  It is found that the staggered $\rm SU(4)$ ferromagnetic spin-orbital model
accommodates highly degenerate ground states arising from spontaneous symmetry
breaking with type-B Goldstone modes. The spontaneous symmetry breaking pattern
is ${\rm SU(4)} \rightarrow {\rm U(1)} \times {\rm U(1)} \times {\rm U(1)}$,
with three type-B Goldstone modes. An abstract fractal underlies the ground
state subspace with the fractal dimension identified with the number of type-B
Goldstone modes. This connection is established by evaluating the entanglement
entropy, which exhibits a universal finite system-size scaling behaviour. The
latter in turn implies that the entanglement entropy scales logarithmically
with the block size in the thermodynamic limit. In addition, the ground state
degeneracies, depending on the boundary conditions adopted, constitute the two
Fibonacci-Lucas sequences. In the limit of large system size their asymptotic
forms become a self-similar logarithmic spiral. As a result, the model has a
non-zero residual entropy $S_{\!r} = -2 \ln R $, where $R=(\!
\sqrt{6}-\!\sqrt{2})/2$.
