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Particles hopping on a two-dimensional hyperbolic lattice feature
unconventional energy spectra and wave functions that provide a largely
uncharted platform for topological phases of matter beyond the Euclidean
paradigm. Using real-space topological markers as well as Chern numbers defined
in the higher-dimensional momentum space of hyperbolic band theory, we
construct and investigate hyperbolic Haldane models, which are generalizations
of Haldane's honeycomb-lattice model to various hyperbolic lattices. We present
a general framework to characterize point-group symmetries in hyperbolic
tight-binding models, and use this framework to constrain the multiple first
and second Chern numbers in momentum space. We observe several topological gaps
characterized by first Chern numbers of value $1$ and $2$. The momentum-space
Chern numbers respect the predicted symmetry constraints and agree with
real-space topological markers, indicating a direct connection to observables
such as the number of chiral edge modes. With our large repertoire of models,
we further demonstrate that the topology of hyperbolic Haldane models is
trivialized for lattices with strong negative curvature.
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