From dynamical scaling to local scale-invariance: a tutorial

• Published in: The European physical journal. ST, Special topics Special topics, 2017-04, Vol.226 (4), p.605-625
• Main Author: Henkel, Malte
• Format: Article
• Language: English
• Subjects: Aging; Algebra; Atomic; Classical and Continuum Physics; Condensed Matter Physics; Equilibrium; Ferromagnetism; Field theory; Invariance; Ising model; Lie groups; Materials Science; Measurement Science and Instrumentation; Molecular; Operators (mathematics); Optical and Plasma Physics; Phase ordering; Physics; Physics and Astronomy; Recent Advances in Phase Transitions and Critical Phenomena; Representations; Review; Scaling
• ISSN: 1951-6355; 1951-6401
• DOI: 10.1140/epjst/e2016-60336-5

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.