  The one-dimensional spin-$s$ ${\rm SU}(2)$ ferromagnetic Heisenberg model, as
a paradigmatic example for spontaneous symmetry breaking (SSB) with type-B
Goldstone modes (GMs), is expected to exhibit an abstract fractal underlying
the ground state subspace. This intrinsic abstract fractal is here revealed
from a systematic investigation into the entanglement entropy for a linear
combination of factorized (unentangled) ground states on a fractal decomposable
into a set of the Cantor sets. The entanglement entropy scales logarithmically
with the block size, with the prefactor being half the fractal dimension of a
fractal, as long as the norm for the linear combination scales as the square
root of the number of the self-similar building blocks kept at each step $k$
for a fractal, under an assumption that the maximum absolute value of the
coefficients in the linear combination is chosen to be around one, and the
coefficients in the linear combination are almost constants within the building
blocks. Actually, the set of the fractal dimensions for all the Cantor sets
forms a {\it dense} subset in the interval $[0,1]$. As a consequence, the
ground state subspace is separated into a disjoint union of countably
infinitely many regions, each of which is labeled by a decomposable fractal.
Hence, the interpretation of the prefactor as half the fractal dimension is
valid for any support beyond a fractal, which in turn leads to the
identification of the fractal dimension with the number of type-B GMs for the
orthonormal basis states. Our argument may be extended to any quantum many-body
systems undergoing SSB with type-B GMs.
