Tunable band topology in gyroscopic lattices

Gyroscopic metamaterials, mechanical structures composed of interacting spinning tops, have recently been found to support one-way topological edge waves. In these structures, the time-reversal symmetry breaking that enables their topological behavior emerges directly from the lattice geometry. Here we show that variations in the lattice geometry can give rise to more complex band topology than has been previously described. A “spindle” lattice (or truncated hexagonal tiling) of gyroscopes possesses both clockwise and counterclockwise edge modes distributed across several band gaps. Tuning the interaction strength or twisting the lattice structure along a Guest mode opens and closes these gaps and yields bands with Chern numbers of $\left|C\right|>1$ without introducing next-nearest-neighbor interactions or staggered potentials. A deformable honeycomb structure provides a simple model for understanding the role of lattice geometry in constraining the effects of time-reversal symmetry and inversion symmetry breaking. Last, we find that topological band structure generically arises in gyroscopic networks, and a simple protocol generates lattices with topological excitations.

Figure 1

A spring-coupled gyroscopic metamaterial is composed of spinning gyroscopes that hang from a pinned pivot point. (a) $\overrightarrow{{\ell }_f}$ is the vector from the pivot point to where a force acts. When the only force is gravity, $\overrightarrow{{\ell }_f}=\overrightarrow{{\ell }_{cm}}$. (b) Gyroscopes in the metamaterial are coupled to their neighbors in the lattice via a spring which is attached to the free end. (c) The linearized equation of motion for our system relates the displacements via angles between bonds and the local gyroscope's local $x$ axis (indicated by dotted lines in this view from below).

Figure 2

The spindle lattice shares features of both the honeycomb and kagome lattices while supporting a Guest mode in which each triad of gyroscopes is locally rotated. (a) Taking the size of the red triangles in the spindle lattice to zero returns a honeycomb configuration, while taking the length of the blue bonds connecting each red triangle of gyroscopes to zero transforms the spindle lattice into the kagome configuration. The associated band structures are shown below each of the three configurations. (b) Locally twisting the triangles of a spindle lattice preserves bond lengths while globally deforming the lattice.

Figure 3

The gyroscopic spindle lattice contains chiral edge modes of either chirality as well as band with a Chern number of $\left|C\right|>1$. Direct simulation of Eq. (5) reveals clockwise (left) and counterclockwise (right) edge modes in the same structure when shaken at different frequencies $\mathrm{[}\mathrm{\Omega }=1.8{\mathrm{\Omega }}_g$ (left) and $3.62{\mathrm{\Omega }}_g$ (right)]. The displacement of each gyroscope is represented as a circle with a radius proportional to the displacement's magnitude. The color of each circle represents the phase of the displacement, as depicted in the color wheel on the bottom right. Computing the Chern numbers for each band confirms the topological origin of the chiral edge modes, as shown by the colored band structure in the middle panel. A single gyroscope on the edge is shaken at a fixed frequency with an amplitude varying in time; the spectrum of the excitation is indicated by red (left) and blue (right) curves overlying the density of states $D\left(\mathrm{\Omega }\right)$. The density of states, shown above each lattice, is given for the case with periodic boundary conditions. For these simulations, the interaction strength was set to ${\mathrm{\Omega }}_k=3{\mathrm{\Omega }}_g$.

Figure 4

Phononic band structures for the spindle and twisted spindle lattices show the opening and closing of gaps with topological edge states. (a) Simply increasing the interaction strength enables the closing and opening of band gaps, creating and annihilating protected chiral edge modes. (b) Band gaps with chiral edge modes are highlighted in green for two different interaction strengths. (c) and (d) As the structure is twisted through a bond-length-preserving deformation, three of the five gaps close and reopen, leading to three or four gaps with chiral edge modes, depending on the value of the twist deformation angle $\alpha$. In (c) and (d), we set ${\mathrm{\Omega }}_k=3{\mathrm{\Omega }}_g$.

Figure 5

Band gaps and topology in the deformed honeycomb lattice. (a) The angles $\varphi$ and $\delta$ control the deformation of the honeycomb lattice. (b) The $\varphi \text{−}\delta$ phase diagram shows that the Chern number of the lower band changes when straight lines of bonds appear in the lattice, which occurs on the white diagonal lines in the left corners and on the white vertical line at $\delta =\pi$. (c) The bricklayer configuration $\left(\delta =\pi \right)$ band structure is plotted along paths in the Brillouin zone. The gap is closed at two Dirac points. (d) No band gap opens in the canted brick layer $\left(\delta =\pi ,\varphi \ne \pi /2\right)$, even though time-reversal symmetry is broken in this configuration.

Figure 6

Inversion and time-reversal symmetries compete in a gyroscopic lattice. (a) The phase diagram for a deformed honeycomb lattice (without shear, so that $\varphi =\pi /2)$ with varying ${\mathrm{\Omega }}_g$ values on sites A and B shows an interplay between inversion and time-reversal symmetries. (b) In a simulation of the honeycomb lattice with inversion symmetry breaking ${\mathrm{\Delta }}_{AB}=0.15$, driving a gyroscope on the edge at a gap frequency results in a clockwise wave packet. (c) When the lattice is deformed to a brick-layer geometry, the Chern number vanishes. This configuration is gapped due to the inversion symmetry breaking $\left({\mathrm{\Delta }}_{AB}=0.15\right)$. The gap contains modes which are localized on the edge, but these unprotected edge waves propagate in both directions and are not robust against disorder. (See also Videos 3 and 4 in the Supplemental Material.) (d) In the bow-tie geometry, edge modes propagate counterclockwise, as predicted by the calculations shown in (a).

Figure 7

Coordination number and topological phases. (a) A lattice with four lattice sites per unit cell, where sites A and B have two neighbors and sites C and D have two. (b) The topological phase diagram for varying the gravitational precession frequencies on sites A and B shows that because of the different coordination numbers for the lattice sites, the band structure is trivial when ${\mathrm{\Omega }}_{gA}={\mathrm{\Omega }}_{gB}={\mathrm{\Omega }}_{gC}={\mathrm{\Omega }}_{gD}$. (c) The band structure in the nontrivial phase for a strip which is infinite along $y$ and 120 unit cells wide in $x$.

Figure 8

Examples of topological lattices created balancing coordination by varying on-site precession frequencies. Each lattice is generated by placing triangulated points in a square unit cell, then deleting some bonds randomly. (a) An example of a deformed kagome lattice structure exhibits two topological gaps. (b) A mechanically stable lattice with one topological gap (upper gap) demonstrates that gyroscopic lattices need not be undercoordinated to be Chern insulators. The propagation of edge modes is in the same direction as the kagome lattice. (c) A three-site-per-unit-cell lattice structure with one topological gap (lower gap). The propagation of edge modes is in the same direction as the honeycomb lattice. (d) An example of a four-site-per-unit-cell lattice structure with three topological gaps. The propagation of edge modes is in the same direction as the kagome lattice for all three gaps.