A Hierarchy of Fibonacci Forbidden-Word Hamiltonians: From the Golden Chain to the Plastic Chain and Aperiodic Order

Abstract

We introduce an infinite, scale-aligned hierarchy of one-dimensional, frustration-free Hamiltonians by forbidding the minimal forbidden factors of the Fibonacci word up to length $F_K$, the $K$-th Fibonacci number. The ground-state languages have exponential growth constants $\lambda_K$ that decrease monotonically, starting from the value associated with the ``golden chain'' (approximately 1.618) and progressing toward 1. This process yields a staircase of topological-entropy plateaus that flows to an aperiodic fixed point, also known as the Fibonacci subshift. The first nontrivial rung ($K=4$) is the ``Plastic chain,'' which forbids \texttt{SS} and \texttt{LLL}. We prove its ground-state counts follow a specific four-term linear recurrence relation and provide a closed-form solution governed by the plastic constant $\rho\approx 1.3247$. We propose an energy-entropy scaling where the energy penalty for each new forbidden pattern is proportional to the logarithmic ratio of the growth constants from the previous and current rungs, turning the sequence of projectors into an explicit renormalization-group flow from the initial high-entropy phase to the zero-entropy aperiodic fixed point. Algebraically, exact Temperley-Lieb braiding compatibility holds only at the base rung, $K=3$ (which forbids only \texttt{SS}); higher rungs define constrained aperiodic Hamiltonian codes rather than Temperley-Lieb representations. Small instances realized on a D-Wave quantum annealer match these predictions: $K=3$ is trivial, $K=4$ resolves a unit gap with moderate success, and $K\ge 5$ instances require reverse annealing to exceed $99\%$ success, clarifying reduction penalties and embedding variability.