Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a model of q interacting fermions. Gross and Rosenhaus have proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, those flavors are reminiscent of the colors used in random tensor theory. This gives us the opportunity to apply some modern, yet elementary, tools developed in the context of random tensors to one particular instance of such colored SYK models. We illustrate our method by identifying all diagrams which contribute to the leading and next-to-leading orders of the 2-point and 4-point functions in the large N expansion, and argue that our method can be further applied if necessary. In a second part we focus on the recently introduced Gurau-Witten tensor model and also extract the leading and next-to-leading orders of the 2-point and 4-point functions. This analysis turns out to be remarkably more involved than in the colored SYK model.

The bulk dual of SYK: cubic couplings

“…The simplest is to make the couplings Ji 1 ...iq be nearly static quantum variables [28]. A better way of eliminating disorder is to turn SYK into a tensor model [29] (see also, [30][31][32]), analogous to the ones previously studied [33][34][35][36][37]. To leading order in 1/N all these approaches agree.…”

SYK-like tensor models on the lattice

“…The SYK models have been generalized in various directions (e.g., flavor, supersymmetry etc.) [18][19][20][21][22][23][24][25]. The SYK model has also been realized on a lattice in higher dimensions [26][27][28][29][30].…”

“…The 1 N corrections are much harder, since now a host of other nonmelonic diagrams as well as other class of diagrams 3 starts contributing. In fact, it was shown [24] that the leading non-melonic contribution scales like O(N 3 ). Hence, one cannot immediately conclude whether Pillow channel will also saturate the chaos bound or not.…”

“…It is convenient to understand the rank-D KT model via "uncoloring" the GW model of rank-D which has been more explicitly studied in the literature [24,40,42].…”

Correlators in the N = 2 $$ \mathcal{N}=2 $$ supersymmetric SYK model

“…The model develops an emergent (approximate) reparametrization symmetry at low energy [6,[17][18][19] that is also present in dilaton gravity theories on AdS 2 [6,[20][21][22][23]. It has intimate relations with wellstudied random matrix models [6,18,[24][25][26][27][28][29][30][31][32], it further boosts the study of a different type of the large-N limit [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48], and it is closely related to vector models [49]. The SYK model can be generalized to include extra symmetries [50] or to live in higher dimensions [51][52][53][54][55][56].…”