Periodic boundary conditions on the pseudosphere

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.

30 pages, minor corrections, accepted to J. Phys. A

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• Two-point correlation functions of scaling fields in the Dirac theory on the Poincare disk
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• One-loop effective potential on hyperbolic manifolds
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• Symmetry and topology of hyperbolic Haldane models
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• Hyperbolic Lattice for Scalar Field Theory in AdS${}_3$
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• Phase transition of $q$-state clock models on heptagonal lattices
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• Extracting Hidden Hierarchies in 3D Distribution Networks
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