Comments on the Random Thirring Model

TL;DR

The paper analyzes a 1+1D random Thirring (SYK-like) model at large N by deriving and solving the Schwinger-Dyson equations, uncovering an IR where q=2 exhibits logarithmic running and a beta function β(J)=4π^2 J^3. Through conformal perturbation theory and ensemble-based RG analysis, it shows that the quadratic beta term vanishes on average while the cubic term universally drives the flow, rendering the ensemble marginally irrelevant on average. This leads to an effective, regulator-sensitive IR that is not truly conformal and suggests smoothing of chiral transitions under disorder averaging, with potential connections to holography and BTZ-black hole physics. The work delineates the interplay between randomness, conformal perturbation theory, and large-N dynamics in a translationally invariant 1+1D setting, highlighting both renormalizability questions and possible avenues for further exploration of higher-point functions and finite-temperature behavior.

Abstract

The Thirring model with random couplings is a translationally invariant generalisation of the SYK model to 1+1 dimensions, which is tractable in the large N limit. We compute its two point function, at large distances, for any strength of the random coupling. For a given realisation, the couplings contain both irrelevant and relevant marginal operators, but statistically, in the large N limit, the random couplings are overall always marginally irrelevant, in sharp distinction to the usual Thirring model. We show the leading term to the $β$ function in conformal perturbation theory, which is quadratic in the couplings, vanishes, while its usually subleading cubic term matches our RG flow.