Functional Renormalization Group and the Field Theory of Disordered Elastic Systems

TL;DR

This paper develops a two-loop functional renormalization group theory for disordered elastic systems, overcoming previous ambiguities, and provides precise predictions for roughness exponents and universal amplitudes that align well with numerical and exact results.

Contribution

It introduces a unique, renormalizable 2-loop field theory for disordered elastic systems, resolving ambiguities from non-analytic disorder correlators and extending previous one-loop analyses.

Findings

Calculated roughness exponent for interfaces: ζ ≈ 0.2083ε + 0.0069ε²

Derived universal amplitudes for random field disorder to order ε²

Validated predictions against numerical and exact results, showing improvement over one-loop results

Abstract

We study elastic systems such as interfaces or lattices, pinned by quenched disorder. To escape triviality as a result of ``dimensional reduction'', we use the functional renormalization group. Difficulties arise in the calculation of the renormalization group functions beyond 1-loop order. Even worse, observables such as the 2-point correlation function exhibit the same problem already at 1-loop order. These difficulties are due to the non-analyticity of the renormalized disorder correlator at zero temperature, which is inherent to the physics beyond the Larkin length, characterized by many metastable states. As a result, 2-loop diagrams, which involve derivatives of the disorder correlator at the non-analytic point, are naively "ambiguous''. We examine several routes out of this dilemma, which lead to a unique renormalizable field-theory at 2-loop order. It is also the only theory consistent with the potentiality of the problem. The beta-function differs from previous work and the one at depinning by novel "anomalous terms''. For interfaces and random bond disorder we find a roughness exponent zeta = 0.20829804 epsilon + 0.006858 epsilon^2, epsilon = 4-d. For random field disorder we find zeta = epsilon/3 and compute universal amplitudes to order epsilon^2. For periodic systems we evaluate the universal amplitude of the 2-point function. We also clarify the dependence of universal amplitudes on the boundary conditions at large scale. All predictions are in good agreement with numerical and exact results, and an improvement over one loop. Finally we calculate higher correlation functions, which turn out to be equivalent to those at depinning to leading order in epsilon.