Published for SISSA by Springer Received: February 6, 2019 Accepted: March 5, 2019 Published: March 14, 2019 Towards a full solution of the large N double-scaled JHEP03(2019)079 SYK model Micha Berkooz,a Mikhail Isachenkov,a;b Vladimir Narovlanskya and Genis Torrentsa aDepartment of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel bInstitut des Hautes Etudes´ Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, France E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We compute the exact, all energy scale, 4-point function of the large N double- scaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL(2), that generalizes the Schwarzian to the complete range of energies. Keywords: Holography and condensed matter physics (AdS/CMT), Matrix Models, Quantum Groups, Random Systems ArXiv ePrint: 1811.02584 Open Access, c The Authors. https://doi.org/10.1007/JHEP03(2019)079 Article funded by SCOAP3. Contents 1 Introduction and summary of results1 1.1 Summary of results, and outline3 2 Computing using chord diagrams3 2.1 Chord diagrams4 2.2 Observables6 2.3 Analytic evaluation8 JHEP03(2019)079 2.4 Relation to quantum groups 11 3 Bi-local observables in double-scaled SYK 11 3.1 Chord construction for bi-local operators 12 3.2 The 2-point function 16 3.3 Time translation invariance of the bi-local operator 17 4 4-point function 18 4.1 Uncrossed 4-point function hM1M1M2M2i 21 4.2 Crossed 4-point function hM1M2M1M2i 22 5 The diagrammatic rules of the full model 23 5.1 q ! 1− and low energy limit of the correlation functions 25 6 Chaos and the R-matrix 27 6.1 Integral form of the R-matrix 27 6.2 Chaos 29 7 Quantum deformation of SL(2) 35 8 Regimes of large N and large p in SYK-like models 40 8.1 Partition function at high temperature 40 8.2 Fermion propagator at high temperature 43 9 Summary and conclusion 44 A Poisson distribution of intersections and q-binomial coefficients 45 A.1 Poisson distribution of intersections of sets in the large N limit 45 A.2 Weighted separation of chords into two sets 46 B Special functions 46 C Series forms for observables and q ! 1− limits 51 C.1 Partition function 51 C.2 2-point function 51 C.3 Uncrossed 4-point function 52 C.4 2n-point functions 53 C.5 q ! 1− limit of the partition function 54 C.6 q ! 1− limit of the 2-point function 55 { i { D Asymptotic forms of Gamma functions 56 E Quantum groups 57 E.1 Hopf algebra of functions on a Lie group 57 ext E.2 Quantum group SUq (1; 1) 62 E.3 Quantum group Uq(su(1; 1)) 63 E.4 Locally compact quantum groups 66 JHEP03(2019)079 1 Introduction and summary of results The Sachdev-Ye-Kitaev (SYK) model [1{5] is a quantum-mechanical model in 0+1 dimen- sions, constructed out of N Majorana fermions with all-to-all interactions, and random (disordered) couplings. Denote the N Majorana fermions by i, i = 1; ··· ;N, satisfying the algebra f i; jg = 2δij. The Hamiltonian of the system is given by p=2 X H = i Ji1i2···ip i1 ··· ip ; (1.1) 1≤i1<···<ip≤N with the Ji1···ip being random couplings (whose statistics is specified below). Over the last few years this model has attracted considerable attention as a toy testing ground of quantum gravity, since it is a rare case which is both tractable, in the IR and other limits, and has a maximal chaos exponent. In particular, it possesses a conformal regime in the IR, and a pattern of symmetry encoded by the Schwarzian low energy effective action, which is the effective dynamics of gravity on AdS2 coupled to a scalar field [6{8]. Many additional interesting models related to SYK were also studied. Higher dimen- sional SYK-like models were considered in [9{13] (see also [14, 15] for disorder effects relevant to these theories). There are also theories with similar behavior to SYK, but without disorder [16{18]. The relation of SYK to random matrix theory was discussed in [19, 20]. Higher correlation functions were studied as well [21, 22], and results beyond the leading order in N were obtained [23{25]. One usually studies the SYK model in the limit N ! 1 and p fixed. This large N limit is solved by observing that the Feynman diagrams are dominated by melonic diagrams, leading to Schwinger-Dyson consistency equations for the 2-point function h i(t) i(0)i; these equations can then be solved in the IR using a conformal Ansatz [1{5]. Consistency equations were obtained also for the 4-point function [3{5]. As mentioned above, at low energies, an emergent reparametrization symmetry (which is broken both spontaneously and explicitly) selects the Schwarzian theory as the low energy limit of the SYK model. This fact was used to get results on the low energy limit of SYK, including the 4-point function [26{29]. We will find it particularly useful to compare our results to [28]. In this work, our goal is to provide a full all-energy solution to the model in a slightly different scaling, which is the double-scaled SYK [19, 30]. In this limit, p is not held fixed { 1 { p (as N ! 1) but rather is proportional to N (and is always assumed to be even), i.e., 2p2 N ! 1; λ = = fixed: (1.2) N We will also use interchangeably λ and q, which is defined by q ≡ e−λ: (1.3) Despite the different scaling, the model can still be used to study gravitational physics. In fact, in [19] this model was triple-scaled along with E − Emin ! 0 to obtain a model with the same IR physics as the usual SYK scaling. We will be interested in computing JHEP03(2019)079 the exact correlation functions in the full double-scaled model rather than just in the IR (using combinatorial tools, and bypassing the Schwinger-Dyson equations). The random couplings Ji1···ip are usually assumed to be independent and Gaussian. However, it is enough for us to take them to be independent random variables, with zero mean and uniformly bounded moments independent of N [30].1 They even need not be identically distributed, as in this limit only their standard deviation matters. They are normalized according to N−1 hJ 2 i = J 2; (1.4) i1···ip J p where h∗iJ stands for the ensemble average over the random couplings J. The variance (1.4) is compatible with the one used in [30], but in the scaling (1.2) it differs from the usual SYK conventions [5] by a constant factor of λ only. Without loss of generality, we will 2 2 set J = 1, corresponding to normalizing the couplings such that htr H iJ = 1; J can be restored easily by dimensional analysis. Actually, there is a large class of models with the same reduction to chord diagrams, as the SYK model in the limit (1.2). I.e., they all boil down to a very similar counting problem after averaging over the random couplings in this limit. One of such different microscopic realizations of this universality class is analyzed in [30, 31]. It consists of N sites, with a spin 1=2 variable (a qubit) at each site, on which the Pauli matrices σ(a) (a = 1; 2; 3) act. The Hamiltonian is given by an all-to-all, length p, random Hamiltonian 3 X X (a ) (a ) H~ = α σ 1 ··· σ p : (1.5) a1;··· ;ap;(i1;··· ;ip) i1 ip 1≤i1<···<ip≤N a1;··· ;ap=1 The random couplings α are again independent, have zero mean, uniformly bounded mo- ments, and variance N−1 hα2 i = 3−p : (1.6) a1;··· ;ap;(i1;··· ;ip) p The analysis of this model is completely analogous to that of (1.1) in the double-scaling limit (1.2); the results are the same when expressed in terms of λ or q, with the only 4p2 difference being that λ = 3N in this case. 1 N1=2 More precisely, the moments of p Ji1···ip are uniformly bounded by a number independent of N. 2The trace here is normalized by tr 1 = 1. { 2 { 1.1 Summary of results, and outline The main results in the paper are the following • We set up the more general formalism of bi-local operators in the entire double-scaled SYK model (and not just in its low energy Schwarzian limit), • We compute the exact 4-point function in the model, • We compute the corrections to the chaos exponent as a function of the temperature, • We recast the formulas in a way that suggests that all correlation functions can be JHEP03(2019)079 computed using a quantum group symmetry. In section2 we review the class of operators under consideration and the reduction of their correlation functions to counting chord diagrams. Then, in section3 we notice that in the averaged theory, the operators appear as bi-local operators (this being manifest in the chord diagrams).