  Spontaneous symmetry breaking (SSB) with type-B Goldstone modes is
investigated in the macroscopically degenerate phase for a quantum spin-1
many-body system with competing dimer and trimer interactions. The SSB involves
three distinct patterns. The first occurs at the dimer point, with the pattern
from staggered ${\rm SU}(3)$ to ${\rm U}(1)\times{\rm U}(1)$. The second occurs
at the trimer point, with the pattern from uniform ${\rm SU}(3)$ to ${\rm
U}(1)\times{\rm U}(1)$. The third occurs in the dimer-trimer regime, with the
pattern from uniform ${\rm SU}(2)$ to ${\rm U}(1)$. The number of type-B
Goldstone modes is thus two, two and one for the three patterns, respectively.
The ground state degeneracies arising from the three patterns are exponential
with the system size, which may be recognized as sequences of integers relevant
to self-similar logarithmic spirals. This in turn is attributed to the presence
of an emergent symmetry operation tailored to a specific degenerate ground
state. As a consequence, the residual entropy is non-zero, which measures the
disorder present in a unit cell of highly degenerate ground state generated
from a generalized highest weight state. An exact Schmidt decomposition exists
for the highly degenerate ground states, thus exposing the self-similarities
underlying an abstract fractal, described by the fractal dimension. The latter
is extracted from performing a universal finite system-size scaling analysis of
the entanglement entropy, which is identical to the number of type-B Goldstone
modes. The model under investigation thus accommodates an exotic scale
invariant quantum state of matter.
