Exact multilocal renormalization on the effective action : application to the random sine Gordon model statics and non-equilibrium dynamics

TL;DR

This paper develops an exact multilocal renormalization group method for the effective action, applying it to the random sine Gordon model to analyze static and dynamic properties near the glass transition.

Contribution

It introduces a novel multilocal RG approach for the effective action and applies it to the random sine Gordon model, revealing new insights into glassy phases and non-equilibrium dynamics.

Findings

Recovered the $ta$ exponent and scaling functions for the O(N) model.

Derived static correlations and equilibrium dynamical exponent $z$ for the glass phase.

Computed non-equilibrium response and correlation functions with novel scaling forms.

Abstract

We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the $\eta$ exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent $z$ are recovered and several new results are obtained. The equilibrium two-point scaling functions are obtained. The nonequilibrium, finite momentum, two-time $t,t'$ response and correlations are computed. They are shown to exhibit scaling forms, characterized by novel exponents $\lambda_R \neq \lambda_C$, as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be non trivial and of the form $X(q^z (t-t'), t/t')$. Analogies and differences with pure critical models are discussed.