arXiv:1701.03385v2 [cond-mat.mes-hall] 21 Aug 2017

A dynamically tunable topological mattress and its edge modes Ching Hua Lee,1, ∗ Guangjie Li,2 Guliuxin Jin,2 Yuhan Liu,2 and Xiao Zhang2, †

arXiv:1701.03385v1 [cond-mat.mes-hall] 12 Jan 2017

1

Institute of High Performance Computing, A*STAR, Singapore, 138632. 2 School of Physics, Sun Yat-sen University, Guangzhou 510275, China (Dated: January 13, 2017)

Of late, there has been intense interest in the realization of topological phases in very experimentally accessible classical systems like mechanical metamaterials and photonic crystals. Subjecting them to a time-dependent driving protocol further expands the diversity of possible topological behavior. We introduce a very simple experimental proposal for a mechanical Floquet Chern insulator using a lattice of masses equipped with time-varying electromagnets. This mechanical realization affords an extremely intuitive visualization of topological edge state dynamics that can be appreciated with just basic classical mechanics. We present a honeycomb lattice realization that contains minimally dispersive and hence extremely robust edge states. With key parameters controlled electronically, our setup has the further advantage of being dynamically tunable for applications beyond simple Floquet modulations.

Since topological concepts were first invoked to explain the curious formation of resistivity plateaus in quantum Hall systems[1–3], they have lead to the discovery of various novel condensed matter phases from topological insulators to Weyl semimetals and hourglass fermions[4–12]. Such topological states are characterized by robust boundary modes with potentially revolutionizing technological applications[13–19], and have attracted substantial efforts in their search. Traditionally, the main focus had been on electronic topological materials, but the requisite precise bandstructure engineering[20– 27] and sample control[28, 29] made realistic topological materials rather elusive. As such, there has been a recent paradigm shift towards more tunable realizations in optical lattices[30–33] as well as photonic[34–44] and acoustic/mechanical[45–52] systems. Macroscopic mechanical realizations of topological states are not only particularly experimentally accessible, but also possess unprecedented levels of tunability[53, 54]. Mechanical analogs of quantum anomalous Hall and spin Hall states have been respectively demonstrated in experiments involving lattices of gyroscopes[46, 47] and coupled pendula[45, 48]. The simplicity and ubiquity of such systems have expanded the realm of potential applications of topology to emerging technologies like heat diodes, adaptive materials, vibration isolation and acoustic waveguides[55–58]. The ease of tunability of a mechanical system also opens it up to further possibilities of topological phases through Floquet engineering. By dynamically varying one or more parameters in the system, one may implement a time-periodic potential that provides an extension of the periodicity of the Bloch states into the time domain. With an extra dimension in the configuration space, new topological invariants may be defined[59–62], leading to certain novel topological edge modes with no static analog[59, 63]. Indeed, chiral edge modes can emerge with appropriate Floquet modulation even if the undriven system is topologically trivial[64].

This has been demonstrated in the realization of the Haldane phase in Graphene through suitably designed irradiation[65–68], and is also purportedly useful for the realization of local and hence more realistic fractional Chern insulators (FCIs)[83]. In this work, we shall first introduce a mechanical realization of the simplest paradigmatic topological system, the SSH model. This, we will show, can be naturally generalized to a 2D mechanical honeycomb Chern lattice (mattress) harboring chiral edge modes whose detailed dynamics can be easily visualized and understood with basic mechanics. Building upon the particular robustness of this lattice, we next show how dynamical modulation on it can also give rise to chiral edge modes, some without static analogs. We provide a rigorous study on the exact mechanism behind this driving protocol and, most significantly, provide a realistic experimental proposal for the realization of this topological Floquet mattress with electromagnets connected to AC currents. Mechanical SSH lattice– To motivate the design of our mechanical Floquet lattice, we first describe how topological boundary states arise in a very simple mechanical version of the Su-Schrieffer-Heeger (SSH) model, the latter which has ubiquitously described systems from polyacetylene to Majorana wires[4, 69–72]. It consists of a semi-infinite chain of identical masses m connected by springs of alternating stiffness k1 and k2 (Fig.1). Suppose that k1 > k2 , where k1 is the stiffness of the spring connected to a fixed boundary support. There exists an exponentially localized boundary mode with displacements of masses taking the form ~y = (yA , 0, −t0 yA , 0, t20 yA , ...), where t0 = k2 /k1 < 1 and yA is the displacement of the mass connected to the boundary. This is an eigenmode because a stationary mass at position 2j experiences zero net force from exactly balanced restoring forces of k1 t0j−1 yA and k2 tj0 yA from either spring, and can thus remain stationary. The ensuing oscillation of q odd-numbered masses occur at frequency ω =

k1 +k2 m ,

2 which is a “mid-gap” mode well-separated from the continuum of bulk modes. To uncover the topological origin of this extremely simple boundary mode, we first write down Newton’s law ¨ = ω 2 ~y = K~y , where as an eigenvalue equation: −M~y the mass-normalized stiffness matrix M −1 K is the tightbinding “Hamiltonian”, and ~y is the vector of displacements. The key insight is that non-interacting topological phases do not require the non-commutativity of the density algebra, and can hence be defined in a classical configuration space. In the basis of odd/even (A/B) sublattices, the (momentum space) K matrix takes the form ~ · ~σ = (k2 + k1 cos p) σx + k1 sin p σy , where σx , σy are d(p) the Pauli matrices. Due to inversion symmetry, there can be no σz term and the eigenmodes define a mapping S 1 → S 1 from the periodic 1D Brillouin zone (BZ) to ~ 2 = 1. The regime t0 = k2 /k1 < 1 correthe circle |d(p)| sponds to a nonzero winding number of this mapping.

A

k1

Mechanical chiral mode dynamics– The abovementioned 2D SSH mattress can be driven into a Chern insulator phase with chiral edge modes by breaking timereversal symmetry. This is most easily done by introducing a Lorentz-type force F~B = γ ~r˙ × zˆ, or by introducing an explicitly time-dependent driving protocol. The former, which we shall first discuss, can either physically arise from a perpendicular magnetic field of strength B = γ/Q, where Q is the charge on a mass, or the reaction torque of a gyroscope attached to an oscillating mass. As detailed in the Supplementary online material (SOM) or Ref. 46, γ = IΨ h2 , where I, Ψ and h are the momentum of inertia, spin and length of a gyroscope respectively. Since magnetic forces are much weaker than electrostatic repulsion at nonrelativistic velocities, this gyroscopic interpretation is much more experimentally viable, and has indeed been successfully demonstrated[46]. Nevertheless, we shall adhere to the equivalent but more intuitive magnetic field interpretation in the following discussions. Together with F~B , the equation of motion of the masses

t2 yA

t2 yA

B

t vx

B

k2

k1

B A

k2 |y|2

A

k1

B

vx

vx

A B

A

vx

yA Γ=0.2 HNNNL

Γ=0.5 ÈΩÈ 2.0

Ω

-t yA

B

A

t0 = k2 / k1 yA

Γ=0

t vx

B

-t yA A

A

-t0 yA

yA

B

Chern

SSH

Γ=1.3

ÈΩÈ

1.5

ÈΩÈ 2.5

1.5

2.0

1.5 1.0

1.5

1.0

1.0

1.0 0.5

-3 -2 -1

One can generalize this 1D SSH system to a 2D mechanical lattice, i.e. mattress with analogously protected edge modes by adding an additional dimension such that the total vertical component of the spring stiffness between the A and B sublattices remain as k1 and k2 (Fig. 1 Upper panel). For instance, a 2D honeycomb extension with identical springs k and angle 2θ between the BAB masses (as illustrated) has k1 = k and k2 = k cos2 θ, where θ = π/3 for the rest of this paper. This 2D inversion symmetric system exhibit characteristic dispersionless edge modes inherited from the 1D SSH model, as shown in the γ = 0 case in Fig. 1. Just like their 1D counterparts, such edge modes consist of yˆ-polarized vibrations of A-type masses accompanied by a stationary B sublattice.

(t0)2yA

0.5

0.5

1 2 3

kx

-3 -2 -1 0 1 2 3

0.5 kx

-3 -2 -1 0 1 2 3

kx

-3 -2 -1 0 1 2 3

kx

FIG. 1: (Color online) Top panel) A 2D Chern mattress (right) may be derived from a 1D SSH mattress (left) by introducing an additional perpendicular (horizontal) dimension, such that the vertical components of the spring constants remain the same. The topological SSH edge state consist of stationary green (B-type) masses and vertically oscillating blue (A-type) masses with exponentially decaying amplitude from the boundary. The Chern mattress inherits similar mechanical behavior, but with the green masses oscillating horizontally as well, to balance the magnetic Lorentz force. Bottom) Dispersion plots of a regular honeycomb lattice with identical springs k = 1, masses m = 1, and Lorentz/gyroscopic coupling γ. The intensity of the red/blue lines represent the extent of localization of the left/right edge modes. At γ increase from 0, the SSH-type edge modes become chiral Chern edge modes with almost uniform dispersion. The third figure with γ = 0.2 was calculated with NNN couplings that break the inversion symmetry, lifting the degeneracy of mode of opposite momenta. The middle edge modes disappear at p γ > 3/2, when the ω > 0 bands acquire Chern numbers −1, 1, −1, 1, in ascending order.

takes the form M ~r¨ − iγ(σ2 ⊗ I)~r˙ + K~r = 0

(1)

in the space spanned by the phonon polarizations and the sublattices. Eq. 1 can be recast as an eigenvalue problem by rewriting it in the configuration space spanned by U = (~r˙, ~r). For each mode ~r at frequency ω, we have   iM −1 γσ2 ⊗ I −M −1 K ω U = −i U = Heff U (2) 1 0 which is unitary equivalent to the time-dependent Schrodinger’s equation considered in Ref. 73, but qualitatively distinct from that studied in Ref. 47, where the effects of the gyroscopes enter the mass term. Due to the presence of both first and second order time derivatives, the configuration space for Hef f has to be doubled.

3 yields[77] Fmag

vx

B

B

Fmag

vy

vx Frod

Frod Fspring

γ y˙ A +

Frod

Frod 2Frod

Fmag

FIG. 2: (Color online) Snapshots of the chiral edge state of the Chern mattress differing by a quarter cycle. Left) The downward force of the blue masses on the green masses are exactly canceled by the Lorentz force due to the latter’s motion, thereby leading to purely horizontal motion for the blue mass. Right) Analogous scenario with roles of blue and green masses reversed.

Indeed the effect of the Lorentz force/gyroscopic torque cannot be absorbed into an effective stiffness matrix, unlike the case of spin-orbit coupling appearing in the firstorder quantum mechanical Schrodinger equation. Eq. 2 can be diagonalized to yield phonon dispersion bandstructures, shown in Fig.1 for the regular honeycomb mattress with identical springs k. Before the introduction of a magnetic field (γ = 0), it exhibits SSH-type dispersionless edge modes, p with a similarly dispersionless bulk band at ω = 3k/m reminiscent of a Landau level with cyclotron orbits. With the introduction of γ, the edge modes become chiral-propagating modes linking bulk bands with nonzero total Chern numbers. Landau level-like √ flat bands also appear at special γ = 1/2 and γ = 3/2. Given a bulk gap, we define P to be the projection operator onto all bands below the gap. From well-known spectral flow arguments[59, 74–76], there exR 1 ist C = 2πi T r[P dP ∧ dP ] edge modes traversing this gap. In our case, there are eight bulk bands, with only the two bands of largest |ω| having Chern number C = ±1. When extra next-nearest neighbor hoppings are added to break the inversion symmetry, modes traveling in opposite directions are no longer degenerate. This is evident from the third bandstructure plot in Fig. 1, where a doubling of the edge modes (and less conspiciously, the bulk modes too) can be observed. Our SSH model-inspired Chern mattress possess edge modes that behave in a particularly simple way. As illustrated in Figs. 1 and 2, an uniform edge mode (at the Γ point) consists of A-type (blue) masses moving vertical, and B-type (green) masses moving horizontally. Although oscillations as such are attenuated in the bulk, they survive near the edge due to the interplay of the SSH mechanism and the Lorentz force. Elementary application of Newton’s law on masses A, B in both directions

3k xB = 0 2

3k yA,n 2 3k m¨ xB = − xB 2 γ x˙ B,n+1/2 = k(1 + t)yA m¨ yA,n = −

(3) (4) (5) (6)

where xA and yB are the horizontal and vertical amplitudes of the oscillations of masses A and B respectively, and t the attenuation per unit cell into the bulk. t agrees with t0 for the SSH system at zero Lorentz force coupling γ = 0. From these equations, the edge mode is seen to exist due to the balance between the spring restoring forces and Lorentz forces: mass A moves only vertically because any horizontal force due to the magnetic field is canceled by the π/2 out-of-phase motion of mass B; reciprocally, mass B moves only horizontally because its Lorentz force and the spring restoring forces from all three A masses around it conspire to cancel. It is important to realize, however, that this seemingly intricate balance is actually topologically robust, existing continuously over a large range of spatial modulations px [77]. In this simple mechanical picture, the breakdown of the topological edge mode occurs when |t| = 1, i.e. when an overwhelmingly large oscillation amplitude from within the bulk is required to maintain the balance of forces. |t| can be easily solved from Eq. 6 to be 1 γ2 − (7) mk 2 which impliesqthat a topological phase transition oc3mk curs at γ = 2 . There the ratio of the amplitudes q 2γ 2 xB yA = 3mk = 1, which is manifestly at the limit of the compensatory ability of the Lorentz force. A more detailed analysis that takes into account spatial modulations yields[77] r √ 3k 3 γk ω(px ) = − px + O(γ 2 )p2x (8) 2m 2 3mk + 2γ 2 |t| =

with very√ small higher power corrections to the group veγk locity − 23 3mk+2γ 2 . This leads to very weakly dispersive edge modes that can travel robustly over long distances and across sharp bends. q When γ is further set to around

the special value γ = mk 2 , |t| vanishes for px = 0 and the edge mode is perfectly localized at the edge, involving only the edgemost A and B masses. Such modes possess superior robustness in the presence of spatial disorder, and can circumnavigate complicated paths without disintegration (Fig. 3). Realistic Floquet topological mattress– More interestingly, chiral edge modes can also be realized by subjecting a (non-magnetic) mattress to suitably chosen timedependent modulations. We shall introduce a remarkably

4

zHtL

G=1,W=4

G=1,W=3.1

Ω

Ω 3.0 2.5

1.5

FIG. 3: (Color online) The robust propagation of a welllocalized and minimally dispersive chiral edge mode on the honeycomb topological mattress, with white regions being topologically trivial. The intensity in red corresponds to vibration amplitude. Due to the near-perfect localization, the edge modes do not interfere even along thin regions a few unit cells across.

simple experimental proposal for such, based on the previously mentioned honeycomb lattice. The combination of proximity to an SSH phase, minimal dispersiveness and excellent locality of its edge modes make it particularly immune to imperfections. The set-up consists of a honeycomb lattice of mobile electromagnets connected by springs k, such that each mobile electromagnet is surrounded by three fixed electromagnets at relative displacements dˆ nj = d(− sin 2πj/3, cos 2πj/3)T , j = 1, 2, 3. (Fig 5). When subject to AC currents, these electromagnets acquire time-dependent magnetic moments and hence fluctuating attractive or repulsive forces. A notion of chirality can be introduced by synchronizing the currents such that the moments of the fixed solenoids oscillate at a Floquet frequency Ω with relative phase offsets. Assuming a symmetric distribution of these offsets, this adds a time-dependent part Kt (t) to the stiffness matrix:   3 X Kt (t) = Γ Re  e2πij/3 [ˆ nj n ˆ Tj ]eiΩt  ⊗ I j=1

 3Γ  = − Im (σx + iσz )eiΩt ⊗ I 4

(9)

which corresponds to a potential Ut (t) = 21 ~uT Kt (t)~u = u|2 cos(Ωt + 2φ), φ the polar angle of the displace− 3Γ 8 |~ 3µ M M 0

0 0 0 ment ~u. Here Γ = in the limit of small osπd5 cillations, with d the solenoid separation and M0 , M00 the magnetic moment amplitudes of the mobile and fixed solenoids respectively. In other words, a fixed point ~u is “swept” by a sinusoidal potential moving with frequency Ω. This has the effect of nudging the mass in the direction of the sweep, since the mass generally receives more impulse when traveling in the direction of the sweep than vice versa. To see this more rigorously, we consider a simplified 1D version of this problem, where φ(t) is replaced by z(t) moving on a straight line:

3Γ|~u|2 sin(2z(t) + Ωt) 4 (10) which has an analytic solution (plotted in Fig. 4) is given
 Ω 0 by z(t) = − Ωt 2 + J 2 − z (0) t, A , where J[w, A] is m¨ z (t) = −∇z Ut (Ωt + 2z(t)) = −

4

2

2.0 1.0

t 1 2 3 4 5 6 7W

1.5 1.0

0.5

-2

0.5 -3 -2 -1 0 1 2 3

kx

-3 -2 -1 0 1 2 3

kx -4

FIG. 4: (Color online) Left) Chiral edge mode of the Floquet mattress with Γ = 1 and Ω = 4 in units of k, with the intensity of red corresponding to vibrational amplitude. Middle plots) Quasifrequency dispersions with Floquet frequencies Ω = 4 and 3.1, with the latter case exhibiting inter-floquet BZ topological modes that have no static analog. Right) Exact solutions to the simplified 1D model of a time-modulated spring stiffness (Eq.10), with intial velocity z 0 (0) = 1 opposite to the apparent motion of the potential. m = 1, 0.5, 0.45, 0.25, 0.1, 0.01 from top to bottom and the rest of the parameters are set to unity. A large mass (m = 1) essentially continues to move affected, but smaller mass are yanked significantly by the potential. Very light masses (m = 0.01) are overwhelmingly carried by the potential, and oscillate at the same period Ω.

the Jacobi amplitude i.e.

R J[w,A] 0

√

dt 1−A2 sin2 t

= w, and

2

12Γ|~ u| A = m(Ω−2z 0 (0))2 . The mass is ”pushed” by the potential when the latter is increasing and therefore, in its moving frame, experiences a potential with modified periodicity A EllipticK[ A−1 ] 8π √ . This reduces to T = 2π T 0 = Ω−2z 0 (0) Ω in 1−A the limit of large mass or small A, where the mass is hardly movable. It diverges at Ω = 2z 0 (0), where the mass is able to follow the moving potential exactly. To see that our periodically driven mattress indeed possess chiral edge modes, we employ the Floquet approach which gives the effective Hamiltonian for a outof-equilibrium system averaged over one cycle. Consider a generic Hamiltonian with periodicity T = 2π Ω , i.e. H(t) = H(t + 2π/Ω). In analogy to Bloch’s theorem, the eigensolutions to the Schrodinger’s equation (H(t) − i∂t )|Φ(t)i P = 0 (Eq. 2) can be decomposed as |Φ(t)i = eit m eimΩt |mi, where φm are the Fourier components of the time-periodic part |Φ(t)i, and  is its Floquet quasienergy which lives in the ”energy” Brillouin zone (BZ). Substituting this decomposition into the Schrodinger’s equation and integrating over a period, one obtains the (time-independent) effective Floquet Hamiltonian[78] X HF = (Hm−m0 + mΩδmm0 ) |mihm0 | (11) mm0

RT where Hj = T1 0 H(t)eijΩt dt are the Fourier components of the original time-dependent Hamiltonian. The

5 Floquet eigenenergies, which we shall also denote as ω for vibrational systems, are the eigenvalues of this HF defined with an additional Floquet dimension. Alternatively, Eq. 11 may be viewed as a Wannier-Stark problem with Ω taking the role of a constant field strength, and Fourier terms Hm−m0 6=0 causing jumps between different |mi. Solving for the Floquet dispersion with the timedependent stiffness matrix given by the zero-field (γ = 0) honeycomb K-matrix plus Kt (t), we obtain plots as shown in Fig. q 4. At sufficiently large driving frequency k , the various copies of the bandstructure i.e. Ω = 4 m are well-separated and the dispersion resembles that of a static system with nonzero γ, with analogous chiral edge modes connecting the bands. As shown in the leftmost panel, the edge mode consists of masses literally ”pushed along” by the periodic modulation on an otherwise dispersionless band. In the rightmost panel, the driving frequency is lowered qto twice of that of the ”optical” bands

k , showcasing the interplay between around ω = 1.5 m the two frequency scales. Interestingly, additional edge modes are seen connecting different copies of the Floquet bandstructure, even though no edge modes connect the optical bulk bands when they are well-separated. These emergent edge modes are the result of the topological phase transition due to the hybridization of the optical bands, and have no static analog. It is worth pointing out that the notion of frequency becomes ill-defined when |ω| > Ω/2, since processes happening below that timescale have been integrated over. An experimental setup for the Floquet topological mattress is achievable with basic lab equipment. With an AC current source of amplitude 2.5A connected to N = 400-loop solenoids of radius R = 2cm, we obtain magnetic moments M0 , M00 = πI0 N R2 = 1.26A · m2 at their maximum. These moments give rise to effective 3µ M M 0 spring constants amplitudes Γ = 0πd05 0 = 6.1N/m, which can be matched with real springs of similar stiffness to reproduce Fig. 4. This is a realistic stiffness for springs of around a = 20cm long, the lattice spacing between the solenoids they connect. Furthermore, we also require that the electromagnets between different sites interact negligibly. This is easy, since Γ falls off as the inverse fifth power of distance. Setting d = 5cm (Fig.5), for instance, we find that the electromagnets between neighboring sites couple with a negligible (5/20)5 ≈ 0.1% strength compared to those producing the dynamical driving. Finally, with mobile electromagnets of about m = 230g, we should q expect topological edge modes at 3k ≈ 1Hz, which are easily observfrequencies of ω ≈ 2m able by an oscilloscope or even the naked eye. Conclusions.– From a mechanical SSH model, we have built a honeycomb Chern mattress with very easily visualizable edge modes that are also extremely robust. By

𝑘1

a 𝑘2′

𝑘3′ 𝑀3

𝑀2

𝑀2 𝑀0

𝑘2

𝑘3 𝑘1′

d

𝑀1

FIG. 5: (Color online) Left,Middle) Top/3D configuration of the mobile electromagnet (red) at each site surrounded by fixed electromagnets (black). The mobile electromagnets are mutually connected by springs, and take the role of the masses of the mattress. Right) Arrangement of each of these electromagnetic setups in the honeycomb lattice.

removing its gyroscopic coupling and introducing dynamical modulation through AC electromagnets, we arrive at a very experimentally realistic Floquet topological Chern insulator (mattress) whose chiral edge dynamics can be directly observed at macroscopic scales. Acknowledgements. Xiao Zhang is supported by the National Natural Science Foundation of China (No.11404413) and the Natural Science Foundation of Guangdong Province (No.2015A030313188).

∗ †

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Supplemental Online Material for “A dynamically tunable topological mattress and its edge modes” In this supplementary material, we detail: 1) The mechanics of the restoring torque from gyroscopes and 2), The easily visualizable edge mode behavior on a gyroscopic/Lorentz force Chern mattress at arbitrary wavevector px .

EQUATION OF MOTION OF MASSES WITH GYROSCOPES ATTACHED

Gyroscopes attached to moving masses provide an experimentally accessible route to the realization of large Lorentztype forces[46] for topological chiral edge modes. Each mass is attached to the free end of a gyroscope that its fixed below its equilibrium position. These gyroscopes break time-reversal symmetry by providing a “reaction torque” in response to movements of the mass. Consider a gyroscope with moment of inertia I and angular spin speed of Ψ attached to a mass moving with velocity r r is taken to be the pivot of the gyroscope. The rate of change of the gyroscopic angular ~r˙ = d~ dt , where the origin of ~ ~ r) dL momentum L = IΨˆ r is dt = d(IΨˆ = IΨ~r˙ /|~r|, which must be compensated by a reaction torque ~r × F~react . Assuming dt small oscillations so that ~r is almost perpendicular to the plane of oscillation, we find that IΨ F~react = −i 2 σ2~r˙ h

(S1)

where −iσ2 implements a π/2 rotation, and h is the length of the gyroscope. In the space spanned by the phonon polarizations and the sublattices, the equation of motion of the masses hence take the form M ~r¨ − iγ(σ2 ⊗ I)~r˙ + K~r = 0

(S2)

where ~r denote the small displacement of each mass about its equilibrium. Here M = I ⊗ diag(m1 , m2 , ...) is the mass matrix consisting of masses m1 , m2 , ... within each unit cell, K is the stiffness matrix, and γ = IΨ h2 is the gyromagnetic ratio. Note that Eq. 1 is of the same form as that of charges q in a magnetic field B, if γ is replaced by qB. In other words, the gyroscopic reaction force behaves exactly like the electromagnetic Lorentz force, at least for small displacements. Eq. S2 can be rewritten in first-order form cf. Eq. 2, which is unitarily equivalent to the alternative phase space formulation of a related problem in Ref. 73. Also, a qualitatively different gyroscopic coupling had been considered in Ref. 47, where the gyroscopic spin depending dynamically with the phonon mode, resulting in a gyroscopic term second order in the time derivative. S1. ANALYSIS OF EDGE STATE OF GYROSCOPIC HONEYCOMB LATTICE

Consider a gyroscopic honeycomb lattice of identical masses m connected by identical springs with stiffness k. Consider the case of a zigzag edge (Fig.1) with A-type masses sitting on the “dangling bonds”. Denote the edge direction as being along x ˆ, so that the momentum px is still a good quantum number. An edge eigenmode is characterized by a decay factor |t| = e−1/L < 1, such that the amplitudes of its oscillations decay exponentially perpendicularly from the edge like |t|y = e−y/L . As explained in the main text, the edge modes of a honeycomb lattice consists of A-type masses moving around in small ellipses, and B-type masses oscillating entirely along x ˆ. We expand an edge mode in terms of plane waves indexed by momenta px , with phase velocity given by ω/px .

S2 For each unit cell n along the x ˆ direction, denote the displacements of masses A and B from their equilibria due to a px -eigenmode by ~rA,n = (xA,n , yA,n )T = (xA sin(px n − ωt), yA cos(px n − ωt))T

(S3a)

~rB,n = (xB,n , yB,n )T = (−xB sin(px n − ωt), 0)T

(S3b)

i.e. the A atoms make small clockwise elliptical oscillations while B atoms strictly vibration in the x ˆ plane. The honeycomb lattice consists of A and B sites displaced by half a unit cell spacing in the x ˆ direction. In addition to ˙ × zˆ, where γ is the gyroscopic coupling. By the spring restoring forces, each mass experiences a Lorentz force of γ ~r balancing the accelerations and forces on masses A and B in the x ˆ and yˆ directions respectively, we obtain  3k  (xB,n−1/2 − xA,n ) + (xB,n+1/2 − xA,n ) 4 √  3k 3k  = −γ x˙ A,n − yA,n − (xB,n−1/2 − xA,n ) − (xB,n+1/2 − xA,n ) 2 4 √  3k  3k = (xA,n − xB,n+1/2 ) + (xA,n+1 − xB,n+1/2 ) + [(yA,n − 0) − (yA,n+1 − 0)] 4 √ 4  3k  k (xA,n − xB,n+1/2 ) − (xA,n+1 − xB,n+1/2 ) = [(yA,n − 0) + (yA,n+1 − 0)] + k[tyA,n+1/2 ] + 2 4

m¨ xA,n = γ y˙ A,n +

(S4)

m¨ yA,n

(S5)

m¨ xB,n+1/2 γ x˙ B,n+1/2

(S6) (S7)

Since it was assumed that masses of type B do not move in the yˆ direction i.e. yB = 0, there is no Lorentz force contribution to m¨ xB in line 3. This further implies that in the yˆ direction, the Lorentz force exactly cancels the spring restoring forces on a mass of type B (line 4). Each of such a mass couples to exactly one A-type mass situated an unit cell further from the edge, whose oscillations are attentuated by a factor of ±t, the ± sign allowing for a possible reversal in oscillation direction. Simplifying the above, we obtain (ω 2 − ω02 )ηA = −Γω + ω02 ηB cos

px 2

ω2 px ω 2 − ω02 = −ΓωηA − √0 ηB sin 2 3 2 ω px px − √0 sin (ω 2 − ω02 )ηB = ω02 ηA cos 2 2 3  2  2 ω ω px px ΓωηB = 0 cos ± 2t + √0 ηA sin 3 2 2 3

(S8) (S9) (S10) (S11)

q γ 3k B where ηA = xyAA , ηB = xyA , Γ = m is the normalized gyroscopic coupling and ω0 = 2m denotes the resonant frequency of each tri-bond. Eqs. S8 to S11 can be simultaneously solved to yield ω, ηA , ηB and L in term of px and Γ. In particular, ω(px ) agrees exactly with the numerical dispersion of the edge modes displayed in Fig.1. Long wavelength (px = 0) limit

It is instructive to study the spatially homogeneous edge modes, i.e. those with px = 0 (at the Γ-point). In this limiting case Eqs. S8 to S11 reduce to (ω 2 − ω02 )ηA = −Γω + ω02 ηB 2

ω − 2

(ω −

ω02

ω02 )ηB

(S12)

= −ΓωηA

(S13)

ω02 ηA ω02

(S14)

=

ΓωηB =

3

(1 ± 2t)

(S15)

2 2 2 2 In particular, q Eqs. S12 to S14 can be combined to yield (ω − ω0 )(ηA − ηB − 1) = 0, with an obvious solution 3k ω = ω0 = 2m , ηA ∝ xA = 0. This corresponds to type-A and B masses oscillating orthogonally, along the yˆ and x ˆ

S3

vx

vx

vy

vx

vx vy

FIG. S1: (Color online) Snapshots at every quarter cycle of the chiral edge mode oscillation at the Γ point (px = 0). The Lorentz force on each mass cancels the exactly synchronized restoring forces from the neighboring masses, thereby maintain strictly vertical/horizontal motion for the A/B-type masses.

axes respectively. From Eq. S12, their relative oscillation amplitudes are given by r xB γ 2γ 2 = ηB = = yA mω 3mk

(S16)

In other words, with nonzero gyroscopic coupling γ (or magnetic force), mass B has to oscillate horizontally to compensate the Lorentz force on mass A. This can also be easily seen by equating the Lorentz force on the latter, which is proportional to γωyA , with the x ˆ-direction inertia force from b, which is proportional to mω 2 xB . Together with Eq. S15, the result Eq. S16 gives the edge mode decay length as −1  −1  2 2 γ |3ηB − 1| 1 L = −(log |t|)−1 = − log − = − log 2 mk 2

(S17)

2 2 A topological phase transition occurs when the gap closes and L diverges, which occurs at 2γ − 1| = 2. − 1 = |3ηB mk In particular, this means that the gap around ω = ω0 becomes topologically trivial at sufficiently large gyroscopic q

coupling γ i.e. γ > 3mk 2 , when the Lorentz force is too strong to be compensated by an oscillatory mode decaying into the bulk (Eq. S15). Such a phase transition arises due to the interplay between the magnetic force and the lattice springs, and does not exist in a continuum quantum Hall system. Interestingly, q it is also possible to tune γ such that the edge mode is perfectly localized at the edge. At the special value γ = mk 2 , L = 0 and there is totally no vibration beyond the boundary A and B sites. This occurs when the Lorentz force on mass B can be completely compensated by the restoring motion of the edgemost A masses alone.