Fibonacci-lucas Degeneracies Reveal Golden Ratio in One-Dimensional Ising Model at Criticality

The behaviour of magnetic materials at critical points, where fundamental properties change dramatically, remains a key area of condensed matter physics, and recent work sheds new light on the underlying mathematical structures governing these systems. Bastian Castorene, from the Instituto de Física at Pontificia Universidad Católica de Valparaíso, alongside Francisco J. Peña and Patricio Vargas from the Departamento de Física at Universidad Técnica Federico Santa María, investigated the one-dimensional antiferromagnetic Ising model, exploring how the arrangement of magnetic spins affects its behaviour at a critical point. Their analysis reveals a surprising connection between the number of possible ground states and well-known mathematical sequences, specifically the Fibonacci and Lucas numbers, depending on whether the system forms an open chain or a closed ring. This discovery establishes a fundamental link between topology, critical degeneracy, and the golden ratio, offering a new framework for understanding and predicting the behaviour of one-dimensional magnetic systems and potentially informing the design of devices operating near critical regimes.

This work examines the one-dimensional antiferromagnetic Ising model at a critical point, investigating how ground state degeneracies relate to mathematical sequences. Researchers focus on systems subjected to a magnetic field, comparing open-chain and closed-ring geometries. By analysing the energy spectrum and counting degenerate configurations, the team demonstrates that the number of ground states follows the Fibonacci sequence for open chains and the Lucas sequence for periodic rings, establishing a surprising connection between topology, critical behaviour, and the golden ratio.

The research provides new insight into how topological constraints influence residual entropy and modify the multiplicity of critical states. The enhanced degeneracy observed in the ring topology highlights the importance of boundary conditions in determining system properties. These findings offer a simplified, yet effective, combinatorial method for describing how degeneracy scales in low-dimensional spin systems and expand understanding of critical behaviour in one-dimensional systems.

Ground State Degeneracy in Ising Chains

The study investigates the degeneracy of ground states in the one-dimensional antiferromagnetic Ising model at a quantum critical point. Researchers explore how the topology of the system, whether it is an open chain or a closed ring, affects the number of possible ground states. Through analytical calculations and combinatorial arguments, the team determines the number of degenerate ground states for both configurations, revealing a fundamental connection between quantum criticality, topology, and number theory.

This discovery highlights the importance of topology in shaping the properties of quantum systems and provides a new perspective on the emergence of complex patterns in physical systems. The observed connection between the ground state degeneracy and mathematical sequences suggests that number theory may play a more significant role in understanding quantum phenomena than previously thought. High degeneracy, often associated with phase transitions, is explored in the context of these low-dimensional systems.

Fibonacci and Lucas Sequences in Critical States

This work investigates the one-dimensional antiferromagnetic Ising model, examining both open-chain and closed-ring configurations subjected to a magnetic field. Researchers determined the structure and degeneracy of the ground states at a specific critical point by analysing spin distributions and interactions within a microcanonical framework. A key finding is that the number of degenerate ground states follows the Fibonacci sequence for open chains and the Lucas sequence for closed rings, establishing a clear link between topology, critical degeneracy, and the golden ratio.

The enhanced degeneracy observed in the ring topology highlights the influence of boundary conditions on residual entropy and demonstrates how topological constraints can modify the multiplicity of critical states. These results reveal an underlying number-theoretic organization governing quantum criticality and provide a simplified, yet effective, combinatorial method for describing how degeneracy scales in low-dimensional spin systems. While the analysis focuses on the one-dimensional Ising model, the authors suggest that these findings may contribute to future studies of quantum critical manifolds, statistical models with constrained configurations, and potentially the operation of quantum thermodynamic devices operating near critical regimes.