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

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.