Remarks on the Sachdev-Ye-Kitaev model

Abstract

We study a quantum mechanical model proposed by Sachdev, Ye and Kitaev. The model consists of N Majorana fermions with random interactions of a few fermions at a time. It it tractable in the large N limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies where it develops an emergent conformal symmetry. We study two and four point functions of the fundamental fermions. This provides the spectrum of physical excitations for the bilocal field.The emergent conformal symmetry is a reparametrization symmetry, which is spontaneously broken to SL(2, R), leading to zero modes. These zero modes are lifted by a small residual explicit breaking, which produces an enhanced contribution to the four point function. This contribution displays a maximal Lyapunov exponent in the chaos region (out of time ordered correlator). We expect these features to be universal properties of large N quantum mechanics systems with emergent reparametrization symmetry.This article is largely based on talks given by Kitaev [1], which motivated us to work out the details of the ideas described there.

Higher dimensional generalizations of the SYK model

“…We will be interested in two filters: the standard gaussian filter 4) whereD sets the scale of the filter, and the "scaling" filters…”

“…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].…”

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

“…This correction is small provided χ U/N; longer-range interactions relax the criterion further. Tunneling into the dot provides an appealing benchmark of proximity to SYK physics: the conductance approaches a constant at zero bias if bilinears dominate but diverges as V −1/2 for the large-N SYK model [19].…”

“…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.…”