Precise Low-Temperature Expansions for the Sachdev-Ye-Kitaev model

TL;DR

This paper numerically solves the SYK model's Dyson-Schwinger equations to accurately determine its low-temperature energy expansion, revealing a non-integer power law consistent with conformal perturbation theory predictions.

Contribution

It provides a highly precise numerical method for low-temperature expansions of the SYK model and clarifies the nature of its non-integer temperature powers.

Findings

First non-integer power of T is T^{6.54}

Coefficient matches conformal perturbation theory

Expansion accuracy reaches 10^{-36}

Abstract

We solve numerically the large $N$ Dyson-Schwinger equations for the Sachdev-Ye-Kitaev (SYK) model utilizing the Legendre polynomial decomposition and reaching $10^{-36}$ accuracy. Using this we compute the energy of the SYK model at low temperatures $T\ll J$ and obtain its series expansion up to $T^{7.54}$. While it was suggested that the expansion contains terms $T^{3.77}$ and $T^{5.68}$, we find that the first non-integer power of temperature is $T^{6.54}$, which comes from the two point function of the fermion bilinear operator $O_{h_{1}}=\chi \partial_{\tau}^{3}\chi$ with scaling dimension $h_{1}\approx 3.77$. The coefficient in front of $T^{6.54}$ term agrees well with the prediction of the conformal perturbation theory. We conclude that the conformal perturbation theory appears to work even though the SYK model is not strictly conformal.