Infrared singularities of gapless Goldstone modes preclude magnetic long-range order at finite temperature in conventional two-dimensional systems. By studying the spin-$S$ Heisenberg model on regular tilings of the hyperbolic plane, we show that this obstruction is absent in negatively curved space. Using spin-wave theory, we find that the zero-energy collective modes required by symmetry carry vanishing local spectral weight and are separated from the thermodynamic bulk magnon continuum by a finite gap in the bulk local spectral density. As a result, local transverse correlations remain short-ranged, with a finite correlation length, despite the presence of Goldstone modes associated with the broken $\mathrm{SO}\left(3\right)$ spin-rotation symmetry. Stronger negative curvature is found to suppress quantum fluctuations in bulk thermodynamic quantities, pushing the ordered state toward “mean-field-like” behavior. We further estimate the ordering temperature from the thermal spin-wave correction to the ordered moment. These results establish hyperbolic geometry as a route to finite-temperature magnetic order that circumvents the Mermin-Wagner obstruction without breaking or modifying the continuous symmetry.

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• Goldstone bosons
• Magnetic order
• Magnetism
• Magnons
• Hyperbolic lattice
• Heisenberg model
• Holstein-Primakoff method
• Linear spin wave theory
• Nonlinear sigma model
• SO(N) symmetry
• Spin wave theory