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Scale without Conformal Invariance in Dipolar Ferromagnets

Aleix Gimenez-Grau, Yu Nakayama, Slava Rychkov

Abstract

We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding RG fixed point - dipolar fixed point - was first studied in 1973 by Aharony and Fisher. 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 non-renormalization 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.

Scale without Conformal Invariance in Dipolar Ferromagnets

Abstract

We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding RG fixed point - dipolar fixed point - was first studied in 1973 by Aharony and Fisher. 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 non-renormalization 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.
Paper Structure (34 sections, 165 equations, 2 figures, 1 table)

This paper contains 34 sections, 165 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dipolar fixed point: Aharony-Fisher theory
  3. Manifestly local description
  4. Phenomenology and experiments
  5. Microscopic derivations of effective theory
  6. Experimental evidence of the dipolar fixed point behavior
  7. Scale invariance without conformal invariance
  8. Two-point function argument
  9. Stress tensor argument
  10. Virial current dimension and the shift symmetry
  11. Other consequences of shift symmetry
  12. Other interacting models with scale without conformal invariance
  13. Landau-gauge massless QED in d=4-eps
  14. Landau-gauge Banks-Zaks fixed point in d=4
  15. Fixed points not in the Landau gauge
  16. ...and 19 more sections

Figures (2)

  • Figure 1: The RG flow diagram with four fixed points: $G$ - Gaussian, $G'$ - Gaussian dipolar, $H$ - Heisenberg, $D$ - dipolar. Materials with $t_d\sim 1$, like EuS and EuO, correspond to trajectories like 1 which flow straight to $D$. Materials with $t_d\ll 1$, like Fe and Ni, are supposed to correspond to trajectories like 2 which first approach $H$ and then flow to $D$. The shown trajectories correspond to the exact critical temperature $t=0$ (critical flow). For $t$ slightly different from 0, the RG flow initially tracks the critical trajectory, and then deviates from it. Depending on the value of $t$, this deviation may happen, for type 2 trajectories, when the flow is near $H$ or near $D$. This implies the crossover behavior mentioned in the main text.
  • Figure 2: Fcc lattice structure of EuX, X=S,O.

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Remark 4.6
  • Remark C.1