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

Quantum Physics arXiv:2511.10672 (quant-ph) [Submitted on 9 Nov 2025] Title:A Hierarchy of Fibonacci Forbidden-Word Hamiltonians: From the Golden Chain to the Plastic Chain and Aperiodic Order Authors:Marcelo Maciel Amaral View a PDF of the paper titled A Hierarchy of Fibonacci Forbidden-Word Hamiltonians: From the Golden Chain to the Plastic Chain and Aperiodic Order, by Marcelo Maciel Amaral View PDF HTML (experimental) 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. Subjects: Quantum Physics (quant-ph); Materials Science (cond-mat.mtrl-sci) Cite as: arXiv:2511.10672 [quant-ph] (or arXiv:2511.10672v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2511.10672 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Marcelo Maciel Amaral [view email] [v1] Sun, 9 Nov 2025 23:28:22 UTC (13 KB) Full-text links: Access Paper: View a PDF of the paper titled A Hierarchy of Fibonacci Forbidden-Word Hamiltonians: From the Golden Chain to the Plastic Chain and Aperiodic Order, by Marcelo Maciel AmaralView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-11 Change to browse by: cond-mat cond-mat.mtrl-sci References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)