Magnetic long-range order at finite temperature in two-dimensional hyperbolic lattices

Abstract: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 SO(3) 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.

Strongly Correlated Electrons (cond-mat.str-el); Quantum Physics (quant-ph)

arXiv:2606.27422 [cond-mat.str-el]

(or arXiv:2606.27422v2 [cond-mat.str-el] for this version)

https://doi.org/10.48550/arXiv.2606.27422 Focus to learn more arXiv-issued DOI via DataCite

Phys. Rev. B 114, 134430 (2026)

https://doi.org/10.1103/bbzf-dzrc Focus to learn more DOI(s) linking to related resources

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