Integrability vs non-integrability: Hard hexagons and hard squares compared

TL;DR

This paper compares the integrable hard hexagon model with the non-integrable hard squares model by analyzing their partition function roots and transfer matrix eigenvalues across various boundary conditions and sizes.

Contribution

It provides a detailed comparison of the spectral properties and root distributions of the two models, revealing structural differences and conjectures at special points.

Findings

Hard squares roots lie in a bounded complex region with a universal line segment.

Density of roots on the line matches phase derivative and shows complex structure.

Eigenvalues at z=-1 all have unit modulus, with several conjectures proposed.

Abstract

In this paper we compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. We consider partition functions for toroidal, cylindrical, and free-free boundary conditions up to sizes $40\times40$ and transfer matrices up to 30 sites. For all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest (and equal) moduli and exhibits much greater structure than the corresponding density of hard hexagons. We also study the special point $z=-1$ of hard squares where all eigenvalues have unit modulus, and we give several conjectures for the value at $z=-1$ of the partition functions.