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Scale invariance implies conformal invariance for the three-dimensional Ising model.

Bertrand Delamotte1, Matthieu Tissier1, Nicolás Wschebor1,2

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Scale invariance implies conformal invariance if a specific operator is absent. This condition is met in the Ising universality class across all dimensions, including 3D.

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Area of Science:

  • Statistical Mechanics
  • Quantum Field Theory
  • Condensed Matter Physics

Background:

  • Scale invariance is a key concept in critical phenomena.
  • Conformal invariance is a stronger symmetry often present at critical points.
  • The relationship between scale and conformal invariance is a fundamental question.

Purpose of the Study:

  • To establish the conditions under which scale invariance implies conformal invariance.
  • To investigate this implication for the Ising universality class.
  • To confirm the result for the three-dimensional Ising model.

Main Methods:

  • Utilizing the Wilson renormalization group.
  • Applying the Lebowitz inequalities.
  • Analyzing operator scaling dimensions.

Main Results:

  • Demonstrating that scale invariance implies conformal invariance in the absence of an integrated vector operator of scaling dimension -1.
  • Proving this condition holds for the Ising universality class in all dimensions.
  • Confirming the implication for the 3D Ising model.

Conclusions:

  • The absence of a specific operator is a sufficient condition for scale invariance to imply conformal invariance.
  • The Ising universality class satisfies this condition universally.
  • This provides a rigorous proof for the 3D Ising model.