Phonon analog of topological nodal semimetals

Topological band structures in electronic systems like topological insulators and semimetals give rise to highly unusual physical properties. Analogous topological effects have also been discussed in bosonic systems, but the novel phenomena typically occur only when the system is excited by finite-frequency probes. A mapping recently proposed by C. L. Kane and T. C. Lubensky [Nat. Phys. 10, 39 (2014)], however, establishes a closer correspondence. It relates the zero-frequency excitations of mechanical systems to topological zero modes of fermions that appear at the edges of an otherwise gapped system. Here we generalize the mapping to systems with an intrinsically gapless bulk. In particular, we construct mechanical counterparts of topological semimetals. The resulting gapless bulk modes are physically distinct from the usual acoustic Goldstone phonons and appear even in the absence of continuous translation invariance. Moreover, the zero-frequency phonon modes feature adjustable momenta and are topologically protected as long as the lattice coordination is unchanged. Such protected soft modes with tunable wave vector may be useful in designing mechanical structures with fault-tolerant properties.

Figure 1

Typical spectrum of a 2D TNS protected by chiral symmetry hosting a pair of topological nodes at $\omega =0$ (only the two bands close to $\omega =0$ are shown). The spectrum is computed for the fermionic problem defined in Eq. (2) with $R$ defined in Eqs. (3) and (4). The parameters $({\theta }_1,{\varphi }_1,{\theta }_2,{\varphi }_2)=(1/4,0,1/8,-1/8)\pi$ are used. The inset shows the edge spectrum when open and periodic boundary conditions are enforced for the ${\widehat{\mathit{e}}}_1$ and ${\widehat{\mathit{e}}}_2$ directions, respectively, featuring a line of zero modes connecting the projections of the bulk topological nodes onto the surface Brillouin zone.

Figure 2

A possible realization of the considered phonon problem. The dependence of the spring extensions on the displacements of the masses can be modified by bending the springs with fixed, smooth pegs which serve as a pinning potential and gap out the acoustic modes. The spring extensions are then characterized by Eq. (4) to linear order in mass displacements. The shaded region indicates the unit-cell convention adopted.

Figure 3

Linearized phonon spectrum for the smooth-peg model along different paths in the 2D BZ. The insets show the dispersion of the two independent phonon modes around the critical quasimomentum ${\mathit{k}}_c$. As in the text, here we take $m={\kappa }_1={\kappa }_2=1$. (a) The parameters $({\theta }_1,{\varphi }_1,{\theta }_2,{\varphi }_2)=(0.1,0.2,0.2,0.1)\pi$ are used. As such, ${\theta }_1+{\varphi }_1-{\theta }_2-{\varphi }_2=0$, which gives rise to an accidental vanishing of $\omega$ along a pair of arcs in BZ. This is seen in the vanishing of $\omega$ at multiple values of $\mathit{k}$ as well as the existence of a quadratic phonon mode around ${\mathit{k}}_c$. (b) The parameters $({\theta }_1,{\varphi }_1,{\theta }_2,{\varphi }_2)=(1/4,0,1/8,-1/8)\pi$ are used (same as Fig. 1), which corresponds to the generic case in which the linearized phonon spectrum contains a pair of isolated bulk nodal points at $\pm {\mathit{k}}_c$ with conical dispersion around them.

Figure 4

Finite-size scaling of the lowest eigenvalue of the disordered dynamical matrix. The linearized dynamical matrix for systems with size $N=N_1{N}_2$ and site-dependent couplings is numerically diagonalized to find the smallest eigenvalue ${\omega }_{\text{min}}^2$. The clean system is defined by the same set of parameters used in Fig. 1. Disorder is incorporated by adding to each parameter at each site a small, independent deviation drawn from a normal distribution with standard deviation $\sigma$ $(\kappa$ and $m$ are kept non-negative by taking absolute values). For each $\sigma$, the mean value $\langle {\omega }_{\text{min}}^2\rangle$ is obtained from 200 disorder realizations. Error bars represent standard deviation in the disorder averages, and solid lines are guides for the eye.

Figure 5

Disorder average of the spatial profile of the lowest-frequency mode. Error bars indicate standard deviation from the disorder averaging.

Figure 6

An alternative model in which the smooth pegs are replaced by pinned rotors, which preserves isostaticity. (a) Schematic of the model indicating the different parameters. The shaded region indicates the choice of the unit cell, and each spring in the cell is labeled by a pair of numbers $(l,m)$. (b) Example phase diagram with $a=0.4,b=-0.15,r=0.1$, and ${\theta }_0,{\varphi }_0\in [-\pi /3,\pi /3]$. The colored region corresponds to the TNS phase, with the color (inset) encoding the value of $\pm {\mathit{k}}_c$. The white regions correspond to a gapped phonon spectrum, which can still have different topological polarization ${\mathit{P}}_T$.