  This work reports topological extraordinary properties of sound transmission
through topological states in sonic crystals denominated Suzuki phase,
consisting of a rectangular lattice of vacancies created in a triangular
lattice. These low-symmetry crystals exhibit unique properties due to the
embedded lattice of vacancies. A generalized folding method explains the band
structure and the quasi-type-II Dirac point in the Suzuki phase, which is
related to the underlying triangular lattice. In analogy to the acoustic valley
Hall effect, the Suzuki phase contains three types of topological edge states
on the four possible interfaces separating two Suzuki phase crystals with
distinct topological phases. The edge states have defined symmetries with
inherent directionality, which affect the topological sound transmission and
are different from chirality, valley vorticity or helicity. Particularly, the
existence of topological deaf bands is here reported. The propagation of
topological eigenmodes on the same interface is also different, which is
quantified using the acoustic Shannon entropy, making the topological transport
dependent on the frequency of the edge states. Based on the abundant
topological edge states of Suzuki phase crystals, a multifunctional device with
acoustic diodes, multi-channel transmission, and selective acoustic
transmission can be designed. Numerical simulations and measurements
demonstrate the topological transmission. Our work extends the research
platform of acoustic topological states to lattices with low symmetry, which
opens new avenues for enriching topological states with broad engineering
applications.
