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.
Versions
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BibTeX
@misc{hickey2026magnetic,
title = {{Magnetic long-range order at finite temperature in two-dimensional hyperbolic lattices}},
author = {Alexander Hickey and Joseph Maciejko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.27422},
eprint = {2606.27422},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27422},
}
RIS
TY - UNPB
TI - Magnetic long-range order at finite temperature in two-dimensional hyperbolic lattices
AU - Alexander Hickey
AU - Joseph Maciejko
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2606.27422
UR - https://arxiv.org/abs/2606.27422
AB - 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.
ER -