Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Complex Systems

Background:

• The Ising model is a fundamental model in statistical mechanics used to study magnetism and phase transitions.
• Understanding the behavior of physical systems on curved manifolds is crucial for various fields.
• Hyperbolic geometry offers a unique framework to explore non-Euclidean physics.

Purpose of the Study:

• To investigate the phase transition of the Ising model on hyperbolic two-dimensional lattices.
• To develop and apply a generalized corner transfer matrix renormalization group method for hyperbolic systems.
• To analyze the impact of negative curvature on critical phenomena and correlation functions.

Main Methods:

• Generalization of the corner transfer matrix renormalization group method.
• Recursive construction of asymmetric transfer matrices for hyperbolic lattices.
• Precise analysis of thermodynamic functions to study phase transitions.

Main Results:

• The Ising model on hyperbolic lattices exhibits mean-field universality.
• Thermodynamic functions precisely capture the phase transition behavior.
• Correlation functions and density-matrix spectra show exponential decay, even at the transition point.
• Absence of finite correlation length in the limit of infinite negative Gaussian curvature confirmed.

Conclusions:

• Hyperbolic geometry fundamentally alters the nature of criticality in the Ising model.
• The generalized corner transfer matrix method is effective for studying systems on hyperbolic lattices.
• The findings provide insights into the role of geometry in statistical mechanics and critical phenomena.