Coriolis force induced topological order for classical mechanical vibrations

We show that topological order and vibrational edge modes can exist in a classical mechanical system consisting of a two-dimensional honeycomb lattice of masses and springs. The band structure shows the existence of Dirac cones and unconventional edge states that are similar to the vibrational modes in graphene. Interestingly, as the system is placed on a constantly rotational coordinate system, the Coriolis force resulting from the non-inertial reference frame introduces time-reversal symmetry breaking and leads to topologically nontrivial band gaps. The nontrivial topological orders are further verified by the calculation of Chern numbers for corresponding bands.

Over the past few decades, the discovery of matter with topological nature, such as quantum Hall effect and topological insulators, has attracted much attention towards these novel phenomena [1][2][3][4][5]. Topological protection, the immunity against scattering of matter waves by disorder, is a main property existing in these kinds of materials. The scattering-immunity states probably are the most impressive concept of these materials because this robustness of electronic waves was previously only known in superconductors. Theoretical studies show that breaking time-reversal symmetry plays a crucial role in opening a nontrivial frequency gap, which is an essential condition to bring about the topologically protected edge states [6]. The condition of time-reversal symmetry breaking can be fulfilled through applying an external magnetic field, or the intrinsic spin-orbit coupling of the material itself. Recently, the thriving progress of topological properties in condensed matter has also led to a great interest in discovering similar phenomenon in other physics systems. Raghu and Haldane [7,8] proposed that the optical analog of quantum Hall effect could be achieved by using periodically arranged gyromagnetic rods. Their idea implies that the nontrivial topological phase is fundamentally a wave phenomenon rather than quantum effect. Following this concept, many research groups have realized topologically protected one-way edge states theoretically [8][9][10] and experimentally in photonic systems [11]. Also, by employing periodically arranged meta-crystals and helical arrays, two approaches of photonic analog are proposed to realize a photonic topological insulator that breaks time-reversal symmetry for each spin but preserves it in the whole system [12,13]. A few recent works focus on the analogs of quantum Hall effect in acoustic or phononic systems [14][15][16][17]. Here, we show that a classical mass-spring crystal in a rotating frame can possess the topologically protected edge modes of classical vibration under the influence of the Coriolis force.
We begin by considering a two-dimensional spring-mass system with honeycomb lattice in the inertial lab frame. The system consists of a series of honeycomb-arranged rigid body spheres with mass M and massless springs with elastic constant C, as shown in figures 1(a) and (b). Let a be the separation between the nearestneighbor spheres (A and B) and the lattice translation vectors are expressed as m n r a a  Let displacements of the A and B spheres be ξ and , η respectively, the difference between them be u, and the dynamic displacement of longitudinal oscillation between the two spheres is expressed as R R u,    (2) The band structure described by equation (2) is shown in figure 1(c). There exist four bands, with the two flat bands of transverse modes at zero frequency and 3 , 0 ω and two dispersive longitudinal modes. The number of zero modes is given by the rank-nullity theorem, which can also be described by an index theorem proposed by Kane and Lubensky [17]. Physically, these two flat bands arise from the assumption of negligible transverse restoring force of the springs (see the appendix for detailed discussion). If the transverse restoring force is present in a similar way as in graphene, both the optical and acoustic transverse modes would become dispersive, as shown in [20]. Six linear degeneracies are seen in the vicinity of K points around the entire Brillouin zone. By expanding equation (2) to the second order at K point ( ) is a linear function with respect to the k vector. Figure 1(d) indicates the zoomed-in plot near the K point. These degeneracies, referred to as Dirac points, provide the chance to exhibit topological edge states known as the integer quantum Hall effect when the time-reversal symmetry is broken.
To investigate the vibrational edge modes, we study an infinitely long ribbon along the x direction, but with a finite width containing N unit cells in the y direction. The band structure of a spring-mass ribbon with honeycomb lattice is shown in figure 2. Similar to a graphene ribbon ending with bearded edges, a flat edge state between two Dirac points is observed in the energy spectrum, shown in figures 2(a) and (b). Due to the existence of the flat energy band, at the corresponding frequency, the particles located on the edge vibrate locally without transporting energy forward. For the bearded edge, the band diagram shown in figure 2(b) also gives a flat band throughout the whole k space. We note that it differs from conventional bearded flat edge mode for electrons in graphene, which only appears in the region between the two K points. In addition, two unconventional edge modes (red curves) are found around the Dirac points in figures 2(a) and (b). These new modes for a beaded or zigzag boundary were recently proposed and observed by Plotnik et al in a photonic analog system of graphene [21] and have been theoretically investigated for phonons in real graphene [22]. Figures 2(c) and (d) show the vibrational amplitudes at points A along the transverse cross-section (N = 20) of the spring-mass ribbon. As predicted by the energy spectrum, the peaks indicate that the vibration of particles is highly confined to the corresponding edge.
Next, we introduce time-reversal symmetry breaking by placing the spring-mass lattice in a rotating frame, as schematically shown in figure 3(a). In classical mechanics, objects moving in a rotating reference frame experience inertial forces-centrifugal force and Coriolis force-resulting in corrections to the equation of motion. For a particle oscillating harmonically on a counterclockwise rotating round plate with a constant angular velocity Ω in the z direction, the Lagrangian in terms of equilibrium position r, elongation Δr, and vibration velocity v can be expressed as × [23]. The first two terms correspond to the normal simple harmonic oscillation, the third term times −1 represents the elastic potential energy when the centrifugal force is present, and the fourth term corresponds to the Coriolis force. Then, except for the centrifugal term, by defining A r ( ) 2 Ω = × as an effective vector potential in the Coulomb gauge, the Lagrangian can be rewritten as ⋅ which describes the dynamics in a nonrotating frame with a constant magnetic field applied in the normal direction while the elastic constant is given as the corrected value, and the relevant 'charge' of the particle is Q = 2M, without any 'Coulomb interaction' between a pair of particles. On the other hand, the presence of centrifugal force will disturb the system and break the translational symmetry of the lattice. However, since the centrifugal force has a quadratic dependence on the angular frequency of the rotation frame, its effect can be neglected for sufficiently small angular frequencies. More detailed discussion of the effect of the centrifugal force will be given later. With only the Coriolis force taken into account, the dynamic equations for the propagating modes can be written as As the angular velocity increases to 4 Hz, the band gaps broaden and the dispersion of the edge modes become more linear. When the angular velocity reaches to C M 3 /8 12.247 Hz, c Ω = ∼ the second and third band touch each other and the gap between them is closed. When the angular velocity is further increased, the gap reopens but is now converted into a trivial one, which does not contain any edge modes due to the band crossing, as shown in figure 3(d). Figures 4(a)-(d) show the bearded edge modes under different speeds of rotation. Similar to the zigzag edge, the bearded edge modes exist in every band gap, as the angular velocity is less than Ω c . Due to band inversion, topological phase transition occurs at the second and third bands after reaching the critical angular frequency.
We now address the influence of equilibrium position correction affected by centrifugal force. To estimate the effect of the centrifugal force, we consider a mass-spring ribbon with characteristic frequency 20 Hz 0 ω = located on a round plate with a width of ten periods (N = 10), a length two times the width, and a lattice constant a = 3 cm. The center of the ribbon is located at the center of the round plate. Under the approximation  , are the position vectors for particle A and B relative to the center of the strip. For angular velocity Ω = 2 Hz, the farthest particle from the center is shifted from its static position by 0.129 cm due to the centrifugal force. It has been shown that the relation between disorder and deviation of band gap is linked by an approximate formula~3 cen Δω ω or 0.017 cen Δω ω ∼ [24], where δ is the deviation parameter defined as a a, Δ Δω is the change of bandgap, and cen ω gives the central frequency of a certain gap. The gap widths for each band are 2.12, 4.01, and 2.12 Hz and their corresponding Δω are 0.05, 0.42, and 0.57 Hz, respectively. For this particular example, the change of bandgap is so small that it can be neglected while the rotating frequency is sufficiently low. Hence, as long as the angular velocity is much less than the characteristic frequency, the influence of equilibrium position shift is insignificant to the band structure. For operations at higher angular velocity, one can always set all particles at corrected positions in advance such that when a round plate rotates at certain angular velocity, all particles move back to the original ordered locations.
To confirm the topological order of edge modes, we have numerically evaluated the Chern number for each band. The formula for the Chern number is expressed asC In summary, we have introduced a simple classical mechanical system-a honeycomb lattice plane consisting of rigid bodies and soft springs, which, when positioned in a rotating frame, can have nontrivial frequency gaps that give rise to topologically protected edge modes. Due to the simplicity of the system, these Coriolis-force-induced topological vibrational modes can be readily observed experimentally for low-frequency oscillations.

Acknowlegments
The work is supported in part by EPSRC. PGL would like to acknowledge the National Science Council of Taiwan (Grant No. NSC 101-2112-M-008 -015 -MY3, NSC 103-2918-I-008-012). YTW would like to thank the Ministry of Education, Taiwan for the financial support.

Appendix. A generalized model including the transverse restoring force
In the main text, we assume transverse restoring force is too weak to be considered. However, due to the centrifugal force, all the springs are stretched and transverse force becomes more and more crucial with the increase of angular frequency. In this appendix, we will include this term to investigate its effect. If the restoring force caused by stretching a spring is to be considered, the original elastic potential energy ( ) Note that we only consider in-plane transverse displacement here. For out-of-plane, transverse vibration can be decoupled from the in-plane vibration modes [19], and hence we will exclude it in the following discussion. The modified Lagrangian becomes