The signed loop approach to the Ising model: foundations and critical point

Abstract

The signed loop method is a beautiful way to rigorously study the two-dimensional Ising model with no external field. In this paper, we explore the foundations of the method, including details that have so far been neglected or overlooked in the literature. We demonstrate how the method can be applied to the Ising model on the square lattice to derive explicit formal expressions for the free energy density and two-point functions in terms of sums over loops, valid all the way up to the self-dual point. As a corollary, it follows that the self-dual point is critical both for the behaviour of the free energy density, and for the decay of the two-point functions.

38 pages, 7 figures, with an improved Introduction. The final publication is available at link.springer.com

Cited by in corpus (8)

• The planar Ising model and total positivity
• Phase transition free regions in the Ising model via the Kac-Ward operator
• The fermionic observable in the Ising model and the inverse Kac-Ward operator
• Ising Model Observables and Non-Backtracking Walks
• Derivation of the free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals
• A Statistical Model of Current Loops and Magnetic Monopoles
• Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function
• Kac-Ward solution of the 2D classical and 1D quantum Ising models