Non-Local Meta-Conformal Invariance, Diffusion-Limited Erosion 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. 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.

Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions

“…5 They differ also from Cardy's proposal (t, r) → (t ′ , r ′ ) = (b(r) z t, b(r)r) [21]. 6 The same algebra of dynamical symmetries also arises for diffusion-limited erosion in 1D [63,64].…”

“…If that is so, then the two-time correlator C n (τ + s, s), see eq. (5.5), will be of the same form as in (5.7), with a prefactor C (s) which might still depend on the waiting time s. While we gave here an example of 1D meta-conformal invariance, we point out that the Lie algebra of 2D meta-conformal transformations is isomorphic to the dynamical symmetry [64] of the spatially non-local stochastic process of 1D diffusion-limited erosion [72] or the terrace-step-kink model [83,71].…”

Meta-conformal algebras in $d$ spatial dimensions

“…The algebra (4.17) is identical to the non-local meta-conformal algebra found recently for the diffusion-limited erosion process in one spatial dimension [38,39]. Therefore, we define the new generators…”

“…It will be possible to adapt constructions from local scale-invariance [34,40] such that this Lie group can also describe the relaxation of single-time correlators, which is work in progress. Remarkably, the Lie algebra of dynamical symmetries in 2D is isomorphic to the one of the spatially non-local stochastic process of 1D diffusion-limited erosion [39].…”

“…Here, we shall be interested in Lie algebras of time-space transformations, which as Lie algebras are still isomorphic to the ortho-conformal Lie algebra or at least contain it as a sub-algebra, but which no need to be angle-preserving. Such transformations will be called meta-conformal transformations [38,39]. To make this idea explicit, consider the infinitesimal generators, acting on a two-dimensional time-space with points (t, r) ∈ R 2 [32]…”