σ [1] = X c = c NLO c c c + X l ≥ X c ,c = c l =0 ⇒ c = c c c . . . length l ≥ c c c c c + X l ≥ X c ,c = c l =0 ⇒ c = c c c . . . length l ≥ c c c c c + X l ≥ X c ,c ,c = c c = c = c l =0 ⇒ c = c ,c = c c . . . length l ≥ c c c c c c c + X l ≥ X c ,c ,c = c c = c = c l =0 ⇒ c = c ,c = c c . . . length l ≥ c c c c c c c (2.18)where the edges with gray discs insertions represent dressed leading order propagators.We give a few indications on the correspondence between these terms and the graphs describedin [58]. In the language of [58], the two-point function is obtained by cutting an edge of an NLO6acuum graph. NLO vacuum graphs are ladder diagrams closed on themselves. They can be closedwith an even or odd number of crossing. The even number of crossing class is equivalent to thenon crossing case, while the odd number of crossing class is equivalent to the one crossing case. Asdescribed in [58] the graphs contributing to G NLO exist in two types A and B , which are themselvesseparated in two subtypes ∅ or not ∅ . The second subtype always contributes to the first term ofthe right hand side of equation (2.18), while the type A, ∅ (respectively B, ∅ ) case accounts forthe second and third (resp. fourth and fifth) terms of the right hand side of equation (2.18). Theseequations can be rewritten using the further defined color space matrix Q . To this aim we firstdefine a matrix in the color space, whose elements K c,c ( t , t ; t , t ) are defined by the equation K c,c ( t , t ; t , t ) = − g (1 − δ c,c ) G [0] ( t , t ) G [0] ( t , t ) G [0] ( t , t ) q − , (2.19)for c, c ∈ { , . . . , q } . The analogue of this operator re-appears with slight modifications in thetensor model context as well. One defines the matrices Q and Q , whose elements are, Q ,c,c ( t , t ; t , t ) = δ c,c (cid:0) G [0] ( t , t ) G [0] ( t , t ) − G [0] ( t , t ) G [0] ( t , t ) (cid:1) (2.20) Q c,c ( t , t ; t , t ) = X n ≥ (cid:2) K n ∗ Q (cid:3) c,c ( t , t ; t , t ) (2.21)= (cid:2) ( δ ⊗ ⊗ − K ) − Q (cid:3) c,c ( t , t ; t , t ) (2.22)where δ ⊗ = δ ( t − t ) δ ( t − t ) and the ∗ product here means both matrix and convolution productof the form [ K ∗ Q ] c,c ( t , t ; t , t ) = X i Z d t d t K c,i ( t , t ; t, t ) Q ,i,c ( t, t ; t , t ) , (2.23)and the powers n of K are taken with respect to this product.Notice here that equation (2.20) is singular if K admits an eigenvector with eigenvalue 1, whichis the case in the large coupling limit. As explained in the introduction (section 1) this signalsa spontaneous breaking of the conformal symmetry and for this reason this contribution can beignored: it is an artifact of the limit which can be handled by including subleading corrections inthe coupling constant. Since the latter break the conformal symmetry any statement about theconformal symmetry assumes that one is considering the large coupling limit with the divergentcontribution removed [4].If we consider the 1PI counterpart of Q , Γ, we find that it satisfies Schwinger–Dyson-likeequations of the form Γ( t , t ; t , t ) = Γ ( t , t ; t , t ) + [Γ ∗ K ]( t , t ; t , t ) (2.24)and Γ writes element-wiseΓ ,c,c ( t , t ; t , t ) = g (1 − δ c,c ) δ ( t − t ) δ ( t − t ) G [0] ( t , t ) q − . (2.25)We can rewrite the equation on Σ NLO NLO ( t , t ) = X c = c NLO c c c + X c ,c = c c c c Q c ,c c c c c + X c ,c = c ,c c = c c c Q c ,c c c c c c c . (2.26)Equation (2.26) rewrites formally asΣ NLO ( t , t ) = ( q − g G [0] ( t , t ) q − G NLO ( t , t )+ g Z d t d t X c ,c = c Q c ,c ( t , t ; t , t ) G [0] ( t , t ) q − G q − ( t , t ) G [0] ( t, t )+ g Z d t d t X c = c X c ,c = c ,f Q c ,c ( t , t ; t, t ) G [0] ( t , t ) q − G [0] ( t , t ) G [0] ( t , t ) G q − ( t, t ) . (2.27)We assume that in the large coupling limit, ¯ G NLO = O ( λ − q ) in λ . Then we notice that ¯ G [0] is oforder O ( λ − q ) in the large coupling limit. Moreover since ¯ K is of order O (1) in this limit, ¯ Q is oforder O ( λ − /q ). From these considerations one finds the large coupling equation for ¯Σ NLO ¯Σ NLO ( t , t ) = g Z d t d t X c ,c = c ¯ Q c ,c ( t , t ; t , t ) ¯ G [0] ( t , t ) q − ¯ G q − ( t , t ) ¯ G [0] ( t, t )+ g Z d t d t X c = c X c ,c = c ,c ¯ Q c ,c ( t , t ; t, t ) ¯ G [0] ( t , t ) q − ¯ G [0] ( t , t ) ¯ G [0] ( t , t ) ¯ G q − ( t, t ) . (2.28)We now want to study the scaling dimension of the NLO. To this aim we come back to theequations (2.24). These imply that, in the conformal sector, the 1PI counterpart of Q → ¯ Q hasscaling dimension ( q − /q . Indeed, it is easy to check that the terms ¯Γ has ( q − /q as scaling This is true in the 0-dimensional bosonic case. Then this assumption is reasonable as we expect that both thetime dependence, and the fermionic character of the fields change the values of the coefficient of the expansion in1 /g but not its main properties. σ i = f ( t i ) for i = 1 , . . . ,
4, we have for ¯Γ ¯Γ ,c,c ( σ , σ ; σ , σ ) = ( q − g (1 − δ c,c ) δ ( t − t ) δ ( t − t ) | f ( t ) | ( q − /q | f ( t ) | /q | f ( t ) | ( q − /q | f ( t ) | /q ¯ G [0] ( t , t ) q − | f ( t ) f ( t ) | ( q − /q (2.29)= ¯Γ ,c,c ( t , t ; t , t ) | f ( t ) f ( t ) f ( t ) f ( t ) | ( q − /q . (2.30)Let us consider the terms of the form (cid:2) ¯Γ ∗ K (cid:3) ( t , t ; t , t ). We have by definition of K that¯ K c,c ( σ, σ ; σ , σ ) = ¯ K c,c ( t, t ; t , t ) | f ( t ) f ( t ) | /q | f ( t ) f ( t ) | ( q − /q . (2.31)Therefore using equations (2.29) and (2.31) it is simple to check that if ¯Γ( t , t ; t , t ) is a solutionof (2.24) in the conformal sector, then the equation satisfied by ¯Γ( σ , σ ; σ , σ ) transforms into(2.24) provided σ i = f ( t i ) and¯Γ( t , t ; t , t ) = | f ( t ) f ( t ) f ( t ) f ( t ) | ( q − /q ¯Γ( σ , σ ; σ , σ ) . (2.32)One is then interested in the scaling dimension of ¯ Q . We have¯ Q ( t , t ; t , t ) = ¯ Q ( t , t ; t , t ) + Z d t d t d τ d τ (cid:16)(cid:2) ¯ G [0] ( τ, t ) ¯ G [0] ( τ , t ) − ¯ G [0] ( τ, t ) ¯ G [0] ( τ , t ) (cid:3) × ¯ G [0] ( t , t ) ¯ G [0] ( t , t )¯Γ( t, t ; τ, τ ) (cid:17) , (2.33)where the integration is done element-wise. From this last equality, one shows that the scalingdimension of ¯ Q ( t , t ; t , t ) is 1 /q by using the scaling properties of ¯ G [0] as well as the ones of¯Γ. Then, as we know that ¯ Q has scaling dimension 1 /q , a simple computation using equation(2.28) shows that ¯Σ NLO has scaling dimension q − q . We now come to the two-point function atNext-to-Leading order. We have G NLO ( t , t ) = Z d t d t G [0] ( t , t )Σ NLO ( t, t ) G [0] ( t , t ) . (2.34)This is obtained from the fact that in Fourier space, equation (2.12) rewrites G e ( ω ) = − iω (cid:18) ω ) iω (cid:19) − (2.35)= G f ( ω ) X p ≥ (cid:18) − Σ( ω ) iω (cid:19) p . (2.36)Since G e ( ω ) = P ‘ m ≥ N − ‘ m G [ ‘ m ] ( ω ) and Σ( ω ) = P ‘ m ≥ N − ‘ m Σ [ ‘ m ] ( ω ), we have G NLO ( ω ) = " G f ( ω ) X q ge (cid:18) − Σ [0] ( ω ) iω (cid:19) q Σ NLO ( ω ) G f ( ω ) X p ≥ (cid:18) − Σ [0] ( ω ) iω (cid:19) p (2.37)= (cid:0) G f ( ω ) − − Σ [0] ( ω ) (cid:1) − Σ NLO ( ω ) (cid:0) G f ( ω ) − − Σ [0] ( ω ) (cid:1) − (2.38)= G [0] ( ω )Σ NLO ( ω ) G [0] ( ω ) . (2.39)9his is the Fourier space form of equation (2.34). Thanks to equation (2.34), we can deduce thescaling dimension of ¯ G NLO . Indeed, if we set σ , = f ( t , ) and σ, σ = f ( t ) , f ( t ), then¯ G NLO ( σ , σ ) = Z d σ d σ ¯ G [0] ( σ , σ ) ¯Σ NLO ( σ, σ ) ¯ G [0] ( σ , σ ) (2.40)= 1 | f ( t ) f ( t ) | /q Z d t d t | f ( t ) f ( t ) | ¯ G [0] ( t , t ) ¯Σ NLO ( t, t ) ¯ G [0] ( t , t ) | f ( t ) | /q | f ( t ) f ( t ) | ( q − /q | f ( t ) | /q (2.41)= ¯ G NLO ( t , t ) | f ( t ) f ( t ) | /q . (2.42)As a consequence the scaling dimension of ¯ G NLO is 1 /q in the conformal sector. This is the samescaling dimension than ¯ G [0] , thus the conformal symmetry is not altered at NLO in N in the largecoupling limit. In this part, we consider one dimensional fermionic quantum field tensor models. The first one isbuilt out of real fermionic fields, while the second one is built from complex fermionic fields. Eachfield carries a color index c plus D additional indices denoting the component of the tensor.The real model is the Gurau–Witten model introduced in [12]. Its partition function writes, Z R N,λ = Z D Y c =0 D ψ c e − R d t L [ ψ ] (3.1)where L [ ψ ] = 12 D X c =0 X n c ψ cn c ∂ t ψ cn c + i ( D +1) / λN D ( D − / X n d Y c =0 ψ cn c . (3.2)It is also convenient to define g = λ . (3.3)We now need to explain several points. Let us first start with the notations. As explained abovethe fermionic fields are tensors. As such they are D -fundamentals of O( N ). The tensors carry acolor index c which runs from 0 to D . This means we have a family of D + 1 fermionic tensor fields { ψ c } Dc =0 . Since each ψ c is a tensor, its components write ψ cn c ··· n cD for n cj ranging from 1 to N .We call N the size of the tensor, each field ψ c has N D components. Then P n c ψ cn c ∂ t ψ cn c means X n c ψ cn c ∂ t ψ cn c := X n c ··· n cD ≥ ψ cn c ··· n cD ∂ t ψ cn c ··· n cD . (3.4)The interaction term notation P n Q dc =0 ψ cn c contains P n which is a shorthand for the constraintthat n c = ( n c ( c − · · · n c n cD · · · n c ( c +1) ) and that the indices are constrained to n kl = n lk .In this model the free scalar two-point function G f is G f ( t , t ) = 1 N D D X n i T ψ cn i ( t ) ψ cn i ( t ) E = 12 sign( t − t ) , (3.5)its Fourier transform writes, G f ( ω ) = − iω . (3.6)10eanwhile we have D T ψ c n c ( t ) ψ cn c ( t ) E = δ cc Y c = c δ n c c n cc G f ( t , t ) . (3.7)The exact two-point function G e on the other hand satisfies the same type of relation G e ( t , t ) = 1 N D D X n i T ψ cn i ( t ) ψ cn i ( t ) E (3.8) D T ψ c n c ( t ) ψ cn c ( t ) E = δ cc Y c = c δ n c c n cc G e ( t , t ) . (3.9)The complex model is very similar to the real one. It is constructed out of 2( D + 1) complexfermionic tensor fields ψ cn c ( t ) , ¯ ψ cn c ( t ). c ∈ [0 ..D ] is the color of the tensor, and each subscript n c is an abbreviation of the form n i = { n cc − , . . . , n c , n cD , . . . , n cc +1 } , where each n ij ∈ [1 ..N ] forsome N , again the size of the tensors. The corresponding partition function is Z C N,λ, ¯ λ = Z D Y i =0 D ψ i D ¯ ψ i e R d t L [ ψ ] . (3.10)where L [ ψ ] = D X c =0 X n c ¯ ψ cn c ∂ t ψ cn c + i ( D +1) / λN D ( D − / X n D Y c =0 ψ cn c + i ( D +1) / ¯ λN D ( D − / X n D Y c =0 ¯ ψ cn c . (3.11)The definition of the sum in the interaction term is the same than in the real case. Each fermionfield is a d -fundamental of U( N ) and we will make use of the notation g = λ ¯ λ. (3.12)The two-point functions are defined in similar ways. The free two-point function satisfies G f ( t , t ) = 1 N D D X n i T ¯ ψ cn i ( t ) ψ cn i ( t ) E = sign( t − t ) , (3.13) D T ¯ ψ c n c ( t ) ψ cn c ( t ) E = δ cc Y c = c δ n c c n cc G f ( t , t ) , (3.14)while the exact two-point function satisfies G e ( t , t ) = 1 N D D X n i T ¯ ψ cn i ( t ) ψ cn i ( t ) E (3.15) D T ¯ ψ c n c ( t ) ψ cn c ( t ) E = δ cc Y c = c δ n c c n cc G e ( t , t ) . (3.16)We make a slight abuse of notations here as we use the same notations for both the complex andreal case. In fact this is to avoid introducing too many notations.We now describe the Feynman graphs of these models. The Feynman graphs have the followingproperties: 11 The vertices are ( D + 1)-valent.• Edges carry a color index c ranging from 0 to D in such a way that no two adjacent edgeshave the same color index.• the faces of the graphs are the bicolored edge cycles.• In the complex case, the graphs are bipartite.The free energy of these models has a 1 /N expansion driven by the degree $ , F N,λ, ¯ λ = log Z N,λ, ¯ λ = X $ ≥ N D − D − $ F [ $ ] ( λ, ¯ λ ) , (3.17)where the degree $ of a graph G is computed of the genera of its jackets, see [13], its amplitudeis then A ( G ) = N D − D − $ ( G ) a ( G ) where a ( G ) is a reduced amplitude that depends on integralover positions and the coupling constants but not on N . The main difference between the complexand real case is that, a priori , the degree in the complex case is an integer because all jacketsare ribbon graphs representing surfaces, while in the real case, non-orientable two manifolds canappear among the jackets and thus turn the degree into an half-integer. However, it is easy toshow that the degree is an integer in both cases.The fixed degree free energies F [ $ ] λ, ¯ λ can be computed by summing the amplitudes of all vacuumconnected Feynman graphs of degree $ .These considerations imply that the two-point function also has a 1 /N expansion. This expan-sion writes in both the real and complex cases G e ( t , t ) = X $ ≥ N − D − $ G [ $ ] ( t , t ) . (3.18) The leading order of the 1 /N expansion, $ = 0, is described by melon diagrams. They are graphsof degree 0, meaning that all jackets are planar. Thanks to the structural properties of the melonicgraphs, it is easy to infer the equation satisfied by the LO 2-point function. Indeed, one has theusual relation between the self-energy Σ and the exact two-point function: G e ( t , t ) = (cid:0) G f ( t , t ) − − Σ( t , t ) (cid:1) − , (3.19)where the inverse is taken with respect to the matrix-like/convolution product. Recalling that onewrites g = λ in the real case or g = λ ¯ λ in the complex case, one deduces that at leading order, G [0] ( t , t ) = G f ( t , t ) + g Z d t d t G f ( t , t )Σ [0] ( t, t ) G [0] ( t , t ) , (3.20)where G f is the free field two-point function. Then the structural properties of melonic graphsimplies that Σ [0] ( t, t ) = G [0] ( t, t ) D . (3.21) Actually one should be more precise here. By summing all the amplitudes one gets the perturbative freeenergies. However these free energies are likely to have a finite radius of convergence in the coupling constant, andthus be defined only in a disc type domain around λ = 0. As a consequence, if one is interested in large couplingphysics one should find the (possibly many) analytic continuations of these perturbative free energies. Another wayto consider the large coupling case is to find functional equations for the free energies and solve them in the largecoupling regime. These functional equations can sometimes be found using only perturbative arguments, this isexactly what is done for the leading order two-point function. − i ω G [0] ( ω ) = 1 + g Σ [0] ( ω ) G [0] ( ω ) , (3.22)where we introduced the notation Σ [0] ( ω ) for the Fourier transform of the self-energy at leadingorder in N . In the infrared limit, the left hand side drops. If we introduce ¯ G [0] the infrared/largecoupling limit of G [0] then ¯ G [0] satisfies the equation,0 = 1 + g ¯Σ [0] ( ω ) ¯ G [0] ( ω ) , (3.23)which rewrites in position space as, g Z d t ¯ G D [0] ( t , t ) ¯ G [0] ( t , t ) = − δ ( t − t ) . (3.24)The explicit solution in this limit is given by []¯ G [0] ( t , t ) = (cid:18) ( D −
1) tan( π/ ( D + 1))2 π ( D + 1) g (cid:19) / ( D +1) sign( t − t ) | t − t | / ( D +1) . (3.25)Coming back to the equation (3.24) satisfied by ¯ G [0] , one can show that if ¯ G [0] ( t , t ) is asolution, then, ¯ G [0] ( σ , σ ), where σ , = f ( t , ), is a solution as well, provided that ¯ G [0] ( t , t ) = | ∂ t f ( t ) ∂ t f ( t ) | D +1 ¯ G [0] ( σ , σ ). D +1 is the scaling dimension of ¯ G [0] . We want to study the Next-to-Leading Order of the real and complex colored tensor model. Thegoal is to check whether or not these models display the conformal symmetry property at largecoupling. In particular to check if it is true or not, we need to compute the scaling dimension ofthe two-point function at NLO. We then study the two-point function at NLO.As is seen in [58], the NLO of the real and complex model are described by the same family ofFeynman graphs. This means that non bipartite graphs do not appear at NLO. This is a specificityof the NLO that is not recovered at all orders. The complex cases have been investigated in thezero dimensional bosonic tensor model case in [76]. Following [76], it is possible to show that thevalue of the degree at NLO is $ NLO = ( D − D − . (3.26)The NLO 1PI self-energy and two-point functions are defined by G NLO ( t , t ) := G (cid:2) ( D − ( D − (cid:3) ( t , t ) , Σ NLO ( t , t ) := Σ (cid:2) ( D − ( D − (cid:3) ( t , t ) . (3.27)The functional equation for the 1PI self-energy writes graphically13 NLO ( t , t ) = X c = c NLO c c c + X c = c Q c ,c c c c c c c d c c c d c c + X pairs { c ,c } c ,c = c c = c Q c ,c c c c c c d c c c d c c c c . (3.28)The edges with grey disk insertion represent leading order two-point functions. The box representsone of the Q c i ,c j , c i = c j , which are the sum of ladder graphs of even length with ingoing/outgoingcolor c i and transmitted colors both c i and c j (unbroken chains in the language of [50]), so to saywe have Q c i ,c j ( t , t ; τ , τ ) = c i c i + c i c i c i c i c j c j + c i c i c i c i c j c j c j c j c i c i + · · · (3.29)We have the helpful property that Q c i ,c j ( t , t ; τ , τ ) = Q c n ,c m ( t , t ; τ , τ ) for any choice of c i , c j and c n , c m . Then we call Q c i ,c j ( t , t ; τ , τ ) = Q ( t , t ; τ , τ ). Equation (3.28) rewrites formallyΣ NLO ( t , t ) = Dg G [0] ( t , t ) D − G NLO ( t , t )+ Dg Z d t d t G [0] ( t , t ) D − G [0] ( t , t ) D − G [0] ( t, t ) Q ( t , t ; t , t )+ D ( D − g Z d t d t (cid:16) G [0] ( t , t ) D − G [0] ( t , t ) G [0] ( t , t ) × G [0] ( t, t ) D − Q ( t , t ; t, t ) (cid:17) . (3.30)We also have G NLO ( t , t ) = Z d t d t G [0] ( t , t )Σ( t, t ) G [0] ( t , t ) . (3.31)14ndeed, from the relation (3.19), we have in Fourier space, G e ( ω ) = − ω (cid:18) ω )i ω (cid:19) − (3.32)= G f ( ω ) X p ≥ (cid:18) − Σ( ω )i ω (cid:19) p . (3.33)Therefore, using the expansion (3.18) for G e and the fact that the self-energy can similarly beexpanded Σ( ω ) = X $ ≥ N − D − $ Σ [ $ ] ( ω ) (3.34)we have G NLO ( ω ) = G f ( ω ) X q ≥ (cid:18) − Σ [0] ( ω )i ω (cid:19) q Σ NLO ( ω ) G f ( ω ) X p ≥ (cid:18) − Σ [0] ( ω )i ω (cid:19) p (3.35)= (cid:0) G f ( ω ) − − Σ [0] ( ω ) (cid:1) − Σ NLO ( ω ) (cid:0) G f ( ω ) − − Σ [0] ( ω ) (cid:1) − (3.36)= G [0] ( ω )Σ NLO ( ω ) G [0] ( ω ) , (3.37)which when written in position space leads to equation (3.31).We now take care of Q ( t , t ; t , t ) which appears in equation (3.30). Q ( t , t ; t , t ) can beconstructed from Q ( t , t ; t , t ) using the operator K graphically defined below, K ( t , t ; t , t ) = t t t t . (3.38) Q ( t , t ; t , t ) writes Q ( t , t ; t , t ) = G [0] ( t , t ) G [0] ( t , t ) . (3.39)The operator K formally writes as K ( t , t , t , t ) = − g G [0] ( t , t ) G [0] ( t , t ) G [0] ( t , t ) D − . (3.40)We have Q ( t , t ; t , t ) = X n ≥ K n ( t , t ; t, t ) ∗ Q ( t, t ; t , t ) = (cid:0) δ ⊗ − K ∗ K (cid:1) − ∗ Q (3.41)where the (even) powers of K are taken with respect to the convolution product. Again Q is notdefined if K possesses eigenvalues ±
1: as explained in sections 1 and 2 we restrict our discussionto the non-divergent part of Q .The same argument than in the preceding SYK case applies here and at large g we should get15id of the first term of the right hand side of equation (3.30), then the large g NLO equation reads¯ G NLO ( t , t ) = Z d τ d τ ¯ G [0] ( t , τ ) ¯Σ NLO ( τ, τ ) ¯ G [0] ( τ , t ) (3.42)= Dg Z d τ d τ d t d t ¯ G [0] ( t , τ ) ¯ G D − ( τ, t ) ¯ G D − ( t , τ ) ¯ G [0] ( t, t ) Q ( τ, t ; τ , t ) ¯ G [0] ( τ , t )+ D ( D − g Z d τ d τ d t d t (cid:16) ¯ G [0] ( t , τ ) ¯ G D − ( τ, τ ) ¯ G [0] ( τ, t ) ¯ G [0] ( τ , t ) × ¯ G [0] ( t, t ) D − ¯ Q ( τ, τ ; t, t ) ¯ G [0] ( τ , t ) (cid:17) . (3.43)In order to get the conformal scaling of ¯ G NLO we need to understand how the conformal limit of¯ Q ( t , t ; t , t ) behaves. To do so we reduce ¯ Q to its 1PI connected counterpart Γ( t , t ; t , t ). Wehave that ¯ Q = ¯ G ⊗ + ¯ G ⊗ ∗ Γ ∗ ¯ G ⊗ . More precisely,¯ Q ( t , t ; t , t ) = ¯ Q ( t , t ; t , t ) + Z d t d t d τ d τ (cid:16) ¯ G [0] ( t , t ) ¯ G [0] ( t , t )Γ( t, t ; τ, τ ) × ¯ G [0] ( τ, t ) ¯ G [0] ( τ , t ) (cid:17) . (3.44)The scaling dimension of ¯ Q is D +1 as ¯ Q writes solely in terms of ¯ G [0] .Γ satisfies the following Schwinger–Dyson equationΓ( t, t ; τ, τ ) = Γ ( t, t ; τ, τ ) + Z d η d η d ω d ω Γ( t, t ; η, η ) ¯ K ( η, η ; ω, ω ) ¯ K ( ω, ω ; τ, τ ) , (3.45)where Γ ( t, t ; τ, τ ) = = g ¯ G [0] ( t, t ) D − ¯ G [0] ( t, τ ) ¯ G [0] ( t , τ ) ¯ G [0] ( τ, τ ) D − (3.46)and we do not display the colors of the edges as the dependence in the times is not sensible to it.From equation (3.45) we can deduce the scaling of Γ. First notice thatΓ ( σ, σ ; ζ, ζ ) = Γ ( t, t ; τ, τ ) | f ( t ) f ( t ) f ( τ ) f ( τ ) | DD +1 (3.47)for σ, σ , ζ, ζ = f ( t ) , f ( t ) , f ( τ ) , f ( τ ). This is obtained from the scaling of ¯ G [0] . Using the explicitexpression for ¯ K we also deduce that Z d β d β ¯ K ( σ, σ ; β, β ) ¯ K ( β, β ; ζ, ζ )= Z | f ( ω ) f ( ω ) | d ω d ω ¯ K ( η, η ; ω, ω ) ¯ K ( ω, ω ; τ, τ ) | f ( η ) f ( η ) | D +1 | f ( ω ) f ( ω ) || f ( τ ) f ( τ ) | DD +1 = Z d ω d ω ¯ K ( η, η ; ω, ω ) ¯ K ( ω, ω ; τ, τ ) | f ( η ) f ( η ) | D +1 | f ( τ ) f ( τ ) | DD +1 (3.48)where we have set σ, σ , ζ, ζ as before and β, β = f ( ω ) , f ( ω ). Consequently the scaling dimensionof Γ is DD +1 . Indeed if Γ( σ, σ , ζ, ζ ) is a solution of (3.45), then the function Γ ( t, t ; τ, τ ) = | f ( t ) f ( t ) f ( τ ) f ( τ ) | DD +1 Γ( σ, σ ; τ, τ ) with σ, σ , ζ, ζ = f ( t ) , f ( t ) , f ( τ ) , f ( τ ) is also a solution.16e now turn to the scaling dimension of ¯ Q in the conformal sector. We recall its expression interms of Γ¯ Q ( σ , σ ; σ , σ ) = ¯ Q ( σ , σ ; σ , σ ) + Z d β d β d γ d γ (cid:16) ¯ G [0] ( σ , β ) ¯ G [0] ( σ , β )Γ( β, β ; γ, γ ) × ¯ G [0] ( γ, σ ) ¯ G [0] ( γ , σ ) (cid:17) . (3.49)We call F ( σ , σ ; σ , σ ) the second term of the right hand side of (3.49), F ( σ , σ ; σ , σ ) = Z d β d β d γ d γ ¯ G [0] ( σ , β ) ¯ G [0] ( σ , β )Γ( β, β ; γ, γ ) ¯ G [0] ( γ, σ ) ¯ G [0] ( γ , σ ) . (3.50)We re-parametrize σ , σ , σ , σ = f ( t ) , f ( t ) , f ( t ) , f ( t ) and β, β , γ, γ = f ( t ) , f ( t ) , f ( τ ) , f ( τ )so to get the scaling dimension of F . This leads to F ( σ , σ ; σ , σ ) = F ( t , t ; t , t ) | f ( t ) f ( t ) f ( t ) f ( t ) | D +1 . (3.51)This tells us that F indeed scales and the scaling dimension is D +1 . This together with the factthat ¯ Q has scaling dimension 1 / ( D + 1) implies that ¯ Q has scaling dimension 1 / ( D + 1). Let uscompute the scaling of the NLO 2-point function. In the large g limit we have that, G NLO ( σ , σ ) = D g Z d γ d γ d β d β (cid:16) G ( σ , γ ) G ( γ, β ) D − G ( β , γ ) D − × G ( β, β ) Q ( γ, β ; γ , β ) G ( γ , σ ) (cid:17) + D ( D − g Z d γ d γ d β d β (cid:16) G ( σ , γ ) G ( γ, γ ) D − G ( γ, β ) G ( γ , β ) × G ( β, β ) D − Q ( γ, γ ; β, β ) G ( γ , σ ) (cid:17) (3.52)which leads after simplifications to G NLO ( σ , σ ) = G NLO ( t , t ) | f ( t ) f ( t ) | D +1 . (3.53)This shows that the scaling dimension of G NLO is D +1 as for the leading order term. The U( N ) × O( N ) × U( N ) model has been introduced in the tensor model literature in [77]. It wascalled the multi-orientable model. It has then been stated that it should be related to a complexfermions version of the SYK model in [25]. The model is defined as follows. One consider a pairof complex fermionic tensor fields ψ, ¯ ψ of rank 3. The partition function of the model writes Z m.o. λ,N = Z D ψ D ¯ ψ e − R d t L [ ψ ] (4.1)where L [ ψ ] = X n ¯ ψ n ∂ t ψ n + λN / X i,j,k,i ,j ,k ψ ijk ( t ) ¯ ψ kj i ( t ) ψ k ji ( t ) ¯ ψ k j i ( t ) (4.2)and we also define g = λ . (4.3)17 + --- + Figure 1: Propagator and vertex of the multi-orientable model.The fields transform under the natural action of U( N ) × O( N ) × U( N ) and the action is invariantunder this transformation.The Feynman graphs are constructed out of the building blocks represented on fig. 1 with thecondition that a (+) half-edge can only connect to a ( − ) half-edge. It is also possible to define thenotion of jackets for these graphs. This is indeed a non trivial statement as one can find examplesof tensor models for which this is not the case because of the so called tadface graphs, see [27, 29]for a discussion of this topics.Thanks to this notion of jackets the degree can be generalized in this case and it can be shownthat the multi-orientable model has a well defined 1 /N expansion. So to say we have for the freeenergy, F m.o. λ,N = log Z m.o. λ,N = X $ ≥ N − $ F m.o.[ $ ] ( λ ) . (4.4)In this case however, $ ∈ N ≥ , where N ≥ is the set of integer larger or equal to zero. We canagain define the two-point function. Let us start by the free one, G f ( t , t ) = 1 N D X n T ¯ ψ n ( t ) ψ n ( t ) E = sign( t − t ) , (4.5) D T ¯ ψ ijk ( t ) ψ i j k ( t ) E = G f ( t , t ) δ ii δ jj δ kk , (4.6)where n here is a multi-index that labels the components of the tensor. For the exact two-pointfunction we have, G e ( t , t ) = 1 N D X n T ¯ ψ n ( t ) ψ n ( t ) E , (4.7) D T ¯ ψ ijk ( t ) ψ i j k ( t ) E = G e ( t , t ) δ ii δ jj δ kk . (4.8)Consequently we have, G e ( t , t ) = X $ ∈ N N − $ G [ $ ] ( t , t ) . (4.9) As was shown in [27], the leading order in N is once again dominated by melonic graphs. As aconsequence we can write the equation satisfied by the two-point function at leading order, G [0] ( t , t ) = G f ( t , t ) + g Z d t d t G f ( t , t ) G [0] ( t, t ) G [0] ( t , t ) . (4.10)Using now known manipulations we have in the infrared/large coupling limit the approximatedequation g Z d t ¯ G [0] ( t , t ) ¯ G [0] ( t, t ) = − δ ( t − t ) . (4.11)18his equation has a known solution¯ G [0] ( t , t ) = (cid:18) tan( π/ πg (cid:19) / sign( t − t ) | t − t | / . (4.12)Moreover, the large coupling equation has the same re-parametrization symmetry. If ¯ G [0] ( t , t ) isa solution, then, ¯ G [0] ( σ , σ ), where σ , = f ( t , ), is a solution as well, provided that ¯ G [0] ( t , t ) = | ∂ t f ( t ) ∂ t f ( t ) | ¯ G [0] ( σ , σ ). The next-to-leading order of the two-point function of the multi-orientable model has been studiedin [28]. As the combinatorics is unchanged by the fact that we consider fermionic fields on onedimensional space we can easily infer the next-to-leading order in this case. The degree at next-to-leading order is $ NLO = 12 . (4.13)The self-energy Σ NLO ( t , t ) := Σ [1 / ( t , t ) at next-to-leading order writes graphically,Σ NLO ( t , t ) = + GNLO (4.14)where the gray disks represent insertion of the leading order two-point function on the edges. Thistranslates into the formal equationΣ
NLO ( t , t ) = λ δ ( t − t ) G [0] ( t , t ) + 3 g G [0] ( t , t ) G [1 / ( t , t ) . (4.15)We also have, G NLO ( t , t ) := G [1 / ( t , t ) = Z d t d t G [0] ( t , t )Σ NLO ( t, t ) G [0] ( t , t ) . (4.16)We now introduce the analogue of the operator K ( t , t ; t , t ) introduced in earlier sections.It is here defined as K ( t , t ; t , t ) = 3 g G [0] ( t , t ) G [0] ( t , t ) G [0] ( t , t ) . (4.17)We introduce G [0] ( t , t ) = G [0] ( t − t ) = G [0] ( τ ), with τ = t − t . Then, using (4.16) and (4.15),we can write at least formally, G NLO ( t , t ) = λ Z d t G [0] ( t , t ) G [0] (0) G [0] ( t, t ) + Z d t d t K ( t , t ; t, t ) G NLO ( t, t ) (4.18)= λ Z d t G [0] ( t , t ) G [0] (0) G [0] ( t, t ) + [ K ∗ G NLO ] ( t , t ) , (4.19)and thus (cid:2) ( δ ⊗ − K ) ∗ G NLO (cid:3) ( t , t ) := Z d t d t (cid:0) δ ( t − t ) ⊗ δ ( t − t ) − K ( t , t ; t, t ) (cid:1) G NLO ( t, t ) (4.20)= λ Z d tG [0] ( t , t ) G [0] (0) G [0] ( t, t ) . (4.21) Notice however the difference in sign. G [0] (0) = 0, which then implies (cid:2) ( δ ⊗ − K ) ∗ G NLO (cid:3) ( t , t ) = 0 , ∀ t , t . (4.22)This implies that G NLO must lie in the kernel of ( δ ⊗ − K ). This happens as long as G NLO = 0or G NLO is an eigenvector of K with eigenvalue 1. Since there are such eigenvectors ¯ G NLO can bean arbitrary linear combination of them if it does not vanish and, without additional data on thebehaviour of ¯ G NLO it is not possible to conclude about its conformality.
We have found that the NLO in the large N expansion does not modify the dependence of the 2-point function in the coupling and time in the infrared regime. For this reason the 2-point functionis still conformally invariant and the IR dimension of the fermions does not receive any correctionat this order. Nonetheless higher-order correlation functions may deviate from the CFT behaviourand this provides an incentive to study their behaviour. In any case one can consider the NLO asbeing a CFT in any context where the corrections to these higher-order functions can be neglected.This fact may reveal itself to be important in the construction of the bulk dual using theAdS/CFT dictionary [59]: absence of 1 /N corrections in the CFT translates into absence ofquantum corrections in the bulk dual. For example if the scaling dimensions of the single tracesoperators discussed in [4, 59] are identical at NLO this would translate by the fact that the cor-responding bulk field masses do not receive correction at one loop. Hence our result gives a strongindication that the first quantum correction may be absent and this point calls for a deeper study.A natural extension of this work would be to determine how the spontaneous breaking of theconformal symmetry appears in the NLO 4-point function and what are the effects of incorporatingthe NLO correction of the coupling constant. Another point of interest is to push the study evenfurther and see if the NNLO continues to preserve the conformal invariance. The method describedin this paper can be generalized to study the NLO in other models, such as in the supersymmetriccase [39, 40]. Finally it would be useful to settle the question of the conformal invariance of the2-point function in the multi-orientable tensor model. Acknowledgments
The work of S.M., made within the
Labex Ilp (reference
Anr–10–Labx–63 ), is supported byFrench state funds managed by the
Agence nationale de la recherche , as part of the program
In-vestissements d’avenir under the reference
Anr–11–Idex–0004–02 . S.M. is supported by Cefipraunder project 5204-4.
A Composite field effective action
The goal of this section is to describe how composite fields can be used for colored tensor models.We will focus on real tensors for simplicity but the generalization to complex tensors is straight-forward.Recall the action for D + 1 real fermionic tensor fields (section 3) S [ ψ c ] = Z d t X c ψ c ∂ t ψ c + i D +12 Λ Y c ψ c ! (A.1) One can also convince oneself by going into Fourier space and defining an appropriate cut-off for the regular-ization of the integral. c = 0 , . . . , D and Λ is defined byΛ = λN D ( D − (A.2)The associated partition function is Z = Z Y c D ψ c e − S [ ψ c ] . (A.3) A.1 Effective action
In order to introduce a composite field G cn c ,n c ( t, t ) = − ψ cn c ( t ) ψ cn c ( t ) (A.4)where n c and n c are tensor multi-indices, corresponding to the 2-point function G ce ( t, t ) = − N D *X n c ψ cn c ( t ) ψ cn c ( t ) + , (A.5)one first needs to obtain an action with bilinear terms in each color. In the rest of this section thetensor indices will be implicit.This can be achieved by integrating out one of the color, say ψ which is straightforward sincethe action is quadratic in this field, the productΨ = i D +12 Λ D Y i =1 ψ i (A.6)acting as a source for ψ , where i = 1 , . . . , D . Using standard techniques the effective actionobtained after integrating out ψ is S eff [ ψ i ] = 12 X i Z d t ψ i ∂ t ψ i + i D +1 Λ Z d t d t S ( t, t ) Y i ψ i ( t ) ψ i ( t ) . (A.7)after rearranging the fermions (the signs have been traded for i), where S ( t, t ) is the Green functionfor ∂ t .The next step consists in introducing the bilocal tensor fields G i ( t, t ) (A.4) and to use auxiliaryfields Σ i ( t, t ) such that 1 = Z Y i D G i δ (cid:0) G i ( t, t ) + ψ i ( t ) ψ i ( t ) (cid:1) (A.8a)= Z Y i D G i D Σ i e − S aux [ ψ i ,G i , Σ i ] (A.8b)where S aux [ ψ i , G i , Σ i ] = − X i Z d t d t Σ i ( t, t ) (cid:0) G i ( t, t ) + ψ i ( t ) ψ i ( t ) (cid:1) . (A.9)The functional integral (A.3) becomes Z = Z Y i D ψ i D G i D Σ i e − ˜ S eff [ ψ i ,G i ] − S aux [ G i , Σ i ] . (A.10) The composite fields are distinguished from the correlation functions by the absence of any lower index. S eff [ ψ i , G i ] = 12 X i Z d t ψ i ∂ t ψ i + i ( D +1) Λ Z d t d t S ( t, t ) Y i G i ( t, t ) . (A.11)Performing the quadratic integration over ψ i yields the effective action for G i and Σ i W [ G i , Σ i ] = − X i tr ln( ∂ t − Σ i ) − X i Z d t d t Σ i ( t, t ) G i ( t, t )+ i ( D +1) Λ Z d t d t S ( t, t ) Y i G i ( t, t ) . (A.12)The equations of motion are δWδG i = 0 = ⇒ Σ i ( t, t ) = i ( D +1) Λ S ( t, t ) Y j = i G j ( t, t ) , (A.13a) δWδ Σ i = 0 = ⇒ (cid:16) δ ( t − t ) ⊗ D ∂ t − Σ i ( t, t ) (cid:17) − − G i ( t, t ) = 0 (A.13b)where ⊗ D is the tensor identity. The last equation can be rewritten as δ ( t − t ) ⊗ D = ∂ t G i ( t, t ) + Z d t Σ i ( t, t ) G i ( t , t ) , (A.14a)= ∂ t G i ( t, t ) + i ( D +1) Λ Z d t S ( t, t ) G i ( t , t ) Y j = i G j ( t, t ) (A.14b)where the last equality follows from inserting (A.13a).In the long-time regime Λ | t − t | (cid:29) ( D +1) Λ Z d t G i ( t, t ) S ( t , t ) Y j = i G j ( t , t ) = δ ( t − t ) ⊗ D . (A.15) A.2 Fluctuations
The solutions to the equations of motion are denoted by ( G [0] , Σ [0] ) and they are identical for allcolors since the equations are symmetric under exchange of colors (cid:10) G i (cid:11) = G [0] ⊗ D , (cid:10) Σ i (cid:11) = Σ [0] ⊗ D (A.16)where G [0] and Σ [0] are genuine bilocal fields (not tensors). Powers of (cid:10) G i (cid:11) (or (cid:10) Σ i (cid:11) ) will beaccompanied by factors of N due to the contraction of the identities( (cid:10) G i (cid:11) ) k = N k + ( k ) G k [0] = N k ( k +1)2 G k [0] . (A.17)The saddle point equations (A.13) becomeΣ [0] ( t, t ) = i ( D +1) λ S ( t, t ) G [0] ( t, t ) D − , (A.18a) (cid:0) δ ( t − t ) ∂ t + Σ [0] ( t, t ) (cid:1) − − G [0] ( t, t ) = 0 . (A.18b)where the relation (A.2) between Λ and λ has been used.Then one can consider fluctuations ( g i , σ i ) around these solutions G i = G [0] ⊗ D + g i , Σ i = Σ [0] ⊗ D + σ i . (A.19)22lugging these expressions into (A.12) yield W [ g i , σ i ] = 14 X i Z d t · · · d t σ i ( t , t ) k ( t , . . . , t ) σ i ( t , t ) − X i Z d t d t σ i g i + i ( D +1) λ N D − Z d t d t S G D − X i,j g i g j + 12 X i X n ≥ n tr( G [0] σ i ) n + i ( D +1) N D ( D − D X n =3 λ n ! N ( D − n )( D − n +1)2 Z d t d t S G D − n [0] X i ,...,i n g i · · · g i n (A.20)where the dependence in the time ( t, t ) has been omitted and the kernel k is k ( t , . . . , t ) = G [0] ( t , t ) G [0] ( t , t ) . (A.21)Rescaling the fluctuations such that G i = G [0] ⊗ D + (cid:12)(cid:12) G [0] (cid:12)(cid:12) − D g i , Σ i = Σ [0] ⊗ D + (cid:12)(cid:12) G [0] (cid:12)(cid:12) D − σ i (A.22)and absorbing the factors inside the kernel gives the symmetric kernel [4] K sym ( t , . . . , t ) = − λ (cid:12)(cid:12) G [0] ( t , t ) (cid:12)(cid:12) D − k ( t , . . . , t ) (cid:12)(cid:12) G [0] ( t , t ) (cid:12)(cid:12) D − (A.23)which is conjugated to the kernel (3.40).Truncating the action to the quadratic order one can obtain an effective action for the g i onlyby integrating out σ i W eff [ g i ] = − λ X i,j Z d t · · · d t g i ( t , t ) K ij ( t , . . . , t ) g j ( t , t ) (A.24)where K ij ( t , . . . , t ) = K − ( t , . . . , t ) δ ij − i ( D +1) N D − S ( t , t ) δ ( t − t ) δ ( t − t ) . (A.25)The 4-point function for the fermions correspond to the 2-point function of the fluctuations (cid:10) g i g j (cid:11) . At leading order it can be computed using the above quadratic action and one can see thatit is equivalent to the computation done in section 3. The additional propagator in the action is aconsequence of integrating out one of the colors and it is present to connect vertices, very similarto the way one adds an extra line after averaging over disorder in the standard SYK model [4].However this time the extra line represents a dynamical fields. This breaks to the conformalinvariance of the corresponding effective action in the infrared. References [1] A. Y. Kitaev.
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