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

Conformal Invariance and Critical Phenomena

Malte Henkel

Springer eBooksJan 1, 1999387 citations

Scale without conformal invariance in dipolar ferromagnets

Aleix Gimenez-Grau, Yu Nakayama, Slava Rychkov

We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding renormalization group fixed point, dipolar fixed point, was first studied in 1973 by Aharony and Fisher [Phys. Rev. Lett. 30, 559 (1973]. It is distinct from the Heisenberg fixed point, although the critical exponents are close. On the theoretical side, we discuss scale invariance without conformal invariance realized by this fixed point. We elucidate the nonrenormalization of the virial current due to a shift symmetry, and show that the same mechanism is at work in all other known local fixed points which are scale but not conformal invariant. On the phenomenological side, we discuss the relative strength of dipolar and short-range interactions. In some materials, like the europium compounds, dipolar interactions are strong, and the critical behavior is dipolar. In others, like Fe or Ni, dipolar interactions are weaker, and the Heisenberg critical behavior in a range of temperatures is followed by the dipolar behavior closer to the critical point. Some of these effects have been seen experimentally.We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding renormalization group fixed point, dipolar fixed point, was first studied in 1973 by Aharony and Fisher [Phys. Rev. Lett. 30, 559 (1973]. It is distinct from the Heisenberg fixed point, although the critical exponents are close. On the theoretical side, we discuss scale invariance without conformal invariance realized by this fixed point. We elucidate the nonrenormalization of the virial current due to a shift symmetry, and show that the same mechanism is at work in all other known local fixed points which are scale but not conformal invariant. On the phenomenological side, we discuss the relative strength of dipolar and short-range interactions. In some materials, like the europium compounds, dipolar interactions are strong, and the critical behavior is dipolar. In others, like Fe or Ni, dipolar interactions are weaker, and the Heisenberg critical behavior in a range of temperatures is followed by the dipolar behavior closer to the critical point. Some of these effects have been seen experimentally.

arXiv:2309.02514Physical review. B./Physical review. BJul 16, 202411 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.