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Spectral transform for the Ising model

Terrence George

Abstract

We prove a correspondence between Ising models in a torus and the algebro-geometric data of a Harnack curve with a certain symmetry and a point in the real part of its Prym variety, extending the correspondence between dimer models and Harnack curves and their Jacobians due to Kenyon and Okounkov.

Spectral transform for the Ising model

Abstract

We prove a correspondence between Ising models in a torus and the algebro-geometric data of a Harnack curve with a certain symmetry and a point in the real part of its Prym variety, extending the correspondence between dimer models and Harnack curves and their Jacobians due to Kenyon and Okounkov.
Paper Structure (14 sections, 11 theorems, 54 equations, 10 figures)

This paper contains 14 sections, 11 theorems, 54 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background on the Ising model
  3. Zig-zag paths and the Newton polygon
  4. Kramers--Wannier duality
  5. Y-$\Delta$ move
  6. Background on the dimer model
  7. Dimer models in $\mathbb T$
  8. Zig-zag paths
  9. Moves
  10. The mapping from Ising models to dimer models
  11. The spectral transform
  12. Color change
  13. A characterization of $\mathcal{I}_G(\mathbb R_{>0}) \subset \mathcal{X}_{G^\square}(\mathbb R_{>0})$
  14. Ising spectral data

Key Result

Theorem 1.1

A dimer model $(G^\square,[\mathrm{wt}])$ arises from an Ising model if and only if $[\overline \mathrm{wt}] = \mu([\mathrm{wt}])$.

Figures (10)

  • Figure 1: (a) An Ising model $(G,J)$ in a torus $\mathbb T$, (b) the mapping from $(G,J)$ to $(G^\square,[\mathrm{wt}^\square])$ and (c) the dimer model $(G^\square,[\mathrm{wt}^\square])$.
  • Figure 2: (a) The square move and (b) the contraction-uncontraction move. Using gauge equivalence, we can assume that the original weight is as shown on the right. Then the new weight is as shown on the left.
  • Figure 3: (a) Two of the zig-zag paths of the Ising model $(G,J)$ in Figure \ref{['fig:isingintro']}(a) and (b) its Newton polygon.
  • Figure 4: The Ising Y-$\Delta$ move.
  • Figure 5: The local pairing at a trivalent vertex $\rm v$ (which may be black or white) is defined by $\epsilon_{\rm v}(\gamma,\delta)=\frac{1}{2}$ and bilinearity and antisymmetry.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Theorem 2.2: GK
  • Theorem 3.1: KShKOS
  • Theorem 3.2: KO
  • Remark 3.3
  • Example 3.4
  • Theorem 3.5: KO
  • Theorem 3.6
  • ...and 8 more