Magnonic Su-Schrieffer-Heeger Model in Honeycomb Ferromagnets
MMagnonic Su-Schrieffer-Heeger Model in Honeycomb Ferromagnets
Yu-Hang Li and Ran Cheng
1, 2, ∗ Department of Electrical and Computer Engineering,University of California, Riverside, California 92521, USA Department of Physics and Astronomy, University of California, Riverside, California 92521, USA
Topological electronics has extended its richness to non-electronic systems where phonons andmagnons can play the role of electrons. In particular, topological phases of magnons can be enabledby the Dzyaloshinskii-Moriya interaction (DMI) which acts as an effective spin-orbit coupling. Weshow that besides DMI, an alternating arrangement of Heisenberg exchange interactions criticallydetermines the magnon band topology, realizing a magnonic analog of the Su-Schrieffer-Heegermodel. On a honeycomb ferromagnet with perpendicular anisotropy, we calculate the topologicalphase diagram, the chiral edge states, and the associated magnon Hall effect by allowing the relativestrength of exchange interactions on different links to be tunable. Including weak phonon-magnonhybridization does not change the result. Candidate materials are discussed.
Introduction.—
Magnonics, which utilizes the spin-wave excitations ( i.e. magnons) in magnetic insulatorsrather than the spin of electrons to transfer spin angularmomenta, has attracted persistent attentions in physicsand engineering [1–3]. Different from electrons, magnonsare charge neutral so that generating a magnon spin cur-rent does not incur Joule heating, which holds huge po-tential to realize low-dissipation devices [4–6].In the past decade, an emerging direction in magnonicshas been the study of topological properties of magnonbands and the associated exotic transport phenomena.In a ferromagnetic Kagome lattice, the Dzyaloshinskii-Moriya interaction (DMI) plays the role of an effectivespin-orbit coupling (SOC) and opens a topological non-trivial gap in the magnon spectrum [7]. Consequently,a longitudinal temperature gradient can induce a trans-verse magnon current due to the Berry phase effect, lead-ing to the magnonic counterpart of the quantum anoma-lous Hall effect [8–12].The discovery of two dimensional magnets bringsabout new exciting opportunities to explore topologicalmagnons [13–16]. For instance, a honeycomb ferromag-net with perpendicular order displays Dirac-type magnondispersions around the K (and K ) point [17–19]. Mean-while, its special crystal symmetry allows for the second-nearest neighboring DMI, which acts as a Rashaba-typeSOC on magnons, resulting in a magnonic analog of topo-logical insulators [20–23]. In addition, topological ef-fect manifests in honeycomb antiferromagnets as the spinNernst effect, where an in-plane temperature gradientgenerates a pure spin current in the transverse directionwithout a magnon Hall current [24–26].In spite of remarkable progress in topological magnons,phase transitions among magnonic states of distincttopology have so far been restricted to the variation ofDMI [21, 23], temperature[21], and magnon-phonon cou-pling [27, 28]. An important missing ingredient is theanisotropy among nearest-neighboring (NN) exchange in-teractions on different atomic bonds, as illustrated inFig. 1(a). While the DMI amounts to a magnonic SOC, the NN exchange interactions play the role of hoppingparameters for magnons. The well-known Su-Schrieffer-Heeger (SSH) Model of electrons reveals a profound re-lation between an alternating hopping amplitudes andthe band topology in one dimensions [29]. When gen-eralized into a honeycomb lattice, there can be threedifferent NN hopping parameters forming an alternat-ing pattern, giving rise to intriguing topological phasesin two dimensions. At this point, it is tempting to askif a magnonic analog of the two-dimensional SSH modelexists and, more importantly, what non-trivial transportphenomena are enabled in such a system.In this Letter, we study a honeycomb ferromagnet withtunable NN exchange interactions J , J , and J as il-lustrated in Fig. 1(a). By varying the relative strengthof J , J , and J , we demonstrate that the system canundergo phase transitions between a magnon Hall insula-tor (MHI) with a pair of chiral edge modes and a trivialmagnon insulator without any edge modes. This topolog-ical phase transition is characterized by a sharp changeof the magnon Hall coefficient, which can be detectedexperimentally. We also consider weak hybridization ofmagnons and phonons, which, though inevitable in ther-mal transport, does not change our essential conclusion.Finally, our proposed topological phase transition canbe tested in monolayer transition metal trichalcogenidessubject to lattice deformation ( e.g. , by straining). Model.—
Let us consider a collinear ferromagnet on ahoneycomb lattice as schematically shown in Fig. 1 (a).The minimal Hamiltonian of such a system is H = − X h i,j i J ij S i · S j + D X hh i,j ii (cid:15) ij ˆ z · S i × S j − κ X i S iz , (1)where J ij > κ > D is the second-NNDMI with (cid:15) ij = ± J along α , J along α , and J along α di- a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p J J J (a) (b) Γ K ´ K BA M xy α α α β β β (c) D D D AB FIG. 1. (color online). (a) Schematics for the magnonic SSHmodel in a honeycomb ferromagnet. The NN and second-NN bonds are labeled by α i and β i , respectively. The NNexchange interaction along α i direction is J i . (b) The second-NN DMI on different bonds. (c) The first Brillouin zone ofthe reciprocal lattice. rections, respectively. The difference among J can berealized by lattice straining, which is usually controllablevia a voltage. With lattice deformation, the second-NNDMI can also exhibit directional anisotropy as illustratedin Fig. 1(b). However, as discussed in the supplementarymaterials (SI) [30], the anisotropy of DMI does not affectthe band topology, introducing only trivial modificationsto the band dispersion. For simplicity, we will treat theDMI as isotropic throughout this Letter.Using the Holstein-Primakoff transformation [31], weset S + iA = √ Sa i , S − iA = √ Sa † i , S ziA = S − a † i a i for theA sublattice, and S + iB = √ Sb i , S − iB = √ Sb † i , S ziB = S − b † i b i for the B sublattice, where a † i ( b † i ) creates amagnon on the A (B) sublattice in the i unit cell. Underthe basis ψ ( k ) = (cid:2) a k , b k (cid:3) T with a k ( b k ) the Fouriertransformation of a i ( b i ) and k belonging to the firstBrillouin zone depicted in Fig. 1(c), Eq. (1) becomes H = P k ψ ( k ) † H ( k ) ψ ( k ), where, after discarding the zero-point energy, the matrix H ( k ) reads H ( k ) = a I + bσ x + cσ y + dσ z . (2)Here a = ( J + J + J + K ) S with K = κ (2 S − /S , b + i c = − S P i J i e i kα i , d = SD P i sin ( k β i ) where i = 1 , , α i , β i are illustrated inFig. 1(a). In Eq. (2), σ x,y,z are Pauli matrices acting onthe sublattice space and I is the 2 × a in Eq. (2), whichonly causes an overall shift to the entire band structure.When J = J = J , the system becomes a bosonic Hal-dane model if D is nonzero [21–23].To solve the eigenvalue problem, we parameterize b = l sin θ cos ψ , c = l sin θ sin ψ , and d = l cos θ with l = √ b + c + d . Then diagonalizing Eq. (2)gives the eigenvalues ~ ω ± = a ± l and the eigen-vectors ψ + = (cid:2) − sin θ , cos θ e − iψ (cid:3) T / √ ψ − = -1 0 1-101 k y / π (a) -1 0 1-101 (b) -1 0 1 k x /π -101 k y / π (c) ×10 −2 C=0 -4-3-2-10 -1 0 1 k x /π -101 (d) ×10 −2 C=1
FIG. 2. (color online). (a) and (b): the dispersions of thelower magnon band for J = J = 0 . J = J = 2,respectively. (c) and (d): corresponding Berry curvatures,whose Chern numbers are C = 0 and C = 1. The dottedblack lines denote the edges of the first Brillouin zone. (cid:2) cos θ , sin θ e − iψ (cid:3) T / √
2. The two magnon bands ~ ω ± are mirror symmetric about the energy plane ~ ω = a .As magnons are bosonic excitations, the lower band ~ ω − is always more populated than the upper band ~ ω + . Wewill therefore focus on the lower band when querying onthe band topology even though both bands will be consid-ered when calculating the transport coefficient. To sim-plify our discussion, we adopt the scaling convention that J = 1, the hexagon side length a = 1 such that otherquantities are expressed dimensionlessly with respect tothese parameters. In addition, we scale temperature T by the Curie temperature T c = J S ( S + 1) /k B evaluatedby the mean-field theory at J = J = J . In the fol-lowing, unless otherwise specified, S = 5 / D = 0 . κ = 0 .
05 whereas J and J are tunable. Band topology.—
Key to our proposal is identifyingdifferent topological phases by varying the NN exchangeinteractions. It is thus instructive to first consider tworepresentative cases. Figs. 2(a) and (b) plot the lowermagnon bands ~ ω − for J = J = 0 . J = J = 2,respectively. The upper band, as mentioned previously,can be obtained directly by a mirror reflection about the ~ ω = a plane. In both cases, the band structure exhibitsa C rotational symmetry ascribing to the anisotropy ofthe NN exchange interactions. The corresponding Berrycurvature is Ω ± ( k ) = − Im h∇ ψ ± ( k ) | × |∇ ψ ± ( k ) i = ∓ sin θ ( ∇ θ × ∇ ψ ), which points to the z direction. InFig. 2 (c) and (d), we plot Ω − ( k ) associating with thelower magnon bands in (a) and (b). The Berry curva-ture exhibits sharp peaks around where the band gap ~ | ω + − ω − | reaches local minima. Integrating Ω − ( k )over the Brillouin zone gives the Chern number of thelower band, which is 0 and 1 for the two cases, respec-tively. This confirms that the system can indeed exhibitdifferent topological phases by varying the NN exchangeinteractions.To better understand the underlying physics of the twotopological phases, we turn to two limits: J = J → J = J (cid:29)
1. If we turn off two of the NN exchangeinteractions ( e.g. , by setting J = J = 0) as well as D ,the system breaks up into an array of isolated diatomicpairs each bonded only along the α direction. Sinceneighboring pairs do not talk, the whole system has twoexact flat bands separated by a large trivial gap of ∆ =2 J S . Reintroducing D is inadequate to close this gap,and in turn, change the topology. By contrast, when J = J (cid:29)
1, the system can be effectively viewed as multipleone dimensional SSH chains along the β direction thatare weakly coupled through J . Even though individualSSH chains are gapless, the DMI will open a band gapand give rise to a nontrivial band topology. Edge states.—
Guaranteed by the universal bulk-edgecorrespondence, a non-trivial band topology is always ac-companied by chiral edge states [32]. The phase charac-terized by C = 1 [Fig. 2 (b) and (d)] corresponds to aMHI, which supports chiral edge modes on open bound-aries. To better visualize the edge states, we now wrapup the honeycomb sheet into a ribbon so that the systemis periodic in one direction, leaving open boundaries inthe other. Based on the crystal structure in Fig. 1(a),the ribbon has a zigzag boundary along x while an arm-chair boundary along y [30]. Figure. 3 (a) and (b) showthe topologically-protected edge modes associated witheach type of open boundaries. Although the bulk banddispersion in the zigzag direction is manifestly differentfrom that in the armchair direction, the edge modes al-ways connect the lower and the upper bulk bands, whichcannot be broken perturbatively. Moreover, regardless ofthe type of boundaries, the edge modes always appear inpairs and intersect the ~ ω = a line with opposite slops v g = ∂ω/∂k (marked by red and blue asterisks). In otherwords, they propagate in opposite directions.To see why the edge states are chiral, we plot the wave-functions of the edge states in Fig. 3 (c) and (d). In bothtypes of open boundaries, it turns out that edge statesof opposite group velocity are localized on opposite sides.Consequently, the propagation direction of edge magnonsis locked to the side of the open boundaries. The spatialextension of an edge state can roughly be evaluated as w ∼ | v g | /δ with δ being the bulk band gap [33], whichexplains why the chiral edge states on the zigzag bound-ary [Fig. 2 (b) and (d)] penetrate deeper into the bulkcompared to those of the armchair boundary [Fig. 2 (a)and (c)]. In sharp contrast to the C = 1 phase, the C = 0phase is not accompanied by any topologically-protectededge states as shown in the SI [30]. Phase diagram and transport properties.—
Havingdemonstrated two topologically distinct phases for differ- √ x a ħ ω (a) ħω = aSES1 ES2 0 1 2 y a (b) ħω = aSES1 ES20 50 100 y | ψ | (c) ES1ES2 x (d) ES1ES2
FIG. 3. (color online). Band structure with a pair of chiraledge modes lying in the bulk gap for the armchair edge (a)and the zigzag edge (b). (c) and (d) are the spatial profile(magnon density) of the chiral edge states at the intersectionwith the ~ ω = a line, labeled as ES1 and ES2. The width ofthe ribbon is taken as W = 100, and J = J = 2. ent NN exchange interactions, it is nature to ask where isthe phase boundary and, more importantly, what is thefull phase diagram by varying J and J arbitrarily? InFig. 4, we plot the Chern number of the lower magnonband on the J − J plane. The system turns out to bea MHI ( C = 1) when J + J > | J − J | < C = 0) otherwise. The phaseboundaries are independent of the DMI so long as D isnonzero [30].Even though the magnon Chern number is not directlyrelated to quantized transport because of the bosonicstatistics, different topological phases and phase tran-sitions can still be characterized by the magnon Hall ef-fect [21, 23]. In this regard, we now calculate the magnonHall current driven by an in-plane temperature gradient,which originates from the non-zero Berry curvatures ofboth bands [9–11]. Expressed in unit of a number currentdensity, the magnon Hall current is j H = k B ~ ˆ z × ∇ T X n = ± Z d k (2 π ) Ω n ( k ) { ρ n ( k ) ln ρ n ( k ) − [1 + ρ n ( k )] ln [1 + ρ n ( k )] } , (3)where k B is the Boltzmann constant and ρ n ( k ) =1 / (cid:0) e ~ ω n ( k ) /k B T − (cid:1) is the Bose-Einstein distributionfunction. Since magnons carry both spins and heat, themagnon Hall effect involves simultaneously a spin Hallcurrent a thermal Hall current.Fig. 4(b) plots the magnon Hall coefficient κ xy ≡ ∂j Hy /∂ x T on the J − J plane. Besides showing the J J (a) J = 0.4 C = 0C = 0C = 0C = 1 J J (b) III III IV * * y | ψ | J κ xy / ( k B / ħ ) C=1 (c)
T = 0.1T c T = 0.2T c T = 0.3T c T = 0.4T c Γ K M Γ momentum ħ ω (d) MP FIG. 4. (color online). (a) Chern number of the lower magnonband on the J − J plane for D = 0 .
1; the phase boundaryis independent of D . (b) Thermal Hall coefficient κ xy as afunction of J and J with temperature T = 0 . T c . Theinset shows the edge states with energy ~ ω = a for the twoasterisked points in region IV. (c) κ xy as a function of J at different temperatures for J = 0 .
4, corresponding to thedotted line in (a). (d) Renormalized band dispersion alongthe Γ- K - M -Γ loop [see Fig. 1 (c)] by taking into account themagnon-phonon coupling for g A ± = g B ± = 0 .
08 and
C/m =1 . S . Magnons and phonons hybridize and open tiny gapsnear avoided crossing. The insert is a zoom-in of gap 3. same phase boundaries as Fig. 4(a), the phase diagramfor κ xy has several unique features. First, even though C = 0 in region I, κ xy does not vanish, which ascribesto the bosonic statistics that weights the Berry curvaturenon-uniformly in the Brillouin zone. Second, the systemis a MHI in region IV, but κ xy decreases with an increas-ing J + J . To understand this character, we plot thewavefunctions of the edge states at two different points inregion IV (marked by asterisks) in the inset, from whichwe can tell that the edge states from opposite bound-aries strongly overlap at higher J + J . As a result, thebackscattering of edge magnons is significantly enhanced,thus diminishing the magnon Hall effect. Furthermore,as shown in Fig. 4(c) for J = 0 . κ xy undergoes a sharp change acrossthe phase boundaries, which becomes more prominent athigher temperatures due to enhanced magnon density.Such a mutation of the magnon Hall effect can be at-tributed to the emergence of chiral edge states when thesystem enters the MHI phase. Those distinctive featuresprovides an unambiguous way to identify the topologicalphase transition experimentally. Magnon-phonon coupling.—
Phonons, the quanta oflattice vibrations, may also contribute to the thermalHall effect at finite temperatures [34]. Since phonons do not carry spin angular momenta, their contribution canin principle be separated from magnons by spin-resolvedmeasurements. However, due to the magnon-phonon in-teractions, the two types of quasiparticles can hybridizenear band crossing, where they can no longer be indi-vidually defined [28, 35, 36]. Therefore, it is imperativeto check if magnon-phonon coupling substantially mod-ify the band topology, hence the magnon Hall effect. Intwo dimensional materials, owing to the absence of thethird dimension, out-of-plane lattice vibrations are muchmore amenable to thermal agitations than in-plane vi-brations. Therefore, the dominant contribution stemsfrom out-of-plane phonon modes, which greatly simpli-fies the problem. Using the augmented basis of Ψ ( k ) = (cid:2) a k , b k , c k − , c k + (cid:3) T with c k ± being the phonon annihi-lation operator, we can express the magnon-phonon cou-pling Hamiltonian as H mp = P k Ψ ( k ) † H mp ( k ) Ψ ( k ),where the matrix H mp ( k ) reads [28] H mp ( k ) = a + d b − ic g A − g A + b + ic a − d g B − g B + g ∗ A − g ∗ B − ω pk − ( k ) 0 g ∗ A + g ∗ B + ω pk + ( k ) . (4)Here, g α ± is the (phenomenological) coupling strengthbetween the α (= A, B ) sublattice and the ± phononmode, and ω pk ± ( k ) = q Cm p ± √ S k is the freephonon dispersion with C the elastic constant, m the ef-fective mass of the lattice site and S k = P i =1 cos ( k · β i ).Using typical magnon-phonon coupling strengths, wediagonalize the total Hamiltonian including Eq. (4) andplot the band structure in Fig. 4 (d), where the magnonand phonon characters of the eigenmodes are indicatedby different colors. The magnon-phonon hybridizationresults in avoided crossings in the lower magnon band.In the vicinity of an avoided crossing, the Berry curva-ture transfers from one band to the other [28]. Sincephonons do not carry spins, the phonon contribution tothe magnon spin Hall current relies solely on the lossof Berry curvature in the magnon band ~ ω − near theseavoided crossings. Comparing Fig. 2(d) with Fig. 4(d),however, we can tell that none of the avoided crossingstakes place near the concentration of the Berry curvatureunless the magnon-phonon coupling becomes unreason-ably large such that the anti-crossings span a wide rangeof momenta in the Brillouin zone. Consequently, phononsonly causes negligibly small modifications to the magnonHall current; the transverse spin current is thus robustagainst weak magnon-phonon coupling. Candidate materials.—
Transition metal trichalco-genides such as CrI [13] and Cr Ge Te [16] exhibit sta-ble ferromagnetism down to the monolayer limit. In thesematerials, magnetic atoms are arranged on a honeycomblattice, pointing perpendicular to the plane. Because theNN exchange interactions depend sensitively on the inter-atomic distances [37], applying mechanical strains to thelattice [38–40] can be a viable way to induce appreciablechanges to the NN exchange interactions. Under latticedeformation, the second-NN DMI may also acquire direc-tional anisotropy as illustrated in Fig. 1(b). Nevertheless,as long as the DMI does not change sign, the topologyof magnon bands remains the same, which is detailed inthe SI [30].In summary, we have theoretically demonstrated amagnonic analog of the SSH model and obtained thetopological phase diagram of magnons characterized byboth the Chern number of the lower band and themagnon Hall coefficient. The chiral edge modes result insharp changes of the magnon Hall effect across the phaseboundaries, providing a smoking-gun signature for ex-perimental detection. Considering weak magnon-phononcoupling does not change the essential conclusion.This work is supported by the startup fund of the Uni-versity of California, Riverside. ∗ [email protected][1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, andB. Hillebrands, Nature Physics , 453 (2015).[2] S. Demokritov and A. Slavin, Magnonics: From Fun-damentals to Applications , Topics in Applied Physics(Springer Berlin Heidelberg, 2012).[3] S. A. Nikitov et al. , Physics-Uspekhi , 1002 (2015).[4] A. R¨uckriegel, A. Brataas, and R. A. Duine, Phys. Rev.B , 081106 (2018).[5] A. A. Awad et al. , Nature Physics , 292 (2017).[6] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, andB. J. van Wees, Nature Physics , 1022 (2015).[7] H. Katsura, N. Nagaosa, and P. A. Lee, Phys. Rev. Lett. , 066403 (2010).[8] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa,and Y. Tokura, Science , 297 (2010).[9] R. Matsumoto and S. Murakami, Phys. Rev. Lett. ,197202 (2011).[10] R. Matsumoto and S. Murakami, Phys. Rev. B ,184406 (2011).[11] D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Phys. Rev. Lett. , 026603 (2006).[12] L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. B , 144101 (2013).[13] B. Huang et al. , Nature , 270 (2017). [14] J.-U. Lee et al. , Nano Letters , Nano Letters , 7433(2016).[15] X. Wang et al. , 2D Materials , 031009 (2016).[16] C. Gong et al. , Nature , 265 (2017).[17] W. B. Yelon and R. Silberglitt, Phys. Rev. B , 2280(1971).[18] E. J. Samuelsen, R. Silberglitt, G. Shirane, and J. P.Remeika, Phys. Rev. B , 157 (1971).[19] S. S. Pershoguba et al. , Phys. Rev. X , 011010 (2018).[20] L. Chen et al. , Phys. Rev. X , 041028 (2018).[21] S. K. Kim, H. Ochoa, R. Zarzuela, and Y. Tserkovnyak,Phys. Rev. Lett. , 227201 (2016).[22] S. A. Owerre, Journal of Physics: Condensed Matter ,386001 (2016).[23] S. A. Owerre, Journal of Applied Physics , 043903(2016).[24] R. Cheng, S. Okamoto, and D. Xiao, Phys. Rev. Lett. , 217202 (2016).[25] V. A. Zyuzin and A. A. Kovalev, Phys. Rev. Lett. ,217203 (2016).[26] Y. Shiomi, R. Takashima, and E. Saitoh, Phys. Rev. B , 134425 (2017).[27] S. Zhang, G. Go, K.-J. Lee, and S. K. Kim, Phys. Rev.Lett. , 147204 (2020).[28] E. Thingstad, A. Kamra, A. Brataas, and A. Sudbø,Phys. Rev. Lett. , 107201 (2019).[29] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev.B , 2099 (1980).[30] See the supplementary materials for more informations.[31] W. Nolting and A. Ramakanth, Quantum theory of mag-netism (Springer, Heidelberg; New York, 2009).[32] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annual Review ofCondensed Matter Physics , 301 (2016).[33] M. K¨onig, H. Buhmann, L. W. Molenkamp, T. Hughes,C.-X. Liu, X.-L. Qi, and S.-C. Zhang, Journal of thePhysical Society of Japan , 031007 (2008).[34] L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev.Lett. , 225901 (2010).[35] G. Go, S. K. Kim, and K.-J. Lee, Phys. Rev. Lett. ,237207 (2019).[36] X. Zhang, Y. Zhang, S. Okamoto, and D. Xiao, Phys.Rev. Lett. , 167202 (2019).[37] C. Ulloa, R. E. Troncoso, S. A. Bender, R. A. Duine, andA. S. Nunez, Phys. Rev. B , 104419 (2017).[38] L. Webster and J.-A. Yan, Phys. Rev. B , 144411(2018).[39] Z. Wu, J. Yu, and S. Yuan, Phys. Chem. Chem. Phys. , 7750 (2019).[40] Z. H. Aitken and R. Huang, Journal of Applied Physics , 123531 (2010). upplemental material for “Magnetic Su-Schriffer-Heeger model in honeycombferromagnets” Yu-Hang Li and Ran Cheng
1, 2, ∗ Department of Electrical and Computer Engineering,University of California, Riverside, California 92521, USA Department of Physics, University of California, Riverside, California 92521, USA (Dated: September 7, 2020)
I. EDGE STATES.
The edge states can be studied in a nanoribbon structure [1]. To this end, we first consider a zigzag boundary andrewrite the Eq.(1) in the main text in terms of the lattice indices m and n , H z = X mn {− J S A ( m, n ) S B ( m, n + 1) − J S A ( m, n ) S B ( m, n ) − J S A ( m, n ) S B ( m − , n )+ D ˆ z · [ S A ( m, n ) × S A ( m, n + 1) − S A ( m, n ) × S A ( m + 1 , n + 1)+ S A ( m, n ) × S A ( m + 1 , n ) − S B ( m, n ) × S B ( m, n + 1)+ S B ( m, n ) × S B ( m + 1 , n + 1) − S B ( m, n ) × S B ( m + 1 , n )]+ κ h S zA ( m, n ) + S zB ( m, n ) i } , (S1)where, J i =1 , , , D and κ share the same meanings as those in the main text. Using the Holstein-Primakoff transfor-mation S + A ( m, n ) = √ Sa mn , S − A ( m, n ) = √ Sa † mn , S zA ( m, n ) = S − a † mn a mn ,S + B ( m, n ) = √ Sb mn , S − B ( m, n ) = √ Sb † mn , S zB ( m, n ) = S − b † mn b mn , (S2)the Hamiltonian in Eq. (S1) can be written as H z /S = − J W X m =1 (cid:16) a mn b † mn +1 + a † mn b mn +1 − a † mn a mn − b † mn b mn (cid:17) − J W X m =1 (cid:0) a mn b † mn + a † mn b mn − a † mn a mn − b † mn b mn (cid:1) − J W X m =2 (cid:16) a mn b † m − n + a † mn b m − n − a † mn a mn − b † m − n b m − n (cid:17) + D i " W X m =1 (cid:16) a † mn a mn +1 − a mn a † mn +1 − b † mn b mn +1 − b mn b † mn +1 (cid:17) − W − X m =1 (cid:16) a † mn a m +1 n +1 − a mn a † m +1 n +1 − b † mn b m +1 n +1 − b mn b † m +1 n +1 (cid:17) + W X m =1 (cid:16) a † mn a m +1 n − a mn a † m +1 n − b † mn b m +1 n − b mn b † m +1 n (cid:17) ]+ 2 κ W − X m =1 (cid:0) a † mn a mn + b † mn b mn (cid:1) , (S3)where the summation of index n has been omitted for simplification. Since we use a periodic boundary conditionin the x direction, after the Fourier transformation, ψ kn ≡ (cid:18) a mn b mn (cid:19) = 1 / √ N P k e ikn (cid:18) a mk b mk (cid:19) , the Hamiltonian can a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p finally be written as H z /S = W X m =1 h ( J + J − κ ) (cid:16) a † mk a mk + b † mk b mk (cid:17) − (cid:16) J e √ ika + J (cid:17) (cid:16) a mk b † mk + H.c. (cid:17)i − D sin √ ka W X m =1 (cid:16) a † mk a mk − b † mk b mk (cid:17) + J W X m =2 h(cid:16) a † mk a mk + b † mk b mk (cid:17) − (cid:16) a mk b † m − k + H.c. (cid:17)i + iD W X m =2 h(cid:16) e −√ ika − (cid:17) (cid:16) a † mk a m +1 k − b † mk b m +1 k (cid:17) − H.c. i , (S4)where the summation of k is omitted. By the same token, the Hamiltonian of a nanoribbon with an armchair boundarycan finally be written as H a /S = W X m =1 h ( J − κ ) (cid:16) a † mk a mk + b † mk b mk (cid:17) − D sin 3 ka (cid:16) a † mk a mk − b † mk b mk (cid:17) − J (cid:16) a mk b † mk + H.c. (cid:17)i + J W − X m =1 h(cid:16) a mk b † m +1 k e ika + H.c. (cid:17) − (cid:16) a † mk a mk + b † mk b mk (cid:17)i + J W X m =2 h(cid:16) a mk b † m − k + H.c. (cid:17) − (cid:16) a † mk a mk + b † mk b mk (cid:17)i + iD W − X m =1 h(cid:0) e − ika + 1 (cid:1) (cid:16) a † mk a m +1 k − b † mk b m +1 k (cid:17) − H.c. i . (S5)Diagonalizing Eqs. (S4) and (S5), the energy dispersions and the corresponding eigenvectors can be derived directly,base on which the density distribution functions in the real space can be obtained straightforwardly. Figs. S1 (a)and (b) show the one dimensional band dispersions of a normal magnon insulator in a nanoribbon with armchair andzigzag boundaries, respectively, where the bands are well gaped without any survived edge states. √ x a ħ ω (a) y a (b) FIG. S1: (color online). (a) and (b) are band dispersions of one dimensional nanoribbon with armchair and zigzag boundaries,respectively. Here, the width of the ribbon is W = 100, and J = J = 0 . J . II. DIFFERENT DZYALOSHINSKII-MORIYA INTERACTIONS.
In the main text, we propose that the topological phase transition can be tested in honeycomb ferromagnets underproper deformations. Nevertheless, such deformations generally change not only the nearest-neighbor exchange inter-actions but also the next nearest-neighbor Dzyaloshinskii-Moriya interactions. In this case, the minimal Hamiltonianreads H = − X h i,j i J ij S i · S j + X hh i,j ii D ij (cid:15) ij ˆ z · S i × S j + κ X i S iz , (S6)where the Dzyaloshinskii-Moriya interactions take D = D · J J along β , D = D · J J along β , and D = D · J J along β , respectively. All other parameters share the same meanings and take the same values as those in the maintext.As a comparison, Fig. S2 (a) plots the Chern number of the lower magnon band on the J − J plane with D = D = D = 0 . κ xy as shown in Fig. S2 (c) may be quantitatively different now. However, the sharp increase acrossthe phase boundary from normal magnon insulators to the magnon Hall insulator remains an obvious signature.Therefore, the thermal Hall effect is still an efficient method to detect this topological phase transition in experiment. J J C = 0C = 0C = 0 C = 1(a) J C = 0C = 0C = 0 C = 1(b) J (c) III III IV
FIG. S2: (color online). (a) Chern number of the lower magnon band on the J − J plane with D = D = D = 0 .
2. (b)Chern number of the lower magnon band on the J − J plane when taking different Dzyaloshinskii-Moriya interactions intoconsideration. (c) Thermal Hall coefficient κ xy corresponding to (b) on the J − J plane with temperature k B T = 0 . J . Allother parameters are exactly the same as those in the main text. ∗ [email protected][1] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109 (2009).[2] A. Bansil, Hsin Lin, and Tanmoy Das, Rev. Mod. Phys.88