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We studied the random-bond Ising model (RBIM) on hyperbolic surfaces. The dual-RBIM shows a first-order phase transition, relevant for correcting errors in hyperbolic surface codes.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Geometric Analysis

Background:

  • The random-bond Ising model (RBIM) is crucial for understanding magnetism with quenched disorder.
  • Hyperbolic surfaces offer unique geometric properties distinct from Euclidean lattices.
  • Kramers-Wannier duality provides a powerful tool for analyzing phase transitions.

Purpose of the Study:

  • To investigate the thermodynamic properties and phase transitions of the RBIM and its dual on hyperbolic surfaces.
  • To clarify the behavior of the dual-RBIM, especially its distinction from the RBIM on self-dual lattices.
  • To explore the connection between the ferromagnetic phase of the dual-RBIM and error correction in hyperbolic surface codes.

Main Methods:

  • Monte Carlo simulations were employed to analyze thermodynamic properties.
  • High-temperature series expansion techniques were utilized for detailed analysis.
  • Kramers-Wannier duality was rederived and applied to understand the models' behavior.

Main Results:

  • The RBIM transitions from paramagnet to ferromagnet or spin-glass via a second-order transition.
  • The dual-RBIM exhibits a strongly first-order paramagnetic-to-ferromagnetic transition.
  • The ferromagnetic phase extent in the dual-RBIM aligns with the correctable phase of hyperbolic surface codes.

Conclusions:

  • The geometric properties of hyperbolic surfaces significantly influence the phase transitions of the RBIM and its dual.
  • The dual-RBIM's behavior, particularly its first-order transition, has implications for quantum error correction codes.
  • This research bridges statistical mechanics on curved manifolds with applications in quantum information theory.