From dynamical scaling to local scale-invariance: a tutorial

Abstract

Dynamical scaling arises naturally in various many-body systems far from equilibrium. After a short historical overview, the elements of possible extensions of dynamical scaling to a local scale-invariance will be introduced. Schrödinger-invariance, the most simple example of local scale-invariance, will be introduced as a dynamical symmetry in the Edwards-Wilkinson universality class of interface growth. The Lie algebra construction, its representations and the Bargman superselection rules will be combined with non-equilibrium Janssen-de Dominicis field-theory to produce explicit predictions for responses and correlators, which can be compared to the results of explicit model studies. At the next level, the study of non-stationary states requires to go over, from Schrödinger-invariance, to ageing-invariance. The ageing algebra admits new representations, which acts as dynamical symmetries on more general equations, and imply that each non-equilibrium scaling operator is characterised by two distinct, independent scaling dimensions. Tests of ageing-invariance are described, in the Glauber-Ising and spherical models of a phase-ordering ferromagnet and the Arcetri model of interface growth.

1+ 23 pages, 2 figures, final form

References in corpus (16)

• Dynamics of glassy systems
• Local scale-invariance and ageing in noisy systems
• Universal correlators and distributions as experimental signatures of 2+1 Kardar-Parisi-Zhang growth
• Two-time autocorrelation function in phase-ordering kinetics from local scale-invariance
• Tutorial on Scale and Conformal Symmetries in Diverse Dimensions
• Scaling of the linear response in simple ageing systems without disorder
• Scale invariance implies conformal invariance for the three-dimensional Ising model
• Ageing phenomena without detailed balance: the contact process
• Zipf's Law from Scale-free Geometry
• Out-of-equilibrium relaxation of the Edwards-Wilkinson elastic line
• Universality of (2+1)-dimensional restricted solid-on-solid models
• Spherical model of growing interfaces
• Dynamical symmetries and causality in non-equilibrium phase transitions
• On non-local representations of the ageing algebra
• Local Scale-Invariance of the 2+1 dimensional Kardar-Parisi-Zhang model
• Conformal quantum mechanics and Fick-Jacobs equation

Cited by in corpus (12)

• Anomalous diffusion: A basic mechanism for the evolution of inhomogeneous systems
• Aging in the long-range Ising model
• The Kardar-Parisi-Zhang exponents for the $2+1$ dimensions
• Schrödinger symmetry: a historical review
• Non-local meta-conformal invariance, diffusion-limited erosion and the XXZ chain
• Dynamical universality of the contact process
• Meta-Schrödinger invariance
• Infinite-dimensional meta-conformal Lie algebras in one and two spatial dimensions
• Dynamical symmetries in the non-equilibrium dynamics of the directed spherical model
• Aging following a zero-temperature quench in the $d=3$ Ising model
• Meta-conformal algebras in $d$ spatial dimensions
• Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance