Non-local meta-conformal invariance, diffusion-limited erosion and the XXZ chain

TL;DR

This paper constructs a novel infinite-dimensional dynamical symmetry for diffusion-limited erosion in 1D, revealing its connection to loop-Virasoro algebras and the XXZ chain, and derives exact response functions.

Contribution

It introduces a new symmetry algebra for diffusion-limited erosion and links it to integrable models and surface growth phenomena.

Findings

Constructed the infinite-dimensional dynamical symmetry algebra for the model.

Derived exact two-time response functions consistent with known solutions.

Established connections between erosion dynamics, surface models, and the XXZ chain.

Abstract

Diffusion-limited erosion is a distinct universality class of fluctuating interfaces. Although its dynamical exponent $z=1$, none of the known variants of conformal invariance can act as its dynamical symmetry. In $d=1$ spatial dimensions, its infinite-dimensional dynamic symmetry is constructed and shown to be isomorphic to the direct sum of three loop-Virasoro algebras, with the maximal finite-dimensional sub-algebra $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$. The infinitesimal generators are spatially non-local and use the Riesz-Feller fractional derivative. Co-variant two-time response functions are derived and reproduce the exact solution of diffusion-limited erosion. The relationship with the terrace-step-kind model of vicinal surfaces and the integrable XXZ chain are discussed.