Approximating the Sachdev-Ye-Kitaev model with Majorana wires
AApproximating the Sachdev-Ye-Kitaev model with Majorana wires
Aaron Chew, Andrew Essin, and Jason Alicea
1, 3 Department of Physics and Institute for Quantum Information and Matter,California Institute of Technology, Pasadena, CA 91125, USA Department of Physics, University of California, Davis, CA 95616, USA Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA (Dated: March 22, 2017)The Sachdev-Ye-Kitaev (SYK) model describes a collection of randomly interacting Majoranafermions that exhibits profound connections to quantum chaos and black holes. We propose a solid-state implementation based on a quantum dot coupled to an array of topological superconductingwires hosting Majorana zero modes. Interactions and disorder intrinsic to the dot mediate the desiredrandom Majorana couplings, while an approximate symmetry suppresses additional unwanted terms.We use random matrix theory and numerics to show that our setup emulates the SYK model (upto corrections that we quantify) and discuss experimental signatures.
Introduction.
Majorana fermions provide buildingblocks for many novel phenomena. As one notableexample, Majorana-fermion zero modes [1, 2] capturethe essence of non-Abelian statistics and topologicalquantum computation [3, 4], and correspondingly nowform the centerpiece of a vibrant experimental effort[5–16]. More recently, randomly interacting Majoranafermions governed by the ‘Sachdev-Ye-Kitaev (SYK)model’ [17–19] were shown to exhibit sharp connectionsto chaos, quantum-information scrambling, and blackholes—naturally igniting broad interdisciplinary activity(see, e.g., [20–34]). The goal of this paper is to exploithardware components of a Majorana-based topologicalquantum computer for a tabletop implementation of theSYK model, thus uniting these very different topics.The SYK Hamiltonian reads H SYK = (cid:88) ≤ i 3! ¯ J N . (2)At large N the model is solvable and exhibits rich be-havior. Most remarkably, for temperatures satisfying ¯ J/N (cid:28) T (cid:28) ¯ J the SYK model enjoys approximate con-formal symmetry and, similar to black holes, is maxi-mally chaotic as diagnosed by out-of-time-ordered corre-lators. These properties are expected for a holographicdual to quantum gravity, and there has been much inter-est in the corresponding bulk theory [31, 35].Laboratory realizations of Eq. (1) face intertwined hur-dles: First, hybridizing Majorana fermions naively yieldsbilinears of the form iM jk γ j γ k as the dominant couplings,yet these are absent from the Hamiltonian. Second, gen-erating all-to-all couplings requires abandoning localityfor the Majorana fermions. And finally, the host platform must carry sufficient disorder to at least approximate in-dependence among the large number of random J ijkl ’s.References 23 and 34 proposed SYK-model platforms us-ing cold atoms and topological insulators, respectively.We instead envision a realization [Fig. 1(a)] that exploitsMajorana zero modes germinated in proximitized semi-conductor nanowires [36, 37]—a leading experimental ar-chitecture for topological quantum information applica-tions [5–9, 11, 13–15].More precisely, we explore an array of such wires in-terfaced with a disordered quantum dot that mediatescoupling among the constituent Majorana modes andrandomizes the corresponding zero-mode wavefunctions.Unwanted Majorana bilinears are suppressed by an ap-proximate time-reversal symmetry [38] that, importantly,is preserved by the dominant sources of disorder expectedin the dot. Interactions intrinsic to the dot instead gen-erate the desired all-to-all four-Majorana couplings, thusapproximating the SYK model up to corrections that wequantify (and which appear generic for any physical real-ization). We discuss several future directions that our ap-proach spotlights, including tunneling experiments thatprovide a natural first probe of SYK physics. Setup. We begin with the Hamiltonian for a clean,single-subband proximitized wire: H wire = (cid:90) x (cid:20) ψ † (cid:18) − ∂ x m − µ − hσ x − iασ y ∂ x (cid:19) ψ + ∆( ψ ↑ ψ ↓ + H.c. ) + · · · (cid:21) , (3)which features Zeeman coupling h generated by a mag-netic field B , spin-orbit coupling α , and proximity-induced pairing ∆ . Together these ingredients allowthe formation of Majorana zero modes γ, ˜ γ at the wireends over a chemical potential window centered around µ = 0 [36, 37]. Crucially, the terms explicitly dis-played above respect a time-reversal transformation T that sends ψ → ψ , i → − i and thus satisfies T = +1 [38]. Additional couplings denoted by the ellipsis can in a r X i v : . [ c ond - m a t . d i s - nn ] M a r (a)(b) 2D disorderedquantum dot Majorana wire arrayMajorana zero modes Majorana zero modesDot levels B ˜ γ i γ i λλ δ(cid:30) typ J (c) (cid:31) FIG. 1. (a) Device that approximates the SYK model us-ing topological wires interfaced with a 2D quantum dot. Thedot mediates disorder and four-fermion interactions amongMajorana modes γ ,...,N inherited from the wires, while Majo-rana bilinears are suppressed by an approximate time-reversalsymmetry. (b) Energy levels pre-hybridization. The dot-Majorana hybridization energy λ is large compared to Nδ(cid:15) typ ,where N is the number of Majorana modes and δ(cid:15) typ is thetypical dot level spacing; this maximizes leakage into the dot.(c) Energy levels post-hybridization. The N absorbed Ma-jorana modes enhance the energy (cid:15) to the next excited dotstate via level repulsion; four-Majorana interactions occur ona scale J < (cid:15) . general violate T since it is not a true microscopic sym-metry. Nevertheless, we will assume that such perturba-tions are negligible, which is not unreasonable at low den-sities appropriate for the topological regime. (See Discus-sion for further comments.) Under the approximate T symmetry the Majorana-zero-mode operators transformas γ → γ and ˜ γ → − ˜ γ . The opposite signs acquiredby γ, ˜ γ ensure that T commutes with the ground-statefermion parity P = iγ ˜ γ , as it must.Consider now N topological wires ‘plugged into’ a 2Ddisordered quantum dot [Fig. 1(a)], such that the Majo-ranas γ ,...,N that are even under T hybridize with thedot while their partners ˜ γ ,...,N decouple completely. Thefull architecture continues to approximately preserve T provided ( i ) the dot carries negligible spin-orbit couplingand ( ii ) the B field orients in the plane of the dot sothat orbital effects are absent. Here the setup falls intoclass BDI, which in the free-fermion limit admits an in-teger topological invariant ν ∈ Z [39, 40] that counts thenumber of Majorana zero modes at each end; interac-tions collapse the classification to Z [41, 42]. In essenceour device leverages nanowires to construct a topologicalphase with a free-fermion invariant ν = N : All bilinearcouplings iM jk γ j γ k are forbidden by T and thus cannot be generated by the dot under the conditions specifiedabove. We exploit the resulting N Majorana zero modesto simulate SYK-model physics mediated by disorder andinteractions native to the dot, similar in spirit to Refs. 21and 34.Figures 1(b) and (c) illustrate the relevant parameterregime. The dot-Majorana hybridization energy λ sat-isfies λ (cid:29) N δ(cid:15) typ , where δ(cid:15) typ denotes the typical dotlevel spacing. This criterion enables the dot to absorba substantial fraction of all N Majorana zero modes asshown below. The dot’s disordered environment then effi-ciently ‘scrambles’ the zero-mode wavefunctions, thoughwe assume that their localization length ξ exceeds the dotsize L . More quantitatively, we take the mean-free path (cid:96) mfp (cid:28) L to maximize randomness and the dimensionlessconductance g = k F (cid:96) mfp > such that L < ξ . Turningon four-fermion interactions couples the disordered Majo-rana modes with typical J ijkl ’s that are smaller than theenergy (cid:15) to the next excited state (which as we will seeis enhanced by level repulsion compared to δ(cid:15) typ ). Thisseparation of scales allows us to first analyze the dis-ordered wavefunctions in the non-interacting limit andthen explore interactions projected onto the zero-modesubspace. We next carry out this program using random-matrix theory, which is expected to apply in the aboveregime [43, 44]. Random-matrix-theory analysis. We model thedot as a 2D lattice composed of N dot (cid:29) N sites hostingfermions c a =1 ,...,N dot [45]. In terms of physical dot param-eters we have N dot ∼ ( L/(cid:96) mfp ) . The Hamiltonian gov-erning the dot-Majorana system is H = H + H int , with H and H int the free and interacting pieces, respectively.We employ a Majorana basis and write c a = ( η a + i ˜ η a ) / ,where η a is even under T while ˜ η a is odd (similarly to γ i , ˜ γ i ). In terms of Γ = [ η · · · η N dot ; γ · · · γ N ] T , ˜Γ = [˜ η · · · ˜ η N dot ] T , (4) H takes the form H = i (cid:2) Γ T ˜Γ T (cid:3) (cid:20) M − M T (cid:21) (cid:20) Γ˜Γ (cid:21) . (5)Time-reversal T fixes the zeros above but allows for ageneral real-valued ( N dot + N ) × N dot -dimensional ma-trix M . (The matrix is not square since we discardedthe ˜ γ i modes that trivially decouple.) One can performa singular-value decomposition of M by writing Γ = O Γ (cid:48) and ˜Γ = ˜ O ˜Γ (cid:48) . Here O , ˜ O denote orthogonal matrices con-sisting of singular vectors, i.e., the matrix Λ ≡ O T M ˜ O only has non-zero entries along the diagonal. Writing Γ (cid:48) = [ η (cid:48) · · · η (cid:48) N dot ; γ (cid:48) · · · γ (cid:48) N ] T and similarly for ˜Γ (cid:48) , theHamiltonian becomes H = i N dot (cid:88) a =1 (cid:15) a η (cid:48) a ˜ η (cid:48) a (6)where (cid:15) a ≡ Λ aa are the non-zero dot energies. Mostimportantly, γ (cid:48) i =1 ,...,N drop out and form the modified N Majorana zero modes guaranteed by T symmetry.We are interested in statistical properties of the asso-ciated Majorana wavefunctions in the presence of strongrandomness. To make analytic progress we assume (fornow) that all elements of M in Eq. (5) are indepen-dent, Gaussian-distributed random variables with zeromean and the same variance, corresponding to the chi-ral orthogonal ensemble [46, 47]. This form permitsCooper pairing of dot fermions—an inessential detailfor our purposes—and also does not enforce the strong-hybridization criterion λ (cid:29) N δ(cid:15) typ . We will see that theMajorana wavefunctions nevertheless live almost entirelyin the dot as appropriate for the latter regime.The probability density for such a random matrix M is [48] P ( M ) ∝ exp (cid:104) − π N dot δ(cid:15) Tr ( M T M ) (cid:105) . Because P ( M ) is invariant under M → O T M ˜ O , the singular-vector matrices O , ˜ O are uniformly distributed over thespaces O ( N dot + N ) and O ( N dot ) , respectively. In par-ticular, the Majorana wavefunctions φ i correspondingto γ (cid:48) i are the final N columns of a random element of O ( N dot + N ) . For large N dot + N the distribution of wave-function components is asymptotically Gaussian [43, 49]: (cid:104) φ i,I (cid:105) = 0 , (cid:104) φ i,I φ j,J (cid:105) = δ i,j δ I,J N dot + N ≈ δ i,j δ I,J N dot . (7)Summing φ i,I over the dot sites thus gives unity up tocorrections of order N/N dot , i.e., the dot swallows theMajorana modes as claimed.Once absorbed by the dot, the N Majorana zero modesrepel the nearby energy levels. Random matrix the-ory allows us to estimate the energy (cid:15) to the first ex-cited dot state. References [50, 51] show that the small-est eigenvalue for the Wishart matrix M T M approaches ( √ a − √ b ) v , where M is an a × b matrix with variance v for each element. The energy (cid:15) is the square root of thiseigenvalue. For our matrix M we thus obtain (cid:15) ≈ π N δ(cid:15) typ . (8)The enhancement compared to δ(cid:15) typ [sketched inFig. 1(c)] isolates the N Majorana modes from adjacentlevels, justifying projection onto the zero-energy sub-space.Let us now examine a general T -invariant four-fermion interaction among dot fermions, H int = (cid:80) abcd U abcd c † a c † b c c c d . Projection follows from c a → (cid:80) i φ i,a γ (cid:48) i , which yields H → (cid:88) ≤ i We now semi-quantitatively validaterandom-matrix-theory predictions using a more physi- FIG. 2. (a) Average absorption of Majorana wavefunctions into the dot versus the hybridization strength λ with N = 16 zeromodes. Inset: probability density for a Majorana wavefunction swallowed and randomized by the dot of size × . (b)Enhanced level repulsion of the first excited dot state (cid:15) by N absorbed Majorana modes; cf. Figs. 1(b) and (c). (c) Histogramof J ijkl couplings obtained from local current-current interactions on a dot of size × , together with a Gaussian fit (solidline). (d) Scaling of the variance ∝ ¯ J of these couplings versus N dot . cally motivated Hamiltonian. Consider first the free part, H = − (cid:88) a (cid:54) = b t ab c † a c b + (cid:88) a V a c † a c a + λ N (cid:88) i =1 γ i ( c a i − c † a i ) . (14)Here V a is an uncorrelated Gaussian disorder landscapewith zero mean and variance ¯ V . In the λ hybridizationterm, Majorana γ i couples to a single dot site a i . Forthe hoppings t ab , we consider uniform nearest-neighbortunnelings of strength t (yielding an Anderson model)and compare results with purely random, arbitrary-rangehopping satisfying (cid:104) t ab (cid:105) = 0 , (cid:104) t ab t a (cid:48) b (cid:48) (cid:105) = t (yielding arandom-matrix model). All data below correspond to ¯ V = t with adjacent Majorana modes separated by twoor three dot sites. Unless specified otherwise λ = t/ ,the dot system size is × , and results are disorder-averaged over many configurations [20 for Fig. 2(a), 50for (b) and (d), and 500 for (c)].Figure 2(a) corresponds to N = 16 and plots thefraction of the Majorana mode wavefunctions absorbedby the dot—averaged over all zero modes—versus λ/ ( N δ(cid:15) typ ) . For both the Anderson and random-matrixmodels the fraction is of order one at λ/ ( N δ(cid:15) typ ) (cid:38) ,eventually saturating to unity as in random matrix the-ory. The inset shows the probability density for a zero-mode wavefunction nearly fully absorbed by the dot, ob-tained from an N = 1 Anderson model; the wavefunctionappears thoroughly randomized and loses all informationabout its original position (in this case, the center). Fig-ure 2(b) illustrates level repulsion of the excitation energy (cid:15) (normalized by the level spacing δ(cid:15) typ ) versus N . Notethat the dot almost completely absorbs all zero modesup to the largest N shown. The random matrix modelyields a slope that agrees within ∼ with Eq. (8) ob-tained from random matrix theory, while the Andersonmodel agrees within ∼ .Next we include a local current-current interaction H int = U (cid:88) (cid:104) ab (cid:105) c † a ∇ c a · c † b ∇ c b , (15)with ∇ a lattice gradient, projected into the zero-mode subspace. Figure 2(c) plots a histogram of the resulting J ijkl couplings (in units of U ) using an Anderson modelwith N = 8 and a × dot. The data agrees well witha Gaussian distribution; see solid line. Finally, Fig. 2(d)illustrates the N dot -dependence of the variance ∝ ¯ J for J ijkl [recall Eq. (2)] with N = 8 . The Anderson modelyields a scaling close /N —slower than /N resultfrom random matrix theory [Eq. (11)]. We attribute thisdifference primarily to localization effects that effectivelyreduce the system area. As a check, the random-matrixmodel, which should not suffer localization due to thenon-local hoppings, indeed yields the expected /N scaling. Discussion. We showed that in certain regimes ourMajorana wire/quantum dot setup can emulate the SYKmodel up to very generic corrections. Chiefly, we invokedan approximate time-reversal symmetry that suppressesbilinears, strong dot-Majorana coupling that delocalizesand randomizes the wavefunctions, level repulsion thatsuppresses pollution of the zero-mode subspace by addi-tional dot levels, and sufficient randomness to approxi-mate independent, random all-to-all couplings J ijkl . Re-garding the last property, Eqs. (11) and (12) imply thatindependence requires ¯ J ∼ /N for a dot with local in-teractions. Since ¯ J (cid:28) (cid:15) excited dot states indeed can besafely ignored. However, increasing N rapidly diminishesthe strong-coupling temperature window T (cid:28) ¯ J —wheremuch of the interesting physics emerges. This challengecan be alleviated with long-range interactions, which leadto slower decay with N . Alternatively, one can intention-ally abandon independence to boost ¯ J , though the fate ofSYK physics in such cases remains to be systematicallyunderstood.To maintain approximate T symmetry graphene-baseddots appear ideal due to their strict two-dimensionalityand extremely weak spin-orbit coupling. In this case thedominant source of T violation will likely originate fromthe Majorana wires. We can crudely assess the impact ofsuch perturbations by adding local T -breaking terms forthe dot in the vicinity of the wires and projecting, e.g., δH = χ N (cid:88) i =1 ( ic † a i c a i +1 + H.c. ) → (cid:88) ≤ j We are indebted to X. Chen,M. Franz, Y. Gu, A. Kitaev, J. Meyer, P. Lee, S. Nadj-Perge, J. Iverson, and D. Pikulin for illuminating discus-sions. We gratefully acknowledge support from the Na-tional Science Foundation through grant DMR-1341822(A. C. and J. A.); the Caltech Institute for Quantum In-formation and Matter, an NSF Physics Frontiers Centerwith support of the Gordon and Betty Moore Founda-tion through Grant GBMF1250; and the Walter BurkeInstitute for Theoretical Physics at Caltech. [1] N. Read and D. Green, Phys. Rev. B , 10267 (2000).[2] A. Y. Kitaev, Sov. Phys.–Uspeki , 131 (2001).[3] A. Y. Kitaev, Ann. Phys. , 2 (2003).[4] C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma, Rev. Mod. Phys. , 1083 (2008).[5] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.A. M. Bakkers, and L. P. Kouwenhoven, Science ,1003 (2012).[6] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, andH. Shtrikman, Nat. Phys. , 887 (2012).[7] A. D. K. Finck, D. J. Van Harlingen, P. K. Mohseni,K. Jung, and X. Li, Phys. Rev. Lett. , 126406 (2013). [8] H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen,M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus,Phys. Rev. B , 241401 (2013).[9] E. J. H. Lee, X. Jiang, M. Houzet, R. Aguado, C. M.Lieber, and S. De Franceschi, Nature Nanotech. , 79(2014).[10] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon,J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yaz-dani, Science , 602 (2014).[11] C. Kurter, A. D. K. Finck, Y. S. Hor, and D. J. VanHarlingen, Nature Communications , 7130 (2015).[12] J.-P. Xu, M.-X. Wang, Z. L. Liu, J.-F. Ge, X. Yang,C. Liu, Z. A. Xu, D. Guan, C. L. Gao, D. Qian, Y. Liu,Q.-H. Wang, F.-C. Zhang, Q.-K. Xue, and J.-F. Jia,Phys. Rev. Lett. , 017001 (2015).[13] S. M. Albrecht, A. P. Higginbotham, M. Madsen,F. Kuemmeth, T. S. Jespersen, J. Nygård, P. Krogstrup,and C. M. Marcus, Nature , 206 (2016).[14] H. Zhang, O. Gul, S. Conesa-Boj, K. Zuo, V. Mourik,F. K. de Vries, J. van Veen, D. J. van Woerkom, M. P.Nowak, M. Wimmer, D. Car, S. Plissard, E. P. A. M.Bakkers, M. Quintero-Perez, S. Goswami, K. Watanabe,T. Taniguchi, and L. P. Kouwenhoven, arXiv:1603.04069(2016).[15] M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon,M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, andC. M. Marcus, Science , 1557 (2016).[16] B. E. Feldman, M. T. Randeria, J. Li, S. Jeon, Y. Xie,Z. Wang, I. K. Drozdov, B. A. Bernevig, and A. Yazdani,Nature Physics , 286 (2017).[17] S. Sachdev and J. Ye, Phys. Rev. Lett. , 3339 (1993).[18] A. Kitaev, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/ (2015).[19] J. Maldacena and D. Stanford, Phys. Rev. D , 106002(2016).[20] J. Polchinski and V. Rosenhaus, arXiv:1601.06768(2016).[21] Y.-Z. You, A. W. W. Ludwig, and C. Xu,arXiv:1602.06964 (2016).[22] W. Fu and S. Sachdev, Phys. Rev. B , 035135 (2016).[23] I. Danshita, M. Hanada, and M. Tezuka,arXiv:1606.02454 (2016).[24] A. Jevicki and K. Suzuki, arXiv:1608.07567 (2016).[25] D. J. Gross and V. Rosenhaus, arXiv:1610.01569 (2016).[26] S. Banerjee and E. Altman, arXiv:1610.04619 (2016).[27] E. Witten, (2016), arXiv:1610.09758.[28] R. Gurau, Nuclear Physics B , 386 (2017).[29] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski,P. Saad, S. H. Shenker, D. Stanford, A. Streicher, andM. Tezuka, (2016), arXiv:1611.04650.[30] R. A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen,and S. Sachdev, (2016), arXiv:1612.00849.[31] K. Jensen, Phys. Rev. Lett. , 111601 (2016).[32] Z. Bi, C.-M. Jian, Y.-Z. You, K. A. Pawlak, and C. Xu,(2017), arXiv:1701.07081.[33] Y. Gu, X.-L. Qi, and D. Stanford, arXiv:1609.07832(2016).[34] D. I. Pikulin and M. Franz, arXiv:1702.04426 (2017).[35] D. J. Gross and V. Rosenhaus, (2017), arXiv:1702.08016.[36] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010).[37] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010). [38] S. Tewari and J. D. Sau, Phys. Rev. Lett. , 150408(2012).[39] A. Kitaev, AIP Conference Proceedings , 22 (2009).[40] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W.Ludwig, New Journal of Physics , 065010 (2010).[41] L. Fidkowski and A. Kitaev, Phys. Rev. B , 134509(2010).[42] L. Fidkowski and A. Kitaev, Phys. Rev. B , 075103(2011).[43] C. W. J. Beenakker, Rev. Mod. Phys. , 731 (1997).[44] I. Aleiner, P. Brouwer, and L. Glazman, Physics Reports , 309 (2002).[45] We assume spinless fermions for simplicity; spin can beintroduced trivially since we impose T = 1 symmetry.[46] J. Verbaarschot, Phys. Rev. Lett. , 2531 (1994).[47] M. A. Stephanov, J. J. M. Verbaarschot, and T. Wettig,(2005), arXiv:hep-ph/0509286.[48] C. W. J. Beenakker, Rev. Mod. Phys. , 1037 (2015).[49] T. Guhr, A. Muller-Groeling, and H. A. Weidenmuller,Physics Reports , 189 (1998).[50] J. W. Silverstein, The Annals of Probability , 1364(1985).[51] Z. D. Bai and Y. Q. Yin, Ann. Probab. , 1275 (1993).[52] C.-K. Chiu, D. I. Pikulin, and M. Franz, Phys. Rev. B , 241115 (2015).[53] D. I. Pikulin, C.-K. Chiu, X. Zhu, and M. Franz, Phys.Rev. B , 075438 (2015).[54] T. Gorin, Journal of Mathematical Physics , 3342(2002), http://dx.doi.org/10.1063/1.1471367.[55] T. Prosen, T. H. Seligman, and H. A. Weiden-muller, Journal of Mathematical Physics , 5135 (2002),http://aip.scitation.org/doi/pdf/10.1063/1.1506955.[56] Here connectedness refers to the way the U tensors arecontracted.[57] E. A. Bender and E. Canfield, Journal of CombinatorialTheory, Series A , 296 (1978).[58] B. Bollobas, Journal of the London Mathematical Society s2-26 , 201 (1982). Corrections to Wick’s theorem In this Appendix we discuss the asymptotically smallcorrections to Wick’s theorem for the J ijkl couplings.Our treatment is quite general and does not rely onour particular proposed realization. We will take thebest-case scenario for randomness, invoking Eq. (12) andassuming completely disordered and independent zero-mode wavefunctions φ i that obey Eq. (7). In practicethe φ i ’s also suffer subdominant correlations and will notbe truly Gaussian; these corrections can be studied us-ing techniques described, e.g., in Refs. 54 and 55 but areneglected for simplicity.Using Eq. (10) we see that correlations among m J ijkl ’ssatisfy (cid:104) m (cid:89) f =1 J i f j f k f l f (cid:105) = (cid:18) (cid:19) m (cid:89) f U as a f b f c f d f × (cid:104) (cid:89) f φ i f a f φ j f b f φ k f c f φ l f d f (cid:105) , (A.17) where repeated indices are summed. The assumption ofi.i.d. Gaussian wavefunction components φ i,I allows us tosimply apply Wick’s theorem to evaluate the right-handside—though this does not mean the J ijkl couplings areindependent. Each U as a f b f c f d f connects to four wavefunc-tion elements φ i f ,a f , etc., and each wavefunction elementcontracts with another. The disorder average on the rightside of Eq. (A.17) thus pairs all the a f , b f , c f , d f indicesin some manner and also forces all of the i f , j f , k f , l f indices to similarly pair together (otherwise the averagevanishes trivially).Wick-theorem-obeying correlations among J ijkl ’s oc-cur when all four indices of each U as a f b f c f d f pair with allfour indices of another. For a local interaction such casesyield (cid:104) m (cid:89) f =1 J i f j f k f l f (cid:105) Wick ∼ N g − m dot , (A.18)where g is the number of connected pieces in the diagramand m must be even [56]. Note that the maximally dis-connected pieces with g = m/ minimize the decay with N dot and reduce to (cid:104) J ijkl (cid:105) m/ .Non-Wick correlations arise when more than two J ijkl ’s share indices. Equation (13) gives one example.Another is (cid:104) J ijkl J inkp J mjol J mnop (cid:105) ∝ N (cid:88) abcdefgh U as abcd U as ebgd U as afch U as efgh ∼ N − , (A.19)where we again used a local interaction. By contrast,Eqs. (13) and (A.19) both vanish in the SYK model.Such non-Wick contributions are generically suppressedby some power of N dot compared to (cid:104) J ijkl (cid:105) m/ . Naively,this analysis suggests that the Wick contributions domi-nate the Feynman-diagram expansion of the model, andthat hence we can use the ‘melon diagram’ formalism[18–20] that yields the large- N SYK solution.However, at increasingly high order in the diagram-matic expansion, non-Wick correlations lead to a prolif-eration of new diagrams. Though each individual contri-bution is small in the above sense, the number of allowedgraphs grows far faster than the suppression in /N dot .References [57, 58] discuss the asymptotic number of sim-ple regular graphs, that is, graphs with each vertex con-nected to a fixed number of edges. In our case we areconcerned with 4-regular graphs: the number scales as P ( m ) ∼ (4 m )!(2 m )!( m )! C m ∼ m m (A.20)for some constant C . Note that this result underesti-mates the number of graphs since it excludes the onesthat have multiple edges that join the same vertices.Thus while each graph is, at best, suppressed exponen-tially in m (i.e., by a factor /N const × m dotdot