Scale without conformal invariance in dipolar ferromagnets

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.