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Critical properties of the Ising model in hyperbolic space
Nikolas P Breuckmann1, Benedikt Placke2,3, Ananda Roy4
1Department of Physics & Astronomy, University College London, WC1E 6BT London, United Kingdom.
Physical Review. E
|March 15, 2020
Summary
The Ising model in hyperbolic space shows mean-field behavior, differing from flat space. This study investigates its thermodynamic properties in 2D and 3D hyperbolic spaces, confirming mean-field characteristics.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
- Geometric Physics
Background:
- The Ising model's behavior differs significantly in hyperbolic spaces due to negative curvature.
- A portion of spins are at the boundary, making boundary conditions crucial even in the thermodynamic limit.
Purpose of the Study:
- Investigate bulk thermodynamic properties of the Ising model in 2D and 3D hyperbolic spaces.
- Determine critical exponents and temperatures for various hyperbolic plane tilings.
- Compare results with the Bethe lattice and analyze 3D hyperbolic space behavior.
Main Methods:
- Monte Carlo simulations
- High- and low-temperature series expansions
- Periodic boundary conditions in hyperbolic space for bulk property extraction.
Main Results:
- Ising model in 2D hyperbolic space exhibits mean-field critical exponents and temperatures.
- Results for different hyperbolic tilings approach those of the Bethe lattice.
- Ising model in 3D hyperbolic space demonstrates mean-field behavior, contrasting with field theory predictions.
Conclusions:
- The Ising model's phase transition in 2D and 3D hyperbolic spaces is predominantly mean-field.
- Boundary effects are significant but do not alter the fundamental mean-field nature in these hyperbolic geometries.
- The study reconciles computational findings with theoretical predictions, highlighting the role of geometry.
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