Supporting Information

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1 Supporting Inforation Nash et al /pnas Equation of Motion If a gyroscope is spinning with a large constant angular frequency, ω, around its principal axis, ^l, then its dynaics are deterined priarily by effect that forces have on its angular oentu. If the gyroscope axis is pivoted at one end and is subject to a force, ~ F at a position ~l fro the pivot as illustrated in Fig. 1 of the ain text, then the gyroscope feels a torque ~τ = d~ L dt = ~l ~ F. The resulting otion is _~l Iω l = ~l ~ F _~l = l2 Iω ^l ~ F. [S1] For sall displaceents fro the vertical (^z) axis (sall tilts), the z-coponent of ^l is changed only to second order. Thus, the orientation of the gyroscope can be represented by a coplex nuber, ψ = δx + iδy. When a gyroscope is subject to a restoring force (e.g., gravity), F = k 0 ðδx + iδyþ, it will precess in a circular orbit at frequency Ω 0 = k 0 l 2 =Iω. For gyroscopes subject to a gravitational force, the effective spring constant is k g = g=l c, producing a precession frequency of Ω g = gl c =Iω, where l c is the distance fro the fixed end to the center of ass. Linearized Equation of Motion To find the linearized equation of otion, we consider first an interaction of two gyroscopes p and q, with p being at the origin and q a distance a fro p at an angle θ pq, as illustrated in Fig. S1. The effective spring constant for the interaction is given by the gradient of the force. In general, these gradients ay be asyetric in the coponents that are parallel and perpendicular to vector fro the bond, which is defined as the line connecting points p and q. In the case of a linear spring with constant k 0, for exaple, the spring constant for displaceents along the bond is k 0 whereas the spring constant for displaceents perpendicular to the bond is 0. As an illustration, we now consider the linearized equation of otion for gyroscopes p and q interacting via this linear spring. In this case, we need only to find the displaceent parallel to the bond (Δ k ) and ultiply it by the spring constant k 0 to find the linearized force. We ay write the unit vector fro p to q as e iθpq in coplex for. The parallel coponent Δ k can be found by first rotating the bond to the x axis, taking the real part of ψ p ψ q (illustrated in Fig. S1) and rotating the bond back to its original position; the resulting force in coplex for is given by i F pq = k 0 e iθpq Re he ψ iθpq p ψ q [S2] = k i 0 hψ 2 p ψ q + e 2iθpq ψ* p ψ* q. [S3] For a lattice, there is a su over the nearest neighbors of each gyroscope and the equation of otion becoes i dψ p = Ω g ψ dt p + Ω k 2 Xn. n. i hψ p ψ q + e 2iθpq ψ* p ψ* q, [S4] q where Ω k = k 0 l 2 =Iω and the cross-product fro Eq. S1 has resulted in the iaginary coefficient on the left hand side of Eq. S4. For a general radial interaction force, ~ F = FðrÞ^r (e.g., agnetically coupled gyroscopes), the equilibriu positions can result fro the cancellation of opposing forces instead of the absence of forces. In this case, we obtain the following equation of otion: i dψ p = Ω g ψ dt p Xn. n. h i Ω + p ψ p + Ω + q ψ q + e Ω 2iθpq p ψ* p + Ω q ψ* q, q [S5] where Ω ± j = l 2 =Iωð F pk = x jk ± F p = x j Þ. For a radially syetric potential of the for ~ FðrÞ = kr n^r, this results in Ω ± p = kl 2 =Iωðn ± 1Þa n 1 and Ω ± q = Ω± p,whereaisthe separation between the two lattice sites. In our experient, the gyroscopes are coupled through sall agnets. The force can be approxiated by treating each gyroscope as a agnetic dipole with strength M; this produces an r 4 radial force between the gyroscopes plus an antirestoring torque fro the agnetic interaction. The total effective force gradients are given by F pk = k 1 a2 x pk 12l a2 F p =+ k x p 4 ; 3l 2 ; F pk =+k 1 + a2 x qk 6l 2 F p = k 1 + a2 x q 4 3l 2, [S6] where k = 3μ 0 M 2 =πa 5 is the agnetic characteristic spring constant, corresponding to a gyroscope precession frequency of Ω = k l 2 =Iω, and we have converted torques between the agnetic dipoles to equivalent forces that depend on the ratio of lattice spacing (a) to pendulu length (l). In a honeycob lattice, the syetry of the lattice allows for the equations of otion to be siplified to i dψ p = Ω ψ g dt p X n.n. q hω + i ψ p ψ q + Ω e 2iθpq ψ* p ψ* q, [S7] where Ω g = Ω g ð3a 2 =8l 2 ÞΩ and Ω ± = ½1 + a 2 =6l 2 ð1=4 + a 2 = 12l 2 ÞŠΩ. Here we see that the agnetically coupled syste is different fro the spring-coupled lattice in two ways: The effective pinning frequency, Ω g is decreased, and there is an asyetry between the Ω + and the Ω ters. The equation of otion for the agnets, Eq. S7, is nearly equivalent to the siple linear spring case, excepting the slight asyetry between the ψ and ψ p ters. (Note that for springs, the pairwise force between the two sites is 0 at equilibriu, which results in Ω ± p = Ω k and Ω ± q = Ω k and recovers a sipler equation of otion, Eq. S4.) The asyetry of these ters has an interesting effect when viewed fro the perspective of tie-reversal syetry breaking: The ψ p ters couple forward and backward propagating odes, and so when Ω > Ω + the syetry breaking ters are relatively stronger. As a result there is a wider band gap for the agnetically coupled gyroscopes in coparison with those coupled by linear springs. Experiental Details The gyroscopic etaaterial was constructed fro 54 hanging gyroscopes, as shown in Fig. S2. Each individual gyroscope consisted of a 3D printed cylindrical rotor (radius 10.4 ) heat 1of10

2 fitted to the shaft of a sall (20 long) brushed dc pager otor. An N40 neodyiu agnet (9- diaeter, 4- thickness) was ebedded with its oent aligned with the z axis in each rotor. The spinning ass of each gyroscope rotor was 6.1 g. The dipole oent of the agnets was deterined fro direct easureent of the force between attracting agnets (with dipole oents aligned) to be M = 0.22 ± 0.02 A 2. Each spinning ass was enclosed in a housing that was glued to the otor shaft and suspended by a weak spring. When the end of the gyroscope is displaced, the bending of the spring gives an effective pendulu length of l = 40 ± 2. The spacing between gyroscopes in the experiental lattice was 30.5, corresponding to an effective spring constant that is roughly equivalent to the agnitude of the effective gravitational pinning spring, Ω Ω g. Fixed agnets (N52, 10- o.d., 5.5- i.d., 3 thick) were placed at the perieter of the syste in the position of each edge gyroscope s issing neighbor. These fixed agnets were approxiately two-thirds the strength of the inner agnets. We note that this detail is not crucial to the topological nature of the syste. The strength of the boundary agnets does not change the frequencies of the gap edges (which depend on the bulk properties), and only slightly affects frequency distribution of odes within the gap. We obtained the experiental noral odes for the frequency range Hz in frequency steps of 0.02 Hz. At each frequency, the syste was excited at one lattice site for 80 s before recording began and subsequently recorded at the excitation frequency for 100 s. We found the elliptical orbits for each gyroscope using the Fourier transfors of the position vs. tie data. Linearity of the Magnetically Coupled Syste It has been shown experientally that lattices of interacting agnets can exhibit nonlinear behavior when excited at sufficiently large aplitudes (33). To ensure that our experients were perfored in a regie where linear analysis can be applied, we tested for the presence of nonlinear effects with three experients that are suarized here and listed in detail below. (i) We characterized the shift in the noral ode frequencies of a pair of interacting gyroscopes as a function of the excitation aplitude. (ii) We verified that the frequency of an edge ode in our syste of 54 interacting gyroscopes does not shift significantly as the aplitude of excitation was changed. (iii) We easured speed of the wave packet shown in Movie S1 and verified that the change in aplitude does not significantly affect the average speed of the wave packet. Our easureents of the effects of nonlinearities (detailed below) cover a range of excitation aplitude beyond the axiu excitation aplitude in the experients of the ain text. i) Fig. S3 shows the easured noral ode frequencies of a pair of interacting gyroscopes, each with two neighboring boundary agnets in a honeycob configuration (as described in Experiental Details). Two noral ode frequencies are expected: a lower frequency ode in which the gyroscopes precess in phase (red dots in Fig. S3) and a higher frequency ode in which the gyroscopes precess out of phase (green dots in Fig. S3). The frequencies of both odes were tracked as the aplitude of an excitation daped during an interaction. The nonlinearity of the interaction is apparent because the ode frequencies are not constant with aplitude. However, at the upper liit of excitation aplitude of the experients presented in the ain text (3 ), the deviations of the frequency are not appreciable. At this aplitude, a frequency shift of only less than 2% was observed for either of the odes. We note that the frequency shift reaches a axiu of 5% for the antisyetric ode when the excitation aplitude reaches 4. ii) The frequency shift of a gap ode was easured at varying excitation aplitudes. For each test a region of 0.06 Hz was tested near a known peak, and each frequency was excited for 80 s before recording and subsequently recorded for 100 s, as described previously. We deterined that for changes in excitation aplitude ranging fro 1.2 to 4.0, the shift in frequency was less than 0.02 Hz. An exaple of the effect is shown in Fig.S5B. The frequencies for the fourth ode nuber are plotted for three aplitudes in the range indicated above. iii) We tracked the position of the edge ode wave packet shown in Fig. S4 and Movie S1 as the aplitude of oscillation decayed. The position of the wave packet was found in each frae using a center-of-ass ethod, in which each lattice point was weighted by the squared displaceent of its gyroscope. As shown in Fig. S4, the average speed of the wave packet is unaffected by the gradual decay in aplitude. Furtherore, we do not observe ode ixing at the aplitudes of the data presented in this work. Noral Mode Analysis For second-order ass and spring systes with n lattice sites in two diensions, we find the noral odes by considering the syste of equations ~ X = K ~ X, [S8] where X ~ avectorofdiension2ncontaining the x and y displaceents of each ass in the network and K is a coupling atrix between sites. The noral odes are found by finding the eigenvalues, Ω 2, and eigenvectors, ~ϕ, of the K atrix: 1 K Ω 2 I ~ϕ = 0. [S9] To find the noral odes of a finite syste of gyroscopes we consider the linearized equations of otion, Eq. S5. As with the spring and ass syste, these equations of otion can be expressed as a atrix: i _ ~ Z = K ~ Z, [S10] where ~ Z again has diension 2n; the basis states are of the for ψ p = Ae iωt + B p e iωt. The noral odes are given by the eigenvalues and eigenvectors of the K atrix: ðk ΩIÞ~ϕ = 0. [S11] To accurately odel the experiental syste, we included the effects of the weaker boundary agnets. First, an energy iniization was perfored on a honeycob lattice with 54 agnetically interacting points surrounded by weaker boundary agnets as described in Experiental Details. The energy iniization gave the equilibriu points that were used in the finite-syste noral ode calculation. The spring constant between gyroscopes was given by k = 3μ 0 M 2 =πr 5, where r was the separation between interacting gyroscopes. The ode frequencies were calculated by assuing coupling of individual gyroscopes with all gyroscopes in the lattice as well as with only nearest-neighbor coupling. A coparison between the expected gap odes fro the odel and the easured edge odes in the experient is shown in Fig. S5. The shaded regions indicate the range of possible gap ode frequencies calculated with the easured experiental values M = 0.22 ± 0.02 A 2 and l = 40 ± 2. The red shaded region was calculated with entire lattice coupling, and the blue region was calculated with only nearest-neighbor coupling. The diaonds in Fig. S5 were calculated with entire lattice coupling and odel paraeters M = 0.21 A 2 and l = 38. 2of10

3 We note that disorder (nuerical and experiental) can greatly alter ode profiles. We find that experientally observed ode profiles are qualitatively siilar to nuerically calculated ode profiles in a syste with rando disorder, as shown in Fig. S6. Tie Doain Siulation We siulate a 2D gyroaterial in the tie doain only for the case of spring interactions with free boundary conditions. We nuerically integrate Eq. S1 considering a spring interaction and gravity. We consider only the x and y displaceents and integrate using a fourth order Runge Kutta ethod. Band Structure and Chern Nuber Calculation We find the band structure for both the spring and agnetically coupled systes on a honeycob lattice using the linearized equations of otion, and assuing the solutions ψ a = Ae i ~k ~x ωt + Ce i ~k ~x ωt [S12] ψ b = Be i ~k ~x ωt + De i ~k ~x ωt, [S13] where a and b refer to the two sites in each unit cell. The resulting equations can be expressed as a 4 4 atrix that is a function of the wave vector, ~ k. (For a 2D lattice the diension of this atrix will be 2n 2n, wheren is the nuber of lattice sites per unit cell.) The four eigenvalues of this atrix give the values of the four dispersion bands at a particular value of ~ k. These dispersion bands correspond to the frequencies obtained fro the finite noral ode analysis. At each value of ~ k, each band has a corresponding eigenvector, ju j ð ~ kþi, which corresponds to the aplitudes of the clockwise and counterclockwise rotating odes on the two lattice sites at that particular value of ~ k. The Chern nuber of each band is given by an integral of the Berry curvature Fð ~ kþ: C j = 1 Z d 2 kf j ~k 2π = i 2π I A j ðkþ dk, [S14] where A j ðkþ = ihu j j k u j i. In this work, Chern nubers are calculated nuerically using a phase invariant forula (27): C j dx dy = i Z d 2 ktr dp j P j dp j, [S15] 2π where P j is the projection atrix defined as P j = ju j ihu j j, and is the wedge product. Mapping to the Haldane Model Our experient is perfored in the regie where Ω g Ω k.however, it is interesting to note that in the weak spring liit, Ω k Ω g, our syste reduces to the Haldane odel (26). In this liit, the equations can be siplified considerably and are aenable to analytical treatent. In particular, the Chern nuber can be deterined analytically as a function of the angle in the honeycob lattice deforation. In the weak spring liit, the otion of each gyroscope is approxiately circular, so there is only one degree of freedo per gyroscope. This can be seen by splitting the displaceent of the gyroscope two polarizations, ψ n = e iωt u n + e iωt v n *, where u n is the aplitude of precession in the direction deterined by gravity, v n is the counter rotating aplitude, and Ω is the frequency of precession for the ode. Because Ω g Ω k, the gravitational precession direction doinates (ju n jjv n j) and all ode precession frequencies, Ω, differ only slightly fro Ω g. Substituting this for of ψ into the linearized equation of otion and atching the coefficients of the exponentials gives Ωu n = Ω g u n Ω X k ðu n u Þ Ω X k ðv n v Þe 2iθn, Ωv n = Ω g v n Ω X k ðv n v Þ Ω X k ðu n u Þe 2iθn. [S16] [S17] Perturbationtheorycanbeusedtofindanequationforthe u s alone, which is equivalent to Haldane s odel of the quantu Hall effect. Because jv n jju n j,eq.s17 iplies v n Ω P k 4Ω ðun g u Þe 2iθn. Substituting this in Eq. S16 gives X ωu n = Ω g u n + Ω k ðu n u Þ Ω2 X k ðu n u Þe2iθ n 8Ω g + Ω2 k X ðu u l Þe 2iθ nl, 8Ω g l [S18] where θ nl = θ n θ l is the angle between the bonds n and l, and the second-to-last su is over all pairs of neighbors and of n, and the last su is over all neighbors of n and neighbors l of. If one expresses the right side as P T nu, then finding the noral odes is the sae as finding the band structure of electrons on a lattice with hopping aplitudes T n between the sites of the lattice. (The large value of Ω g iplies that the polarization of the noral ode is circular, so it is defined just by a single phase and agnitude. Likewise, the wave function of an electron on a site is represented by a coplex nuber that is also represented by a phase and aplitude.) Owing to the coplex exponential ter in Eq. S18, the bond angles lead to a phase shift between next-nearest neighbors on the lattice, siilar to the phase shift on hopping ters fro oving in a agnetic field. For the honeycob lattice this differs fro Haldane s odel only in that the second neighbor ter has a real part. The topological character of Haldane s syste can be quantified by calculation of the Chern nuber, which is an integral of the Berry curvature over the Brillouin zone of the lattice. A nonzero Chern nuber indicates a topologically nontrivial state and iplies the existence of chiral edge currents. Systes with tie-reversal syetry ust have a Chern nuber equal to zero, because tie-reversal syetry iplies zero Berry curvature. However, not all systes with broken tie-reversal syetry ust have a nonzero Chern nuber. In general, two bands separated by a finite gap will not acquire nonzero Chern nuber because of infinitesial perturbations. However, an infinitesial perturbation can produce a large change in Berry curvature at Dirac points. Therefore, even the sall coplex phase ters in the Haldane odel and in Eq. S18 can open a gap and induce a nonzero Chern nuber. To see this effect atheatically, one can expand in powers of displaceent fro the Dirac point, ~ k = ~ k ~ k 0 (where ~ k 0 is the wavevector of the Dirac point). Then the hopping atrix, T, can be written in ters of Pauli atrices. For a syste without next-nearest-neighbor coupling, we find that after rotating the wavevector, ~ k ~ k, the Hailtonian can be written as H eff ð ~ k Þ k x σ x + k y σ y. The Berry curvature of the two bands near this point are both zero. Ters with coplex phases break the degeneracy between the two states, which can be represented by adding a ter eff σ z, where eff is an effective ass. Even if eff 3of10

4 is sall, this changes the Berry curvature very close to the Dirac point, so that the net curvature in the botto band in the vicinity of the Dirac point is R d 2 k F = π sgn eff, and the band above the Dirac point has opposite curvature. For the honeycob lattice there are only two sites per unit cell, so there are two odes with each wave nuber. A basis can be obtained by defining u = 1 on one of the two sites and 0 on the other, and translating to other unit cells while ultiplying by e i~ k ~x (Fig. S7). The degeneracy points are at ~ p k 0 =±ð2π=3a,2π=3 ffiffi 3 aþ, where a is the edge length of the hexagon; Fig. S7 shows the two basis states near ~ p k 0 = ð2π=3a,2π=3 ffiffi 3 aþ. Let us focus on just the leading ters of Eq. S18 and the ters that arise fro hopping along the diagonals (next-nearest neighbors, u l ; see Fig. S8), because these are the ters that produce a gap. The atrix for the hopping along the sides is, to lowest order in ~ k, ð3=2þat 1 ðk σ x x + k σ y y Þ, where t 1 = Ω k is the nearest-neighbor hopping aplitude. To understand the contribution fro the next-nearest-neighbor hopping, we set ~ k = 0. As shown in Fig. S7, the two basis wave functions now reseble vortices circulating around the hexagons in opposite directions. Both wave functions pick up the sae phase under translation, but they transfor oppositely under rotation. Because the phase differences are different, the energies (i.e., frequencies of the noral odes in the gyroscope syste) of the states are different; the one whose phase shifts atch the phase of P the hopping better has the lower energy (i.e., the energies are 2t 3 2 r=1 cosðϕ 2r 2π=3Þ and 2t 2 P 3 r=1 cosðϕ 2r 1 + 2π=3Þ, wheret 2 = Ω 2 k =8Ω g and ϕ r = 2θ nl for the r next-nearest neighbors). If we include the energy splitting and the linear ters in ~ k, we have H ~k = 3 2 at X 6 1 ±kx σ x + k σ y y t2 ð 1Þ r cos ϕ r ± ð 1Þ r 2π 3 σ z, r=1 fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} eff = t 2 2 Pr ffiffi ½ð 1Þr cosϕ r 3 p sin ϕrš [S19] where the top sign refers to the Dirac point we have been considering and the botto sign is for the other one. If the phase shifts have twofold syetry p (i.e., ffiffi ϕ r+3 P= ϕ r ), we can siplify the effective ass to eff = ð 3 =2Þt2 r sin ϕ r. R The total Berry curvature for each of the two Dirac points is d 2 k F = π sgn eff = π sgn½ P r sin ϕ rš. As a result, the Chern nuber of the top/botto band is given by: C ± =±sgn½ P r sin ϕ rš. In general, distorting the honeycob lattice produces different phase shifts along different diagonals (Fig. S8). For a distorted honeycob lattice, there are four angles of the hexagon equal to α and two equal to 2π 2α. The phases are twice this; thus, the Berry curvature near each Dirac point is R d 2 k F = π sgn½2sinð4αþ 4sinð2αÞŠ. This curvature, and hence the Chern nuber, switch sign when the hexagon is distorted into a rectangle, in agreeent with the analysis in the ain text. Fig. S1. A figure illustrating the process of finding the parallel and perpendicular coponents of displaceents to a bond of length a between points p and q. Fig. S2. The gyroscopic etaaterial is coposed of 54 gyroscopes suspended by springs and coupled by agnetic dipole interactions. 4of10

5 Fig. S3. The effect of oscillation aplitude on the two noral ode frequencies in a syste of two gyroscopes. The frequencies are shifted by 5% as the oscillation aplitude increases to above 10% of the gyroscope separation. Fig. S4. The position of a wave packet vs. tie (Top) and the total oscillation aplitude squared as a function of tie (Botto). The tie to travel around the boundary of the etaaterial stays constant as the aplitude of oscillation decays. 5of10

6 e tir en ce tti la rs bo tn res a ne h eig Fig. S5. A coparison between the experient and agnetic odel for gap ode frequencies. The red (blue) shaded region indicates the possible values of gap ode frequencies for ode nubers 1 9 fro calculations using easured and ℓ values with entire lattice (nearest-neighbor) coupling. Mode nubers 1 9 are observed in the experient with a gap extending fro Hz. Modes 8 and 9 show soe ixing with bulk odes. The diaonds show values for a theoretical odel with M = 0.21 A2 and ℓ = 38, corresponding to Ω = 0.86 Hz (with entire lattice coupling). For this syste, Ωg = 0.98 Hz, which was deterined by easuring a single gyroscope. (Inset ) The change in frequency observed when the ode aplitude is increased fro (easured as the largest displaceent of a single gyroscope). Fig. S6. Coparison between ideal nuerical odes, experiental odes, and nuerically calculated odes with 10% disorder. The effect of rando disorder on ode profiles is qualitatively siilar to the ode profiles observed in the experiental syste, which had a siilar aount of disorder. 6 of 10

7 Fig. S7. The two basis wave functions for near the Dirac point at ~ p k 0 = ð2π=3a, 2π=3 ffiffiffi 3 aþ, as a function of displaceent fro the Dirac point: ~ k = ~ k ~ k0.the bases are generated by starting with an arbitrary wave function in a unit cell (indicated with the shaded hexagon), and then repeating the wave function periodically, with wave-vector-induced phase factors. A and B show wave functions with angular oenta of +1 and 1 around the hexagon, respectively, as indicated with the red arrows. Fig. S8. Phase shifts for next-nearest-neighbor hoppings. Although these phase shifts are equal for a hexagon, distorting the lattice will ake the phase shifts nonunifor. In general, adding diagonal hopping ters (dotted arrows) opens gaps at the Dirac points. 7of10

8 Movie S1. Deonstration of unidirectional waveguide odes in experient: A single edge gyroscope is excited for five periods at a gap ode frequency. This causes clockwise propagation around the edge. Movie S1 Movie S2. the edge. A single edge gyroscope is excited for five periods at a frequency that is not in the gap. This does not result in an excitation that propagates around Movie S2 8of10

9 Movie S3. Deonstration of unidirectional waveguide odes in lattice with irregular boundary: A single edge gyroscope is excited for five periods at a gap ode frequency. The resulting excitation propagates clockwise around the disturbance in the lattice due to the topological nature of the edge odes. Movie S3 Movie S4. The direction of propagation of edge odes can be controlled by the geoetry of the lattice. An excitation propagating clockwise propagates counterclockwise when the unit cell of the lattice defors fro a hexagonal to bow-tie shape. Movie S4 9of10

10 Movie S5. A honeycob lattice can support clockwise (left) or counterclockwise (right) propagating odes depending on the degree of deforation. When all angles of the lattice are 90, there is no band gap and there are no edge odes. Movie S5 10 of 10