Acoustic Topological Effects In Hexagonal Lattices

Hexagonal lattice plays an important role in constructing models of novel topological quantum states.Since the hexagonal lattice with nearest neighbor hopping supports the conic Dirac cones at the corners of Brillouin zone,opening gaps at the corners are associated with many peculiar phenomena of physics.Both the quantum anomalous Hall effect and quantum spin Hall effect were first proposed in the two-dimensional hexagonal lattice,for which the gaps for these Dirac cones were opened by means of the external magnetic field and intrinsic spin orbital coupling,respectively.However,the study of topological states in acoustic systems cannot completely follow the way in electronic systems.At present,it is feasible to change lattice symmetries or construct effective degrees.Starting from the Dirac-like degeneracies in the reciprocal space of the hexagonal or triangular lattices,the acoustic topological effects in different spatial dimensions are realized by lifting and extending the band degeneracies.The research contents are listed as followsFirstly,we explore the unidirectional edge transport of sound in a modified hexagonal lattice.Based on the modified hexagonal lattice,we obtain two two-fold degenerate states at the ? point.By recombining the degenerate Bloch states,we construct the pseudospin states By constructing the interface of modified hexagonal lattices of trivial and non-trivial band gaps,we implement the pseudospin dependent sound transports with crystal symmetry protection.In addition,there exist a pair of opposite pseudospins at the boundaries of truncated lattices with nontrivial band gap,therefore we achieve the pseudospin dependent unidirectional edge states at the armchair-shaped boundary of the finite hexagonal latticeSecondly,we investigate the related effects of acoustic valley states in the hexagonal lattice with inversion symmetry breaking.Based on the hexagonal lattice of coupled tube resonators,we introduce height difference for the two triangular sublattices to obtain energy valley states.Comparing the degree of symmetry breaking,we find that the height difference directly determines the size of the band gap at K and K' points.For the largely distorted sonic crystal,the topological protection on the interface mode is not completely lost Furthermore,we demonstrate the physical equivalence between the acoustic boundaries and the interface states with odd and even parities,respectively.Using the acoustic boundaries,we realize the programmable edge transports in a single-crystal topological insulator,which greatly reduces the dimension of the acoustic structureThirdly,we demonstrate the parity difference and selective excitation of the interface states.C3V symmetry scatterers are arrayed at the sites of triangular lattice,and the acoustic valley Hall phase transition is confirmed by rotating scatterers in different directions.For the zigzag interfaces constructed by inverted sonic crystals,each interface corresponds to an interface state with specific parity.Further,we propose the C-shaped channel that contains two types of zigzag interfaces.When the source is excited at the central of the channel,the Odd and Even interface states can be detected at the output ports.When the sound waves is incident along the two interfaces,there exists the selective excitation for the interface states Thus,we observe the unidirectional transport of soundFinally yet importantly,we discuss the nodal line states and drumhead surface states in three-dimensional ball-and-stick sonic crystals.In the three-dimensional layer-stacked hexagonal lattices,we obtain nodal line states in different shapes in the momentum space by tuning the cross section of interlayer slanted pipes.When the interlayer coupling strength weighs more than that of the intralayer coupling,the band crossing points form a closed nodal ring in momentum space.Flat drumhead surface dispersions in the band gap verify the existence of a closed nodal ring.Further,comparing with Tamm-like surface states,we confirm that drumhead surface states are featured with strong field confinement at the truncated surface.In contrast to the site disorders,drumhead surface states are more robust against the hopping disorders.