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Classification of (frustrated) 2D Ising models in genus 1

Béatrice de Tilière, Lucas Rey

Abstract

We prove a complete classification of all 2D Ising models, frustrated or not, whose underlying spectral curve has genus 1. As a specific case, we recover Baxter's $Z$-invariant Ising model, thus extending his class of models to \emph{real} coupling constants. We identify two additional families of models, both having \emph{non}-Harnack spectral curves. We show that in all cases the spectral curve is maximal. Moreover, each family undergoes an algebraic phase transition as the genus of the curve tends to 0, explaining the different behaviors observed in the physics literature. In our proof, we use properties of the spectral curve, and Fock's approach. This yields a natural framework for a further systematic study of the frustrated Ising model, in particular for proving local formulas. As an example of application, we prove a full classification of the frustrated Ising model on the triangular lattice.

Classification of (frustrated) 2D Ising models in genus 1

Abstract

We prove a complete classification of all 2D Ising models, frustrated or not, whose underlying spectral curve has genus 1. As a specific case, we recover Baxter's -invariant Ising model, thus extending his class of models to \emph{real} coupling constants. We identify two additional families of models, both having \emph{non}-Harnack spectral curves. We show that in all cases the spectral curve is maximal. Moreover, each family undergoes an algebraic phase transition as the genus of the curve tends to 0, explaining the different behaviors observed in the physics literature. In our proof, we use properties of the spectral curve, and Fock's approach. This yields a natural framework for a further systematic study of the frustrated Ising model, in particular for proving local formulas. As an example of application, we prove a full classification of the frustrated Ising model on the triangular lattice.
Paper Structure (61 sections, 39 theorems, 214 equations, 10 figures)

This paper contains 61 sections, 39 theorems, 214 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Historical perspective: phase transition in the frustrated Ising model
  3. Main contributions.
  4. Outline of the paper.
  5. Section \ref{['sec:background']}.
  6. Section \ref{['sec:real_dimers']}.
  7. Section \ref{['sec:classification']}.
  8. Section \ref{['sec:examples']}.
  9. Acknowledgments.
  10. Background
  11. The dimer model
  12. Definition and founding tools
  13. Definition.
  14. Kasteleyn matrix.
  15. The dimer model in the ${\mathbb Z}^2$-periodic case
  16. ...and 46 more sections

Key Result

Theorem 1

Consider an Ising model on an infinite, periodic, isoradial graph $G$ with periodic real coupling constants ${\varepsilon}\mathsf{J}$, and the corresponding Ising-dimer model on the graph ${\mathsf{G}}^{{\mathrm{Q}}}$. Suppose that the spectral curve $\mathscr{ C}$ has genus 1 and is generic; and co Conversely, consider Fock's dimer model on an infinite, minimal graph ${\mathsf{G}}^{{\mathrm{Q}}}

Figures (10)

  • Figure 1: Spectral curves of the (frustrated) Ising model on the triangular lattice for the three possible families, represented through their amoebas (first row), the log of the modulus of their real locus (second row); the torus and angle map used to parameterize the spectral curve (third row). Family I is an instance of an isotropic non-frustrated model; Family II is an instance of an anisotropic fully frustrated model; Family III is an instance of an isotropic fully frustrated model. Amoebas were plotted using the package PolynomialAmoebas of Sascha Timme.
  • Figure 2: Left: $G$ is the triangular lattice, and the dual $G^*$ is the hexagonal lattice. Center: corresponding Fisher graph ${G}^{{\mathrm{F}}}$. Right: corresponding bipartite graph ${\mathsf{G}}^{{\mathrm{Q}}}$.
  • Figure 3: Left: train-track angles $\alpha,\beta$ at an edge $\mathsf{w}\mathsf{b}$. Center: train-track angles $\alpha_1,\beta_1,\dots,\alpha_{|\mathsf{f}|/2},\beta_{|\mathsf{f}|/2}$ around a face $\mathsf{f}$ of degree $6$, with vertices $\mathsf{w}_1,\mathsf{b}_1,\dots,\mathsf{w}_{|\mathsf{f}|/2},\mathsf{b}_{|\mathsf{f}|/2}$ in counterclockwise order. Right (blue): computation of the discrete Abel map using $\mathbf{d}(\mathsf{f})$.
  • Figure 4: Comparison of the set of train-tracks $\mathscr{ T}$ (left), $\vec{\mathscr{ T}}_0$ (center) and $\vec{\mathscr{ T}}$ (right).
  • Figure 5: Left: notation around a face corresponding to a dual vertex $f$ of $G$. Right: notation around a face corresponding to a vertex $v$ of $G$.
  • ...and 5 more figures

Theorems & Definitions (104)

  • Theorem 1
  • Definition 2
  • Lemma 3: KOS
  • Remark 4
  • Remark 5
  • Definition 6
  • Remark 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 94 more