Periodic boundary conditions on the pseudosphere

François Sausset (1) , Gilles Tarjus (1)

1 LPTMC - Laboratoire de Physique Théorique de la Matière Condensée

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.

Mots clés: statistical physics; pseudosphere; hyperbolic plane; computer simulation; periodic boundary conditions

Domaines: Mécanique statistique [cond-mat.stat-mech]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

François Sausset, Gilles Tarjus. Periodic boundary conditions on the pseudosphere. Journal of Physics A: Mathematical and Theoretical, 2007, 40, pp.12873-12899. ⟨10.1088/1751-8113/40/43/004⟩. ⟨hal-00136101v3⟩