The energy density in the planar Ising model

TL;DR

This paper derives an exact formula for the energy density in the critical planar Ising model using discrete complex analysis, confirming physics predictions and providing higher precision in the scaling limit.

Contribution

It introduces a novel approach linking energy density to fermionic observables and computes its scaling limit explicitly in terms of hyperbolic geometry.

Findings

Exact formula for the energy field one-point function in the scaling limit.

Confirmation of physics-based predictions with higher precision.

Application of discrete complex analysis to critical Ising model energy density.

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 fermionic observable 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 hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.