Critical properties of the Ising model in hyperbolic space

Nikolas P. Breuckmann, Benedikt Placke, Ananda Roy

Research output: Contribution to journalArticle (Academic Journal)peer-review

11 Citations (Scopus)
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The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two and three dimensional hyperbolic spaces using Monte Carlo and high and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the 'asymptotic' limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.
Original languageEnglish
Article number022124
Number of pages13
JournalPhysical Review E
Issue number2
Publication statusPublished - 20 Feb 2020

Bibliographical note

Funding Information:
We thank Leonid Pryadko for encouraging discussions. N.P.B. is supported by the UCLQ Fellowship. A.R. acknowledges the support of the Alexander von Humboldt foundation.

Publisher Copyright:
© 2020 American Physical Society.


  • cond-mat.stat-mech


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