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Discrete stress-energy tensor in the loop O(n) model

Dmitry Chelkak, Alexander Glazman, Stanislav Smirnov

TL;DR

This work constructs a discrete stress-energy tensor for the loop $O(n)$ model on the honeycomb/hexagonal lattice via infinitesimal non-planar lattice deformations and shows that, for $n\in[0,2]$, the observable partially satisfies discrete Cauchy–Riemann relations and converges to the continuum stress-energy tensor with Schwarzian covariance in the Ising case $n=1$. The authors develop a full fermionic framework on the honeycomb lattice, defining discrete fermionic observables and spinor observables, and express stress-energy correlations through Pfaffian and spinor structures, then transfer these to the continuum via conformal boundary-value problems. They prove rigorous convergence statements for Ising ($n=1$), including two-point and mixed correlators with the energy density and spin fields, and establish Schwarzian covariance and OPE-like singularities from the continuum boundary-value perspective. The results provide a bridge between lattice deformations, discrete holomorphicity, and continuum CFT data, with a clear pathway toward showing interfaces converge to SLE$_\kappa$ in the scaling limit for the loop $O(n)$ model. Overall, the paper advances a geometric, analytic program to realize the stress-energy tensor on lattice models and to connect discrete observables to their conformal field theory limits.

Abstract

We study the loop $O(n)$ model on the honeycomb lattice. By means of local non-planar deformations of the lattice, we construct a discrete stress-energy tensor. For $n\in [0,2]$, it gives a new observable satisfying a part of Cauchy-Riemann equations. We conjecture that it is approximately discrete-holomorphic and converges to the stress-energy tensor in the continuum, which is known to be a holomorphic function with the Schwarzian conformal covariance. In support of this conjecture, we prove it for the case of $n=1$ which corresponds to the Ising model. Moreover, in this case, we show that the correlations of the discrete stress-energy tensor with primary fields converge to their continuous counterparts, which satisfy the OPEs given by the CFT with central charge $c=1/2$. Proving the conjecture for other values of $n$ remains a challenge. In particular, this would open a road to establishing the convergence of the interface to the corresponding $\mathrm{SLE}_κ$ in the scaling limit.

Discrete stress-energy tensor in the loop O(n) model

TL;DR

This work constructs a discrete stress-energy tensor for the loop model on the honeycomb/hexagonal lattice via infinitesimal non-planar lattice deformations and shows that, for , the observable partially satisfies discrete Cauchy–Riemann relations and converges to the continuum stress-energy tensor with Schwarzian covariance in the Ising case . The authors develop a full fermionic framework on the honeycomb lattice, defining discrete fermionic observables and spinor observables, and express stress-energy correlations through Pfaffian and spinor structures, then transfer these to the continuum via conformal boundary-value problems. They prove rigorous convergence statements for Ising (), including two-point and mixed correlators with the energy density and spin fields, and establish Schwarzian covariance and OPE-like singularities from the continuum boundary-value perspective. The results provide a bridge between lattice deformations, discrete holomorphicity, and continuum CFT data, with a clear pathway toward showing interfaces converge to SLE in the scaling limit for the loop model. Overall, the paper advances a geometric, analytic program to realize the stress-energy tensor on lattice models and to connect discrete observables to their conformal field theory limits.

Abstract

We study the loop model on the honeycomb lattice. By means of local non-planar deformations of the lattice, we construct a discrete stress-energy tensor. For , it gives a new observable satisfying a part of Cauchy-Riemann equations. We conjecture that it is approximately discrete-holomorphic and converges to the stress-energy tensor in the continuum, which is known to be a holomorphic function with the Schwarzian conformal covariance. In support of this conjecture, we prove it for the case of which corresponds to the Ising model. Moreover, in this case, we show that the correlations of the discrete stress-energy tensor with primary fields converge to their continuous counterparts, which satisfy the OPEs given by the CFT with central charge . Proving the conjecture for other values of remains a challenge. In particular, this would open a road to establishing the convergence of the interface to the corresponding in the scaling limit.

Paper Structure

This paper contains 28 sections, 29 theorems, 201 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Loop $O(n)$ model: observables and relations
  3. Ising model: convergence statements
  4. Organisation of the paper.
  5. Combinatorial section
  6. Ising model on non-planar rhombic-triangular grids
  7. Stress-energy tensor as an infinitesimal non-planar deformation
  8. Graphical representation of the stress-energy tensor
  9. Fermions in the planar Ising model
  10. Definitions of discrete fermionic observables
  11. S-holomorphicity of discrete fermionic observables
  12. Correlation functions via fermions in discrete
  13. Stress-energy tensor expectations via fermionic observables
  14. Four-point fermionic observables
  15. Two-point expectations via four-point observables
  16. ...and 13 more sections

Key Result

Proposition 1.4

Let $a$, $b$, $c$, $d$, $e$, $f$ be the edges of a hexagon of $\Omega$ (listed in the counter-clockwise order). Then the following holds:

Figures (11)

  • Figure 1: Left: Simply-connected domain $\Omega$ (square) with faces of $\Omega_\delta$ ($\delta$-approximation of $\Omega$) in gray. The sets of boundary vertices $\partial\mathcal{V}_\delta(\Omega)$ (bold dots) and edges $\partial\mathcal{E}_\delta(\Omega)$ (dashed). The Dobrushin conditions are defined by two boundary vertices $b,b'$ (identified with edges of $\mathcal{E}_\delta(\Omega)\setminus \partial\mathcal{E}_\delta(\Omega)$ incident to them). In bold is a sample of a loop configuration in $\mathrm{Conf}^{\mathrm{loop}}_\Omega(b,b')$ with a path (called interface) linking $b$ and $b'$. Right: The (boundary) faces in $\partial\mathcal{F}_\delta(\Omega)$ are in gray. The spins $+/-$ define a spin configuration in $\mathrm{Conf}^{\mathrm{spin}}_\Omega(b,b')$. The domain walls between $+$ and $-$ form a loop configuration from the left figure; this defines a bijection between $\mathrm{Conf}^{\mathrm{loop}}_\Omega(b,b')$ and $\mathrm{Conf}^{\mathrm{spin}}_\Omega(b,b')$.
  • Figure 2: Configurations in a rhombus corresponding to the edge $e=(u,v)$ together with their images in $\mathrm{Conf}^{[e]}_\Omega(\mathfrak{b})$, their weights (at the top) and the coefficients from Definition \ref{['def:t-e-loop-o-n']} (at the bottom).
  • Figure 3: Possible local configurations inside the equilateral triangle and a rhombus with angles $\varphi$ and $\pi-\varphi$ and their weights.
  • Figure 4: Left: Yang--Baxter (star-triangle) transformation; $\varphi+\psi+\theta=2\pi$. Right: Star-segment transformation; $\theta=\frac{\pi}{3}+\varphi$ and $\varphi +\psi = \frac{\pi}{3}$.
  • Figure 5: Left: Hexagon $w:=abcdef$ with a midline $m = w^{\left[\rho \right]}$ (for $\rho=1$) and the six faces around $w$. Right: Deformation of the lattice at the midline $m$.
  • ...and 6 more figures

Theorems & Definitions (97)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1
  • Remark 1.7
  • Remark 1.8
  • Theorem 2
  • ...and 87 more