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Kac-Ward Solution of the 2D Classical and 1D Quantum Ising Models
Georgios Athanasopoulos1, Daniel Ueltschi1
1Department of Mathematics, University of Warwick, Coventry, CV4 7AL UK.
Annales Henri Poincare
|August 4, 2025
Summary
This study rigorously derives the free energy for classical and quantum Ising models, including novel negative coupling constants. It analyzes critical phenomena and phase transitions in these magnetic systems.
Area of Science:
- Statistical Mechanics
- Condensed Matter Physics
- Quantum Field Theory
Background:
- The Ising model is a fundamental model in statistical mechanics for understanding magnetism.
- Previous studies often assumed positive coupling constants, limiting applicability.
- Investigating negative coupling constants and quantum effects is crucial for a complete understanding.
Purpose of the Study:
- To provide a rigorous derivation of the free energy for classical and quantum Ising models.
- To explore the impact of negative coupling constants on model behavior.
- To analyze critical phenomena and quantum phase transitions.
Main Methods:
- Utilizing the Kac and Ward method for rigorous derivation.
- Applying techniques from statistical mechanics and quantum mechanics.
- Analyzing the behavior of specific heat and critical temperature formulas.
Main Results:
- Successful derivation of free energy for both classical and quantum Ising models.
- Description of the logarithmic singularity in the specific heat of the classical model.
- Confirmation of the Cimasoni-Duminil-Copin-Li formula for critical temperature validity.
- Discussion of quantum phase transitions in the one-dimensional quantum Ising model.
Conclusions:
- The Kac and Ward method is effective for Ising models with negative coupling constants.
- Negative coupling constants introduce unique behaviors and singularities.
- The study advances the understanding of critical phenomena and quantum phase transitions in magnetic systems.
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