Exact critical-temperature bounds for two-dimensional Ising models

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Mathematical Physics

Background:

• The classical ferromagnetic Ising model is a fundamental model in statistical mechanics.
• Understanding critical phenomena, such as phase transitions, is crucial in condensed matter physics.
• Periodic tessellations offer a rich framework for studying lattice models.

Purpose of the Study:

• To derive exact critical-temperature bounds for the classical ferromagnetic Ising model on 2D periodic tessellations.
• To identify universal factors determining these bounds.
• To explore the tightness of these bounds for various lattice structures.

Main Methods:

• Application of the Feynman-Kac-Ward formalism.
• Derivation of exact upper bounds for the critical temperature.
• Verification across a diverse set of over 200 lattices.

Main Results:

• A universal upper bound for the critical temperature was derived, dependent only on the maximum coordination number of the lattice.
• The derived bounds were found to be exact for specific lattices, including honeycomb, square, and triangular.
• A novel 2D lattice with a 24-coordinated site and a high critical temperature was constructed.

Conclusions:

• The study provides precise, universal critical-temperature bounds for 2D Ising models on periodic lattices.
• The Feynman-Kac-Ward formalism is a powerful tool for analyzing critical phenomena in such models.
• The findings offer insights into designing lattices with high critical temperatures.