Ising Model on the Fibonacci Sphere

TL;DR

This paper investigates the Ising model on a Fibonacci-triangulated sphere, revealing how the unique lattice connectivity influences phase transitions and exhibits singular behavior as the sphere's size increases.

Contribution

It introduces a novel Fibonacci-based triangulation method for modeling the Ising system on curved surfaces and analyzes its impact on phase transition properties.

Findings

Identifies a critical temperature T_c ≈ 3.33 J for the phase transition.

Discovers a series of connectivity-induced singular transitions with increasing sphere size.

Shows the phase transition temperature is slightly lower than in planar lattices due to curvature effects.

Abstract

We formulate the ferromagnetic Ising model on a two-dimensional sphere using the Delaunay triangulation of the Fibonacci covering. The Fibonacci approach generates a uniform isotropic covering of the sphere with approximately equal-area triangles, thus potentially supporting a smooth thermodynamic limit. In the absence of a magnetic field, the model exhibits a spontaneous magnetization phase transition at a critical temperature that depends on the connectivity properties of the underlying lattice. While in the standard triangular lattice, every site is connected to 6 neighboring sites, the triangulated Fibonacci lattice of the curved surface contains a substantial density of the 5- and 7-vertices. As the number of sites in the Fibonacci sphere increases, the triangular cover of the sphere experiences a series of singular transitions that reflect a sudden change in its connectivity properties. These changes substantially influence the statistical features of the system leading to a series of first-order-like discontinuities as the radius of the sphere increases. We found that the Ising model on a uniform, Fibonacci-triangulated sphere in a large-radius limit possesses the phase transition at the critical temperature $T_c \simeq 3.33(3) J$, which is slightly lower than the thermodynamic result for an equilaterally triangulated planar lattice. This mismatch is a memory effect: the planar Fibonacci lattice remembers its origin from the curved space.