Speaker
Description
Infrared singularities of gapless Goldstone modes preclude magnetic long-range order at finite temperature in conventional two-dimensional systems. We show that this obstruction is avoided on lattices in negatively curved space by considering the spin-$S$ Heisenberg model on regular tilings of the hyperbolic plane. Using spin wave theory, we find that the global symmetry mode is separated from the bulk magnon continuum by a finite spectral gap, eliminating the infrared divergence that destroys order in Euclidean (zero-curvature) two-dimensional space. As a result, local transverse correlations remain short ranged, with a finite correlation length, despite the presence of Goldstone modes associated with the broken $\rm{SU}(2)$ spin-rotation symmetry. Stronger negative curvature is found to suppress quantum fluctuations in bulk thermodynamic quantities, pushing the ordered state toward ``mean-field-like'' behaviour. We further estimate the ordering temperature from the thermal spin wave depletion. These results identify hyperbolic geometry as a mechanism for stabilizing bulk magnetic order at finite temperature in the presence of an exact continuous symmetry.