A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in classical many-body systems after a quench from an initially disordered system to a temperature <math altimg='si1.svg'><mi>T</mi><mo>&#8804;</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math>, at or below the critical temperature <math altimg='si2.svg'><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math>. Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents <math altimg='si3.svg'><msub><mrow><mi>&#955;</mi></mrow><mrow><mi>C</mi></mrow></msub><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><msub><mrow><mi>&#955;</mi></mrow><mrow><mi>R</mi></mrow></msub></math>, the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments <math altimg='si4.svg'><mi>y</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mi>t</mi><mo stretchy='false'>/</mo><mi>s</mi><mo>&#8811;</mo><mn>1</mn></math> to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent &#920; is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures <math altimg='si1.svg'><mi>T</mi><mo>&#8804;</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math>. Relevance to experiments is also discussed.

France

Portugal

Malte Henkel

Physical ageing from generalised time-translation-invariance

The dynamical scaling and ageing in the relaxational dynamics of the quenched directed spherical model is analysed. The exact two-time correlation and response functions display new regimes of ballistic or anisotropic ballistic scaling, at larger distances than probed in the usual regime of diffusive scaling. The role of long-ranged initial correlations on the existence of these scaling regimes is clarified. Their dynamical symmetries are described in terms of extensions of the Schrödinger algebra appropriate to non-equilibrium dynamics in that the anisotropic ballistic scaling regime can be interpreted in terms of meta-Schrödinger invariance while the regime of isotropic ballistic scaling is meta-conformally invariant.

Bulgaria

Stoimen Stoimenov

Dynamical symmetries in the non-equilibrium dynamics of the directed spherical model

The Meta-Schrödinger algebra arises as the dynamical symmetry in transport processes which are ballistic in a chosen ‘parallel’ direction and diffusive in all other ‘transverse’ directions. The time-space transformations of this Lie algebra and its infinite-dimensional extension, the meta-Schrödinger-Virasoro algebra, are constructed. We also find the representations suitable for non-stationary systems by proposing a generalised form of the generator of time-translations. Co-variant two-point functions of quasi-primary scaling operators are derived for both the stationary and the non-stationary cases.

Meta-Schrödinger invariance

Logarithmic representations of the conformal Galilean algebra (CGA) and the Exotic Conformal Galilean algebra ( ecga ) are constructed. This can be achieved by non-decomposable representations of the scaling dimensions or the rapidity indices, specific to conformal Galilean algebras. Logarithmic representations of the non-exotic CGA lead to the expected constraints on scaling dimensions and rapidities and also on the logarithmic contributions in the co-variant two-point functions. On the other hand, the ecga admits several distinct situations which are distinguished by different sets of constraints and distinct scaling forms of the two-point functions. Two distinct realisations for the spatial rotations are identified as well. This is the first concrete example of a reducible, but non-decomposable representation, without logarithmic terms. Such cases had been anticipated before.