The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent $z$ is described and the two distinct types of local scale invariance are presented. The special case $z=2$ and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.The generalization of dynamical scaling to local scale invariance is reviewed. Starting from a recapitulation of the phenomenology of ageing phenomena, the generalization of dynamical scaling to local scale transformation for any given dynamical exponent $z$ is described and the two distinct types of local scale invariance are presented. The special case $z=2$ and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.

arXiv:cond-mat/0307649Jul 25, 20030 citations

On logarithmic extensions of local scale-invariance

Malte Henkel

arXiv:1009.4139Nuclear Physics BDec 27, 201220 citations

Anisotropic Scaling and Generalized Conformal Invariance at Lifshitz Points

Michel Pleimling, Malte Henkel

The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.

arXiv:hep-th/0103194Physical Review LettersAug 31, 200149 citations

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Malte Henkel

arXiv:hep-th/0205256Nuclear Physics BOct 1, 2002195 citations

Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems

Malte Henkel

A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/N$, with $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}$ . Differential equations for the two-point function are derived and explicitly solved for all values of $N$ . Known special cases are conformal invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ ) and Schr\"odinger invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ ). For $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ and $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6$ , the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/N$, with $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}$. Differential equations for the two-point function are derived and explicitly solved for all values of $N$. Known special cases are conformal invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$) and Schr\"odinger invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$). For $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ and $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6$, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.

arXiv:cond-mat/9610174Physical Review LettersMar 10, 1997118 citations

z

is described and the two distinct types of local scale invariance are presented. The special case

z=2

and the associated Ward identity of Schrödinger invariance is treated. Local scale invariance predicts the form of the two-point functions. Existing confirmations of these predictions for (I) the Lifshitz points in spin systems with competing interactions such as the ANNNI model and (II) non-equilibrium ageing phenomena as occur in the kinetic Ising model with Glauber dynamics are described.

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}

. Differential equations for the two-point function are derived and explicitly solved for all values of

N

. Known special cases are conformal invariance (

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2

) and Schr\"odinger invariance (

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1

). For

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4

and

N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6

, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next-nearest-neighbor spherical model at its Lifshitz points of first and second order.