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

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. The ground state degeneracies are characterized as the Fibonacci-Lucas sequences -- an ancient mathematical gem, under open and periodic boundary conditions, with the residual entropy being non-zero. This implies that the ground state degeneracies for this model are asymptotically the golden spiral. In addition, sequences of 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, with the prefactor being half the number of type-B Goldstone modes. The latter in turn is identified to be the fractal dimension.