On the Fibonacci-Lucas Ground State Degeneracies of the One-Dimensional Antiferromagnetic Ising Model at Criticality

TL;DR

This paper investigates the one-dimensional antiferromagnetic Ising model in a longitudinal field at the quantum critical point $B_{\,\mathrm{crit}} = 2$ and analyzes ground-state degeneracy under open-chain versus ring geometries using a microcanonical combinatorial approach. It derives a non-adjacency constraint for up spins, showing that the number of degenerate ground states follows the $N$th Fibonacci sequence for open chains and the $N$th Lucas sequence for rings, highlighting a topology-driven structure in critical degeneracy. The key contribution is establishing a direct link between boundary conditions, criticality, and number-theoretic sequences, providing a simple framework to study degeneracy scaling and residual entropy in 1D spin systems. The results may inform explorations of quantum critical manifolds and potential applications in quantum thermodynamic devices operating near critical regimes.

Abstract

This work examines the one-dimensional antiferromagnetic Ising model in a longitudinal magnetic field, comparing open-chain and closed-ring geometries. At the nontrivial quantum critical point (QCP) $B_{\mathrm{crit}} = B/J = 2$, we perform a microcanonical analysis of the ground-state manifold and explicitly count the number of degenerate configurations. The enumeration reveals that ground states follow the $N$th Fibonacci sequence for open chains and the $N$th Lucas sequence for periodic rings, establishing a clear correspondence between critical degeneracy, topology, and the golden ratio. This combinatorial duality exposes a number-theoretic structure underlying quantum criticality and highlights the role of topological constraints in shaping residual entropy. Beyond its conceptual relevance, the result provides a compact framework for analyzing degeneracy scaling in one-dimensional spin systems and may inform future studies of critical phenomena and quantum thermodynamic devices operating near critical regimes.

Figures (2)

• Figure 1: (a) Schematic representation of the antiferromagnetic Ising chain with nearest-neighbor interactions under a longitudinal magnetic field $B$ applied along the spin direction. (b) Corresponding ring geometry obtained by imposing periodic boundary conditions.
• Figure 2: Illustrative energy-level diagram of the Ising model as a function of the magnetic field $B$. A QCP is observed at $B_{\text{crit}}=B/J=2$, where multiple energy levels become degenerate.