# Goldstone modes and the golden spiral in the ferromagnetic spin-1 biquadratic model

## TL;DR

This paper explores the rich structure of ferromagnetic ground states in the spin-1 biquadratic model, revealing scale invariance, Goldstone modes, Fibonacci-Lucas degeneracy sequences, and logarithmic entanglement entropy scaling.

## Contribution

It uncovers the scale invariance and degeneracy patterns of ferromagnetic ground states, linking them to Fibonacci-Lucas sequences and the golden spiral, with detailed analysis of entanglement entropy.

## Key findings

- Ground state degeneracies follow Fibonacci-Lucas sequences.
- Goldstone modes depend on system size parity.
- Entanglement entropy scales logarithmically with block size.

## Abstract

Ferromagnetic ground states have often been overlooked in comparison to seemingly more interesting antiferromagnetic ground states. However, both the physical and mathematical structure of ferromagnetic ground states are particularly rich. We show that the highly degenerate and highly entangled ground states of the ferromagnetic spin-1 biquadratic model are scale invariant, originating from spontaneous symmetry breaking from ${\rm SU}(3)$ to ${\rm U}(1)\times {\rm U}(1)$ with two type-B Goldstone modes if the system size is even or from ${\rm SU}(2)$ to ${\rm U}(1)$ with one type-B Goldstone mode if the system size is odd, when periodic boundary conditions are adopted. The ground state degeneracies are characterized as Fibonacci-Lucas sequences, under open and periodic boundary conditions, with nonzero residual entropy per site. This implies that the ground state degeneracies for this model are asymptotically the golden spiral. In addition, sequences of atypical (periodic) degenerate ground states generated from highest and generalized highest weight states are constructed to establish that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit. The prefactor is half the number of type-B Goldstone modes, which is identified to be the fractal dimension, if one is restricted to atypical degenerate ground states. We also argue that the same conclusion is valid for typical (non-periodic) degenerate ground states, as long as the block size is sufficiently large.