Periodic boundary conditions on the pseudosphere

TL;DR

This paper develops a mathematical framework for implementing periodic boundary conditions on the hyperbolic plane, enabling simulations of physical systems with negative curvature.

Contribution

It introduces the necessary mathematical tools and classification for periodic boundary conditions on the pseudosphere, extending Euclidean concepts to hyperbolic geometry.

Findings

Framework for periodic boundary conditions on the pseudosphere

Application examples in particle dynamics and spin systems

Potential for studying bulk properties in hyperbolic geometries

Abstract

We provide a framework to build periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the needed mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems and we illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.