Type-B Goldstone modes and a logarithmic spiral in the staggered $\rm SU(4)$ ferromagnetic spin-orbital model

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 $R=(\! \sqrt{6}-\!\sqrt{2})/2$.