Using exact diagonalization and variational tensor network methods, we study ground states near the ferromagnetic $\mathrm{SU}\left(3\right)$ point at $\theta =5\pi /4$ in the spin-1 bilinear-biquadratic model from a finite-size approach. In particular, two types of scale-invariant states, scale-invariant states with type-B Goldstone modes and conformal-invariant states, are involved. Universal finite-size entanglement entropy scaling for the two distinct types of scale-invariant states, which exhibit boundary dependence in conformal-invariant states and boundary independence arising from emergent lattice-permutation symmetry in scale-invariant states with type-B Goldstone modes, enable a clarification of the physics in the vicinity of the ferromagnetic $\mathrm{SU}\left(3\right)$ point. According to the results from exact diagonalization with system size up to $L=16$, entanglement entropy in the vicinity of the point $\theta =5\pi /4$ presents a universal finite-size scaling for scale-invariant states with two type-B Goldstone modes to high precision, independent of the boundary conditions, as a result of an emergent lattice-permutation symmetry in the scale-invariant states. Meanwhile, from both an exact diagonalization method and a variational tensor network method, a unique “conformal-invariant point” is identified near $\theta =5\pi /4$ with central charge $c=2$ under both periodic and open boundary conditions, moving toward the point $\theta =5\pi /4$ as $L$ increases, suggesting a direct ferromagnetic-to-dimerized phase transition in the thermodynamic limit. The fractal nature at the $\mathrm{SU}\left(3\right)$ point is confirmed indirectly by the nearby scale-invariant-like states, following the spontaneous breaking of $\mathrm{SU}\left(3\right)$ symmetry to $\mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$ symmetry, resulting in two type-B Goldstone modes. Such scale-invariant-like states in the vicinity of points relating to type-B Goldstone modes may exist more generally in other models, offering an indirect approach to detecting type-B Goldstone modes.

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• Entanglement entropy
• Phase diagrams
• Quantum criticality
• Quantum spin chains