The scaling limit of the energy correlations in non integrable Ising models

Abstract

We obtain an explicit expression for the multipoint energy correlations of a non solvable two-dimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength $\lambda$, in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in $\lambda$. In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.

75 pages, 11 figures

References in corpus (1)

• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model

Cited by in corpus (7)

• Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit
• Conformal invariance and Renormalization Group
• The Ising model on a cylinder: universal finite size corrections and diagonalized action
• Percolation transition for random forests in $d\ge 3$
• Universality in the 2d quasi-periodic Ising model and Harris-Luck irrelevance
• Non-trivial fixed point of a ${\psi }_d^4$ fermionic theory, II. Anomalous exponent and scaling operators
• The scaling limit of boundary spin correlations in non-integrable Ising models