The spectrum in the Sachdev-Ye-Kitaev model

Abstract

The SYK model consists of N ≫ 1 fermions in 0 + 1 dimensions with a random, all-to-all quartic interaction. Recently, Kitaev has found that the SYK model is maximally chaotic and has proposed it as a model of holography. We solve the Schwinger-Dyson equation and compute the spectrum of two-particle states in SYK, finding both a continuous and discrete tower. The four-point function is expressed as a sum over the spectrum. The sum over the discrete tower is evaluated.The Sachdev-Ye-Kitaev model (SYK) [1, 2] is a 0 + 1 dimensional model of N ≫ 1 fermions with an all-to-all random quartic interaction. SYK has three notable features:Solvable at strong coupling. At large N one can sum all Feynman diagrams, and thereby compute correlation functions at strong coupling.Maximally chaotic. Chaos is quantified by the Lyapunov exponent, which is defined by an out-of-time-order four-point function [3,4]. The Lyapunov exponent of a black hole in Einstein gravity is 2π β [4-6], where β is the inverse temperature. This is the maximal allowed Lyapunov exponent [7], and SYK saturates the bound [1].Emergent conformal symmetry. In the context of the two-point function, there is emergent conformal symmetry at low energies [1,[8][9][10].where χ j are Majorana fermions {χ i , χ j } = δ ij , and the model has quenched disorder with the couplings J ijkl drawn from the distribution,leading to a disorder average of,The expressions for the correlation functions that will follow will always be the result after the disorder average has been performed. The Lagrangian trivially follows from

Higher dimensional generalizations of the SYK model

“…This is achieved by first passing the Majorana fermions via a low-pass filter, and then coupling these via an SYK random interaction. The motivation for this is to have a model which is an ordinary relativistic field theory above some scale (and below a UV cut-off), which may model an asymptotically AdS 3 space, and some complicated IR dynamics encoding an object in the interior of AdS 3 . Besides ensuring that the high momenta modes are filtered out, in this work we also focus on chiral filters, thus only one chiral half of the fermions participate in the interactions.…”

“…3 We have dropped some non-essential constant pieces in obtaining this expression. 4 We will take L not divisible by 4 for simplicity.…”

“…The Sachdev-Ye-Kitaev (SYK) model [1,2] has received much attention recently [3][4][5][6][7][8]. It is a simple model with solvable aspects which exhibits interesting connections to quantum chaos [9][10][11][12][13], black hole physics and quantum gravity in 1+1 dimensions [14][15][16][17][18][19][20].…”

Remarks on the Sachdev-Ye-Kitaev model

“…This allows us to connect analytically the free UV theory to the interacting and nearly conformal IR theory. Further recent work in this or similar models includes [14,15,16,17,7,18,19]. See also [20,21] for a string motivated model with disorder.…”

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

“…As one notable example, Majorana-fermion zero modes [1,2] capture the essence of non-Abelian statistics and topological quantum computation [3,4], and correspondingly now form the centerpiece of a vibrant experimental effort [5][6][7][8][9][10][11][12][13][14][15][16]. More recently, randomly interacting Majorana fermions governed by the "Sachdev-YeKitaev (SYK) model" [17][18][19] were shown to exhibit sharp connections to chaos, quantum-information scrambling, and black holes-naturally igniting broad interdisciplinary activity (see, e.g., [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). The goal of this Rapid Communication is to exploit hardware components of a Majorana-based topological quantum computer for a tabletop implementation of the SYK model, thus uniting these very different topics.…”