Random-bond Ising model and its dual in hyperbolic spaces

Benedikt Placke, Nikolas P. Breuckmann

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

1 Citation (Scopus)


We analyze the thermodynamic properties of the random-bond Ising model (RBIM) on closed hyperbolic surfaces using Monte Carlo and high-temperature series expansion techniques. We also analyze the dual-RBIM, that is, the model that in the absence of disorder is related to the RBIM via the Kramers-Wannier duality. Even on self-dual lattices this model is different from the RBIM, unlike in the Euclidean case. We explain this anomaly by a careful rederivation of the Kramers-Wannier duality. For the (dual-)RBIM, we compute the paramagnet-to-ferromagnet phase transition as a function of both temperature
T and the fraction of antiferromagnetic bonds p. We find that as temperature is decreased in the RBIM, the paramagnet gives way to either a ferromagnet or a spin-glass phase via a second-order transition compatible with mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first-order transition from the paramagnet to the ferromagnet both in the absence of disorder and along the Nishimori line. We study both transitions for a variety of hyperbolic tessellations and comment on the role of coordination number and curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds to the correctable phase of hyperbolic surface codes under independent bit- and phase-flip noise.
Original languageEnglish
Article number024125
Number of pages13
JournalPhysical Review E
Issue number2
Publication statusPublished - 16 Feb 2023

Bibliographical note

Funding Information:
We thank Ananda Roy for many helpful discussions in the early stages of this project. We also thank Leonid Pryadko for many helpfull comments and suggestions on this work. We thank Aleksander Kubica, Sounak Biswas, Rajiv Singh, and Roderich Moessner for fruitful discussions and also Philippe Suchsland, Dmitry L. Kovrizhin, and Peng Rao for helpful comments on the paper. B.P. acknowledges support by the Deutsche Forschungsgemeinschaft under Grant No. SFB 1143 (project-id 247310070) and the cluster of excellence ct.qmat (EXC 2147, project-id 390858490). N.P.B. acknowledges support through the EPSRC Prosperity Partnership in Quantum Software for Simulation and Modelling (EP/S005021/1).

Publisher Copyright:
© 2023 authors. Published by the American Physical Society.


  • cond-mat.stat-mech
  • cond-mat.dis-nn
  • quant-ph


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