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

Marcelo Maciel Amaral

TL;DR

This work introduces an infinite, scale-aligned hierarchy of one-dimensional frustration-free Hamiltonians H_K built by forbidding the minimal forbidden factors up to length $F_K$ of the Fibonacci word, creating a staircase of entropy plateaus that flows toward the zero-entropy Fibonacci subshift. The first nontrivial rung, the Plastic chain at $K=4$, forbids $\texttt{SS}$ and $\texttt{LLL}$ and exhibits a four-term recurrence with growth constant $\lambda_4=\rho\approx1.3247$, along with an exact RG-like energy-entropy scaling $J_{M_{F_k}} \propto \log(\lambda_{k-1}/\lambda_k)$. The base rung $K=3$ preserves an exact Temperley-Lieb braiding representation $TL_N(\varphi)$, while higher rungs generate constrained aperiodic codes with a rank obstruction to TL, enabling two operating modes and novel application protocols. The authors validate their predictions through small-scale D-Wave experiments, showing straightforward ground-state resolution for $K=3$, moderate success for $K=4$, and near-certain ground-state recovery for $K\ge5$ via reverse annealing, highlighting both the practical potential and embedding challenges of this Fibonacci-forbidden hierarchy.

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 $λ_K$ 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 $ρ\approx 1.3247$. 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.

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

TL;DR

This work introduces an infinite, scale-aligned hierarchy of one-dimensional frustration-free Hamiltonians H_K built by forbidding the minimal forbidden factors up to length of the Fibonacci word, creating a staircase of entropy plateaus that flows toward the zero-entropy Fibonacci subshift. The first nontrivial rung, the Plastic chain at , forbids and and exhibits a four-term recurrence with growth constant , along with an exact RG-like energy-entropy scaling . The base rung preserves an exact Temperley-Lieb braiding representation , while higher rungs generate constrained aperiodic codes with a rank obstruction to TL, enabling two operating modes and novel application protocols. The authors validate their predictions through small-scale D-Wave experiments, showing straightforward ground-state resolution for , moderate success for , and near-certain ground-state recovery for via reverse annealing, highlighting both the practical potential and embedding challenges of this Fibonacci-forbidden hierarchy.

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 , the -th Fibonacci number. The ground-state languages have exponential growth constants 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 () 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 . 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, (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: is trivial, resolves a unit gap with moderate success, and instances require reverse annealing to exceed success, clarifying reduction penalties and embedding variability.

Paper Structure

This paper contains 22 sections, 7 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Overview and Main Construction
  3. Hamiltonian hierarchy.
  4. Energy-entropy scale setting (RG view).
  5. Fibonacci MFFs and a "Boundary–Flip" Generator
  6. Combinatorics of the Hierarchy
  7. The Plastic Chain ($K{=}4$ forbidding SS and LLL)
  8. General Counting via Automata
  9. Alphabet and patterns.
  10. Counting and growth constant.
  11. Relationship to and Reuse of the QIC Braid Framework
  12. Base rung ($K{=}3$): exact TL($\varphi$).
  13. Higher rungs ($K{\ge}4$): constrained Hamiltonian codes.
  14. Two operating modes.
  15. Proposed application protocols.
  16. ...and 7 more sections

Key Result

Proposition 2.1

For all $K\ge 4$ and all $N\ge 1$ one has $D_K(N)\le D_{K-1}(N)$, hence $\lambda_K\le \lambda_{K-1}$. Moreover, the inequality is strict ($\lambda_K<\lambda_{K-1}$) whenever the newly added MFF ($M_{F_K}$) is not eventually redundant (i.e., it eliminates at least one valid word for infinitely many $

Theorems & Definitions (12)

  • Proposition 2.1: Monotone growth-rate staircase
  • proof
  • Remark 2.2
  • Proposition 3.1: Boundary-flip recursion; constructive
  • Proposition 4.1: Recurrence, characteristic polynomial, Plastic growth
  • Theorem 4.2: Exact form and rounding identity
  • Definition 4.3: Aho-Corasick automaton AhoCorasick1975
  • Lemma 4.4: Avoidance DFA from Aho-Corasick
  • proof : Proof sketch
  • Proposition 5.1: Local-rank obstruction to $TL_N(\varphi)$ for $K\ge 4$
  • ...and 2 more