Potts models with magnetic field: arithmetic, geometry, and computation

TL;DR

This paper interprets Potts models with magnetic fields using sheaf theory, explores how magnetic fields influence polynomial countability of partition functions, and examines the computational complexity of evaluating these functions.

Contribution

It introduces a sheaf theoretic framework for Potts models with magnetic fields and analyzes their algebraic and computational properties.

Findings

Magnetic fields can change the polynomial countability of the partition function hypersurfaces.

Recursive formulas for Grothendieck classes remain valid with magnetic fields, but specific examples differ.

The complexity of evaluating the polynomial varies, with both tractable and NP-hard cases identified.

Abstract

We give a sheaf theoretic interpretation of Potts models with external magnetic field, in terms of constructible sheaves and their Euler characteristics. We show that the polynomial countability question for the hypersurfaces defined by the vanishing of the partition function is affected by changes in the magnetic field: elementary examples suffice to see non-polynomially countable cases that become polynomially countable after a perturbation of the magnetic field. The same recursive formula for the Grothendieck classes, under edge-doubling operations, holds as in the case without magnetic field, but the closed formulae for specific examples like banana graphs differ in the presence of magnetic field. We give examples of computation of the Euler characteristic with compact support, for the set of real zeros, and find a similar exponential growth with the size of the graph. This can be viewed as a measure of topological and algorithmic complexity. We also consider the computational complexity question for evaluations of the polynomial, and show both tractable and NP-hard examples, using dynamic programming.