SYK model with an extra diagonal perturbation: phase transition in the eigenvalue spectrum

TL;DR

This paper analyzes the SYK model with a diagonal perturbation, revealing a phase transition in the eigenvalue spectrum where a single eigenvalue separates from the bulk, using combinatorial and analytical methods.

Contribution

It provides exact expressions for moments and identifies a phase transition in the eigenvalue spectrum caused by a diagonal perturbation in the SYK model.

Findings

Spectrum can have a gap when >\u0011^c

A single eigenvalue separates from the bulk spectrum

Results recover known random matrix behaviors in certain limits

Abstract

We study the SYK model with an extra constant source, \.i.e. a constant matrix or equivalently a diagonal matrix with only one non-zero entry $\lambda_1$. By using methods from analytic combinatorics, we find exact expressions for the moments of this model. We further prove that the spectrum of this model can have a gap when $\lambda_1>\lambda^c_1$, thus exhibiting a phase transition in $\lambda_1$. 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.