Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions

Henkel, Malte

doi:10.3390/sym7042108

Cited by 15 publications

(27 citation statements)

References 105 publications

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“…The conformal galilean generator Y 0 = −t∂ r − γ ∈ cga(1) is distinct from the ordinary Galilei generator Y 1/2 = −t∂ r − Mr ∈ sch(1) of the Schrödinger algebra, as these imply distinct transformations of the scaling operators.3 For the Schrödinger group, an analogous construction shows that the two-point functions are response functions[56,60,61,62]. The scaling form (1.15) of the meta-conformal correlator is the same as the special case z = 1 for the conformally co-variant two-time response function G(t, r) [21, eq.…”

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confidence: 99%

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

Henkel¹,

Stoimenov²

2019

J. Stat. Mech.

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Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time-and space translations, time-space dilatations with dynamical exponent z = 1 and whose Lie algebras contain conformal Lie algebras as sub-algebras. They act as dynamical symmetries of the linear transport equation in d spatial dimensions. For d = 1 spatial dimensions, meta-conformal transformations constitute new representations of the conformal Lie algebras, while for d = 1 their algebraic structure is different. Infinite-dimensional Lie algebras of meta-conformal transformations are explicitly constructed for d = 1 and d = 2 and they are shown to be isomorphic to the direct sum of either two or three centre-less Virasoro algebras, respectively. The form of co-variant two-point correlators is derived. An application to the directed Glauber-Ising chain with spatially long-ranged initial conditions is described.1 In memoriam Vladimir Rittenberg "Whenever you have to do with a structure-endowed entity, try to determine its group of automorphisms. You can expect to gain a deep insight." H. Weyl, Symmetry, Princeton University Press (1952) 1 See [81] and refs. therein for the considerable recent interest into the case r ∈ R d with d > 2.

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confidence: 99%

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

Henkel¹,

Stoimenov²

2019

J. Stat. Mech.

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“…The constant 1/µ has the dimensions of a velocity. The Lie algebra X n , Y n n∈Z is isomorphic to the conformal Lie algebra [33], see table 2, where it is called meta-1 conformal invariance. If γ = µx, the generators (13) act as dynamical symmetries on the equation Sϕ = (−µ∂ t +∂ r )ϕ = 0.…”

Section: Local Conformal Invariancementioning

confidence: 99%

“…Example 5: Taking the limit µ → 0 in the meta-conformal representation (13) produces the generators Hence the cga is not a meta-conformal algebra, although z = 1. The co-variant two-point correlator can either be obtained from the generators (15), using techniques similar to those applied in the above example of meta-conformal invariance [31,33], or else by letting µ → 0 in (14). Both approaches give…”

Section: Local Conformal Invariancementioning

confidence: 99%

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

Henkel

2016

Symmetry

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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.

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“…Niederer [49] gave a classification of the dynamical symmetry of the diffusion equation with any timespace-dependent potential V = V (t, r). Generalised representations of the ageing algebra for Schrödinger operators with an arbitrary time-dependent potential have been found recently [54,55].…”

Section: Ageing-invariance Of a Generalised Diffusion Equationmentioning

confidence: 99%

From dynamical scaling to local scale-invariance: a tutorial

Henkel

2017

Eur. Phys. J. Spec. Top.

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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 nonequilibrium 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 Schrö-dinger-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.

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Dynamical Symmetries and Causality in Non-Equilibrium Phase Transitions

Cited by 15 publications

References 105 publications

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

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

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

From dynamical scaling to local scale-invariance: a tutorial

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