Sachdev–Ye–Kitaev model with an extra diagonal perturbation: phase transition in the eigenvalue spectrum

Abstract

Abstract We study the SYK model with an extra constant source, i̇.e. a constant matrix or equivalently a diagonal matrix with only one non-zero entry λ 1. By using methods from analytic combinatorics (Flajolet and Sedgewick 2009 Analytic Combinatorics (Cambridge University Press)), we find exact expressions for the moments of this model. We further prove that the spectrum of this model can have a gap when {\lambda }_{1}^{c}$?> λ 1 > λ 1 c , thus exhibiting a phase transition in λ 1. In this case, a single isolated eigenvalue splits off from SYK’s eigenvalues distribution. We located this single eigenvalue by analyzing the singular behavior of a supercritical functional composition scheme. In certain limit our results recover the ones of random matrices with non-zero mean entries.