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, which motivated us to work out the details of the ideas described there.

• Received 8 September 2016

DOI:https://doi.org/10.1103/PhysRevD.94.106002

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Physical Review D 50^th Anniversary Milestones

This collection of seminal papers from PRD highlights research that remains central to developments today in particle physics, quantum field and string theory, gravitation, cosmology, and particle astrophysics.