A generalization of Sachdev-Ye-Kitaev

Journal of High Energy Physics, Feb 2017

The SYK model: fermions with a q-body, Gaussian-random, all-to-all interaction, is the first of a fascinating new class of solvable large N models. We generalize SYK to include f flavors of fermions, each occupying N a sites and appearing with a q a order in the interaction. Like SYK, this entire class of models generically has an infrared fixed point. We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global O(N 1) × O(N 2) × … × O(N f ) symmetry: a subgroup of the O(N) symmetry of SYK; thereby giving a richer spectrum. We also elucidate aspects of the large q limit and the OPE, and solve q = 2 SYK at finite N.

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A generalization of Sachdev-Ye-Kitaev

Received: December Published for SISSA by Springer Santa Barbara 0 1 CA 0 1 U.S.A. 0 1 include f 0 1 0 Open Access , c The Authors 1 Kavli Institute for Theoretical Physics, University of California The SYK model: fermions with a q-body, Gaussian-random, all-to-all interaction, is the rst of a fascinating new class of solvable large N models. We generalize SYK to avors of fermions, each occupying Na sites and appearing with a qa order in the interaction. Like SYK, this entire class of models generically has an infrared We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global O(N1) ArXiv ePrint: 1610.01569 1/N Expansion; AdS-CFT Correspondence; Conformal Field Theory; Matrix - generalization of Sachdev-Ye-Kitaev O(Nf ) symmetry: a subgroup of Contents 1 Introduction 2 Two-point function 2.1 Large qk A generalization of SYK Graphical solution Spectral function E ective action Four-point function Dimensions of composite operators Four-point function Generalized model Equal qa, a Equal qa, a A E ective action A.1 Fluctuations Model with a scalar C Random mass matrix fermions C.1 In nite N C.2 Finite N Dirac fermion Majorana fermion Introduction maximal chaos. While there are models that contain some of these properties, SYK is seminars [2]. Matrix models are closely related to string theories [3{8], with the most calculations for strong coupling. SYK is such a model. understanding both quantum eld theories and string theories. One may hope that the study of SYK-like models will also prove productive. The SYK model consists of N 1 Majorana fermions i, with a q-body Hamiltonian largely applicable. with quenched disorder, H = The model has qualitatively similar properties for any choice of even q 4. The couplings es a simple integral equation which can be explicitly solved near the infrared xed point. The fermions start with dimension 0 in the UV, and ow to dimension = 1=q in the IR. operators, singlets under O(N ), are schematically PN i=1 i @2n+1 i. In the UV, these opsmall n, and approach 2 + 2n + 1 asymptotically for large n. The standard AdS/CFT SYK is, and the extent to which it is nonlocal, remains an open problem. H = of the interaction, qa, can depend on the avor a, as long as Na=N remains nite as N = P the fermions, and nd that the model (1.2) generically has an IR xed point. While the IR dimension for SYK (1.1) was determine the dimensions a. In the limit of large qa, these have simple analytic solutions. the generalized model (1.2) has an O(N1) O(Nf ) symmetry. The singlet bilinear operators are PNa same properties hold for the generalized model. appears with a degeneracy of f 1. Indeed, this model is similar to SYK with N fermions singlet operators exist, allowing for a richer model. 4. This appendix can be read independently of the rest of the paper. Two-point function S = q! i1;:::;iq=1 a Gaussian ensemble. The two-point function of the Ji1;:::;iq is taken to be, i2;:::;iq=1 hJi1i2:::iq Ji1i2:::iq i = J 2 : hJi1i2:::iq Ji1i2:::iq i = (q At zero coupling, the Euclidean two-point function hT i( ) j (0)i G( ) ij is given by, G0( ) = sgn( ) ; G0(!) = > 0 and sgn( ) = < 0) accounts for for the two-point function drastically simplify and are given by, G(!) 1 = G0(!) 1 (!) = ( ) = J 2G( )q 1 of the one-particle irreducible self-energy (!). The second equation, which is written in position space, is a special feature of SYK (see however, an analytic solution to this equation is not known. At strong coupling, jJ j (equivalently, the infrared limit), one can drop the i! in (2.5), to get, with a lled circle is the two-point function. The dashed line is the disorder. One can verify that, is a solution to (2.7) provided one takes, G(!) (!) = ( ) = J 2G( )q 1 G( ) = b bq = The Fourier transform of G( ), given in (2.8), is useful in verifying this, where we de ned, G(!) = d ei! G( ) = b ( )J 2 1sgn(!); 2 ) = 2ip reparameterization of time, 2) ! f 0( 1) f 0( 2) G(f ( 1) f ( 2)). Theretwo-point function by mapping the real line to a circle [2, 35{37]. A generalization of SYK The model we introduce is a generalization of SYK (2.1). It contains f avors of fermions, with Na fermions of avor a, each appearing qa times in the interaction, so that the Hamiltonian couples q = Pf avor a 2 f1; : : : ; f g. Explicitly, the action is, S = a=1 i=1 Qfa=1 qa! I X JI Gaussian distribution, where the disorder average hJI JI i is given by It will be convenient to make the following de nitions, P [JI ] / exp hJI JI i = J a=1 Na Y(qa k = a6=k The class of models (2.12) for large N is characterized by f 1 independent continuous paalized model (2.12) the symmetry is (...truncated)


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David J. Gross, Vladimir Rosenhaus. A generalization of Sachdev-Ye-Kitaev, Journal of High Energy Physics, 2017, pp. 93, Volume 2017, Issue 2, DOI: 10.1007/JHEP02(2017)093