Dualities for Ising networks

TL;DR

This paper explores the deep connection between planar Ising networks and the positive orthogonal Grassmannian, introducing recursive methods for calculating correlators and revealing fractal structures and efficient algorithms.

Contribution

It establishes a microscopic construction linking Ising networks to the Grassmannian and introduces two recursive techniques for correlator computation, including an RG approach and an efficient amalgamation method.

Findings

Duality moves generate networks within the same Grassmannian cell.

Fractal lattices emerge where recursion formulas become exact RG equations.

Correlator computation complexity scales logarithmically with the number of spins.

Abstract

In this note, we study the equivalence between planar Ising networks and cells in the positive orthogonal Grassmannian. We present a microscopic construction based on amalgamation, which establishes the correspondence for any planar Ising network. The equivalence allows us to introduce two recursive methods for computing correlators of Ising networks. The first based on duality moves, which generate networks belonging to the same cell in the Grassmannian. This leads to fractal lattices where the recursion formulas become the exact RG equations of the effective couplings. For the second, we use amalgamation where each iteration doubles the size of the seed lattice. This leads to an efficient way of computing the correlator where the complexity scales logarithmically with respect to the number of spin sites.