[Paper Review] The energy density in the planar Ising model

Clément Hongler, Stanislav Smirnov|arXiv (Cornell University)|Aug 16, 2010

Theoretical and Computational Physics11 references5 citations

TL;DR

This paper establishes the exact scaling limit of the energy density field in the critical planar Ising model on bounded simply connected domains with + and free boundary conditions. Using discrete complex analysis and fermionic spinors, it derives a conformally covariant formula for the one-point function in terms of the hyperbolic metric, confirming conformal field theory predictions with full precision and resolving the lattice-dependent constant $\frac{1}{2\pi}$.

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.

Motivation & Objective

• To rigorously compute the scaling limit of the energy density one-point function in the critical planar Ising model on bounded domains.
• To resolve the long-standing question of the lattice-specific prefactor in the conformal field theory prediction for the energy density.
• To establish full conformal invariance (beyond Möbius invariance) of the energy density correlation function.
• To provide a new method based on discrete complex analysis and fermionic spinors, avoiding SLE or traditional integrability techniques.
• To extend the applicability of discrete fermionic spinors to boundary value problems in general simply connected domains.

Proposed method

• The energy density is expressed in terms of a discrete fermionic spinor satisfying a discrete Riemann boundary value problem.
• The discrete spinor is constructed using the coupling function from Kenyon's dimer model framework, adapted to the Ising model.
• Convergence of the discrete spinor to its continuous limit is proven via discrete holomorphicity, singularity analysis, and boundary value control.
• The scaling limit of the energy density is derived by projecting the discrete spinor onto medial lattice edges, yielding a conformally covariant expression.
• The limit is shown to be proportional to the hyperbolic metric element $ l_\Omega(a) $, with the precise prefactor $ \frac{1}{2\pi} $.
• The method leverages discrete complex analysis tools to bypass the need for explicit summation formulas, enabling a short and exact expression in the continuum limit.

Experimental results

Research questions

• RQ1 What is the exact scaling limit of the energy density one-point function in the critical Ising model on a bounded simply connected domain with + boundary conditions?
• RQ2 How does the energy density scale with lattice spacing $ \delta $, and what is the precise coefficient in front of the hyperbolic metric?
• RQ3 Can the full conformal invariance of the energy density field be rigorously established beyond Möbius invariance?
• RQ4 What is the role of the discrete fermionic spinor in capturing the energy density in the scaling limit?
• RQ5 How does the boundary condition (free vs. +) affect the energy density at the microscopic level?

Key findings

• The energy density one-point function scales as $ \pm \frac{l_\Omega(a)}{2\pi} \delta + o(\delta) $ for + and free boundary conditions, respectively, with $ l_\Omega(a) $ the hyperbolic metric element.
• The scaling limit of the energy field is conformally covariant and matches the prediction of conformal field theory with exact prefactor $ \frac{1}{2\pi} $.
• The discrete fermionic spinor converges to a continuous holomorphic function in the scaling limit, enabling exact computation of the energy density.
• The method confirms the validity of the mirror image technique from CFT with full mathematical rigor and lattice precision.
• The result establishes full conformal invariance of the energy density field, not just Möbius invariance, in the critical Ising model.
• The approach provides a general framework applicable to any simply connected planar domain, not restricted to the full plane or periodic settings.