Super-rough phase of the random-phase sine-Gordon model: Two-loop results

TL;DR

This paper analyzes the super-rough phase of the two-dimensional random-phase sine-Gordon model near its glass transition, deriving two-loop renormalization group equations and revealing universal behavior of the correlation function with a quadratic logarithmic form.

Contribution

It provides the first two-loop renormalization group analysis of the model, identifying universal invariants and correcting previous one-loop predictions for the correlation function amplitude.

Findings

Correlation function behaves as a universal quadratic logarithm at large distances.

Amplitude of the correlation function is a universal function of temperature.

Two-loop order results differ from previous conformal field theory predictions.

Abstract

We consider the two-dimensional random-phase sine-Gordon and study the vicinity of its glass transition temperature $T_c$, in an expansion in small $\tau=(T_c-T)/T_c$, where $T$ denotes the temperature. We derive renormalization group equations in cubic order in the anharmonicity, and show that they contain two universal invariants. Using them we obtain that the correlation function in the super-rough phase for temperature $T<T_c$ behaves at large distances as $\bar{<[\theta(x)-\theta(0)]^2>} = \mathcal{A}\ln^2(|x|/a) + \mathcal{O}[\ln(|x|/a)]$, where the amplitude $\mathcal{A}$ is a universal function of temperature $\mathcal{A}=2\tau^2-2\tau^3+\mathcal{O}(\tau^4)$. This result differs at two-loop order, i.e., $\mathcal{O}(\tau^3)$, from the prediction based on results from the "nearly conformal" field theory of a related fermion model. We also obtain the correction-to-scaling exponent.