Random-bond Ising model and its dual in hyperbolic spaces.

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.