Numerical simulations are traditionally carried out on a flat background geometry, such as Cartesian lattices which posses no inherent curvature. In this example, however, we turn our attention away from the flat Euclidean space and towards geometries with constant curvature. Although these still represent the simplest curved manifolds, one immediately faces a number of crucial implications concerning spatial discretizations. Due to the inherent length scale, regular tilings are restricted to very specific polygons and corresponding dimensions. To be specific, scaling (i.e. shrinking or enlarging) a regular polygon in a hyperbolic/spherical space will also change its shape and the tessellation will in general no longer cover the space without voids or overlappings. Technically speaking, there is no concept of similarity in curved spaces. This severely limits the applicability of traditional finite-size scaling methods and essentially forbids the construction of periodic boundaries which appear to be indispensable when studying bulk systems due to the generically large boundary of hyperbolic spaces.
Figure 1. Spin model on a hyperbolic lattice. Heptagons coloured in red (blue) represent positive (negative) spin values.
Using MARQOV, we simulate an Ising-like model with real spin values on a regular tiling hyperbolic plane by congruent heptagons, as can be seen in Figure 1. Since three heptagons meet at each vertex, the tessellation shown here is denoted by {7,3}. In this particular representation, the hyperbolic plane is projected onto the so-called Poincaré disk. Note that the circumference of the disk corresponds to an infinite distance from the centre of the disk. One readily recognizes a number of magnetic domains of opposite spin orientation (red/blue areas). The domain walls are approximately given by arcs of Euclidean circles contained within the disk and orthogonal to its boundary. These arcs represent straight lines in the Poincare disk representation of the hyperbolic plane. Hence, analogously to the Euclidean case, we find the effect of a surface tension, resulting in approximately smooth domain boundaries.