A lattice model with Fibonacci degree of degeneracy
Athena Wang
TL;DR
The paper constructs a 1D quantum lattice with local space $V=\langle|0\rangle,|1\rangle\rangle$ and a local operator that leaves only $|11\rangle$ nonzero, producing a non-negative Hermitian $H$ with $\dim\ker H=F_{n+1}$. Two approaches—modular arithmetic/matrix analysis and a recursion on qubit basis states—both derive the Fibonacci degeneracy, linking ground-state structure to combinatorics of no-adjacent-ones sequences. The per-site ground-state count converges to the golden ratio, since $(F_{n+1})^{1/n}\to φ$, and the paper generalizes to $k$-step Fibonacci degeneracies where the limit equals the dominant root of $x^k-x^{k-1}-\cdots-1=0$, a Pisot number. The work connects quantum lattice models to algebraic-number theory and raises questions about realizing broader Pisot roots and related conjectures for ground-state counting.
Abstract
In this paper, we explore two different methods of finding the degrees of degeneracy for lattice model systems, specifically constructing one with a Fibonacci degree of degeneracy. We also calculate the number of ground states per site as the golden ratio $(φ)$ for the system that we constructed and extend our results to systems with $k-$Step Fibonacci degrees of degeneracy. Finally, I end with a few open questions that we may examine for future works.