Recently, the team of Professor Yao Yugui, Professor Li Feng and researcher Zhou Di from the Beijing Institute of Technology has realized for the first time a polarized topological mechanical insulator in three-dimensional space, whose upper boundary is topologically rigid, while the lower boundary is topologically flexible。Even if the material is cut in the middle and split into two, then the upper and lower boundaries of the new two pieces of material will still show this strong asymmetric stiffness of "hard on the top and soft on the bottom", and the system is called a polarized "static topological insulator".。The results are published in the journal Physical Review Letters。

Since the 1980s, topological insulators have become one of the key research directions in condensed matter physics。In 2014, Charles Kane and Tom Lubensky, academicians of the American Academy of Sciences, successfully extended the concept of topological insulators to quasi-static mechanical systems, and established static topological insulators with isostatically determined structures as the main research object。Static topological insulators characterized by topological polarization vectors have edge modes called mechanical floppy mode and state of self-stress on their surfaces，Show zero frequency (no loss), anti-interference, anti-defect and other excellent properties。And even more interesting，In static topological insulators，Mechanical soft modes can be completely localized on an open surface，In the opposite direction, there is no mechanical soft mode at all，The surface of the material exhibits a high degree of asymmetric elasticity: the open border on one side is unusually soft，The other side is extremely hard，The difference of soft and hard stiffness between parallel open boundaries can be more than 1,000 times。The "upper hard, lower soft" mechanical state displayed by this statically indeterminate structure is therefore called the "topologically complete polarization" state。Like other topological insulators also described by polarization vectors, fully polarized statics materials are topologically stable: even if the material is split in two, the two new mechanical structures will still exhibit a hard and soft mechanical polarization behavior。

It is easy to realize the topological mechanics of complete polarization by adjusting the geometric parameters of 1-D and 2-D isostatically indeterminate structures, but in 3-D isostatically indeterminate structures, the widespread existence of Weyl lines will destroy the topological polarization property of mechanical structures。Since Weyl lines themselves are topologically stable and extremely difficult to remove, it is very difficult to find static topological insulators in theory。

The Beijing Institute of Technology team proposed a mechanical transfer matrix method.，The transfer behavior of each mechanical soft mode in space is studied separately，The direct relation between each mechanical soft mold and the geometrical parameters of the structure is established，It is possible to localize all mechanical soft modes to a specific surface of the structure by adjusting the parameters。Based on this theoretical innovation, the team designed and 3D printed a topologically fully polarized generalized pyrochlore lattice.。The team innovatively used a freely rotating spherical hinge (shown in Figure 1c) to eliminate bending stiffness at the lattice joints, ensuring that the 3D printed generalized pyrochite lattice is an ideal isostatically indeterminate structure。The protocell of the generalized pyroconite lattice consists of two 3D-printed tetrahedrons with a spherical hinge at each vertex。In Figure 1(d), the vertices A, B, C, and D of the white tetrahedron are connected to the vertices A ', B ', C ', and D 'of the green tetrahedron, allowing free rotation between adjacent tetrahedra。

FIG. 1 Design principle of 3D complete polarization static topological insulator。(a)-(b) The primary cells of the regular pyroconite lattice and the generalized pyroconite lattice。 (c) The white and green rigid bodies represent the bottom tetrahedron and top tetrahedron, respectively, that make up the generalized pyroconite lattice primitive cell。 (d) three-dimensional fully polarized static topological insulator after assembly。

Using the overall shear deformation mode, the team was able to change the geometric parameters of the lattice without reassembling the lattice model, allowing the 3D-printed generalized pyrocholate lattice to easily transition between the fully topologically polarized and Weyl phases。Figure 2 (a,c,e) shows three different topological phases reached by the generalized pyrocholate lattice in a single global shear mode, namely, the fully topologically polarized phase (a), the Weyl phase with two Weyl lines (c), and the Weyl phase with four Weyl lines (e).。As shown in Figure 2(b), the complete topologically polarized phase has the same topological polarization vector throughout the Brillouin region, whereas the topological polarization vector of the Weyl phase depends on the choice of wave vector in the specific Brillouin region。

Figure 2 Weyl lines and corresponding topological polarization vector distributions in three generalized pyroconite lattice configurations obtained from the global shear mode。

The team carried out numerical simulations on the surface mechanics of generalized pyroconite lattices。Figure 3(a) shows that the stiffness ratio between the upper and lower surfaces of the lattice decreases rapidly during the transition of the lattice from a completely topologically polarized phase to a Weyl phase。In Figure 3(b), a completely topologically polarized phase with θ=0° exhibits a strongly asymmetric surface mechanical response to point external forces。FIG. 3(c) shows the obvious difference in spatial distribution between the mechanical soft modes of complete topological polarization and the Weyl soft modes in the Weyl phase. The former is localized to a certain boundary of the lattice, while the latter runs through the entire lattice。

FIG. 3 Surface mechanics numerical simulation of generalized pyroconite lattice under horizontal periodic boundary and vertical open boundary conditions。(a) The local stiffness of the top and bottom open surfaces in a lattice composed of 5×5×8 protocells as Geist's global shear mode Angle θ changes。Different background colors represent different topological phases, and the numbers 0, 2, and 4 in Figure a represent topological complete polarization, the Weyl phase of 2 Weyl lines, and the Weyl phase of 4 Weyl lines, respectively。

Experimentally, the team performed force-displacement measurements on a 3D-printed generalized pyroconite lattice, and FIG. 4(c, d) shows force-displacement curves for a completely topologically polarized phase with θ="0° and a Weyl phase with θ=45°, respectively。Unlike the nearly identical force-displacement curves of the upper and lower surfaces of the Weyl phase, the force-displacement curves of the upper and lower surfaces of the completely topologically polarized lattice show significant differences, indicating that the mechanical soft mode is localized only at the bottom of the lattice, and the upper surface of the lattice therefore has topologically protected stiffness。

The team then designed a three-dimensional "non-inflatable topological tire," as shown in Figure 4(f,g), in which the generalized pyroconite lattice is embedded in a cylinder, behaves as a multi-empty non-inflatable tire, and is able to effectively absorb shocks from rough terrain。In addition, the complete topological polarization phase and Weyl phase, which have strong contrast in stiffness performance, can be easily switched by the global shear mode。Based on this property, materials with different stiffness properties can be designed in different scenarios，An example is the drone landing gear shown in Figure 4(e)，During the landing process, the material can be switched to a Weyl phase with lower stiffness，Absorb shock effectively;During flight, it can be switched to a fully topologically polarized phase，Increase the external stiffness of the drone to increase stability。

Figure 4. Experimental measurement of generalized pyroconite lattice for 3D printing。(a)-(b) are the measured results of the force-displacement curves of the topologically completely polarized phase and the Weyl phase, respectively。(c) Porous topological tire composed of generalized pyroconite lattice。(d) Numerical analysis of the mechanical response of topological tires when rolling on rough terrain。(e) Reconfigurable elastic topological landing gear for UAVs。

In this study, it is proved theoretically and experimentally that the completely polarized phase of a three-dimensional statics topological insulator. The mechanical soft mode that is all localized to a certain boundary indicates that the topologically completely polarized phase has no mechanical Weyl line。This mechanical achievement is similar to the discovery of three-dimensional topological insulators in electronic systems。Complete topologically polarized phases in three dimensions pave the way for physical phenomena that are impossible to achieve in two-dimensional lattices, such as higher order topologically statical soft modes, topologically mechanical stealth, and statical nonreciprocity in full space dimensions。

Beijing Institute of Technology doctoral students Tang Zheng and Ma Fangyuan are co-first authors of the paper, and Beijing Institute of Technology Professor Li Feng and Researcher Zhou Di are corresponding authors。

Article link:http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.133.106101