Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion and the XXZ Chain

Malte Henkel

doi:10.3390/sym9010002

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.

Highlights

Symmetries have since a long time played an important role in the analysis of physical systems.The insight gained can be either calculational, in that a recognised symmetry becomes useful in simplifying calculations, or else conceptual, in that the identification of symmetries can lead to new level of understanding
We shall inquire about dynamical symmetries of the following stochastic Langevin equation, to be called diffusion-limited erosion (DLE) Langevin equation, which reads in momentum space [2]
The physical realisation of Equation (1) in terms of the DLE process makes it convenient to discuss the results in terms of the physics of the growth of interfaces [13,14,15], which can be viewed as a paradigmatic example of the emergence of non-equilibrium collective phenomena [16,17]

Summary

Introduction

Symmetries have since a long time played an important role in the analysis of physical systems. The physical realisation of Equation (1) in terms of the DLE process makes it convenient to discuss the results in terms of the physics of the growth of interfaces [13,14,15], which can be viewed as a paradigmatic example of the emergence of non-equilibrium collective phenomena [16,17] Such an interface can be described in terms of a time-space-dependent height profile h(t, r ). In contrast to the interface width w(t), which shows a logarithmic growth at d = d∗ = 1, logarithms cancel in the two-time correlator C and response R, up to additive logarithmic corrections to scaling This is well-known in the physical ageing at d = d∗ of simple magnets [22,23] or of the Arcetri model [20].

Local Conformal Invariance

Impossibility of a Local Meta-Conformal Invariance of the DLE Process

Riesz-Feller Fractional Derivative

Non-Local Meta-Conformal Generators

Ward Identities for Co-Variant Quasi-Primary n-Point Functions

Co-Variant Two-Time Correlators and Responses