The energy density in the planar Ising model

Abstract

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

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Correspondence to Clément Hongler.

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Hongler, C., Smirnov, S. The energy density in the planar Ising model. Acta Math 211, 191–225 (2013). https://doi.org/10.1007/s11511-013-0102-1

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Keywords

  • Ising model
  • energy density
  • discrete analytic function
  • fermions
  • conformal invariance
  • hyperbolic geometry
  • conformal field theory