Ising Model on the Fibonacci Sphere

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.