Topological magnon bands and unconventional thermal Hall effect on the frustratedhoneycomb and bilayer triangular lattice∗

S. A. Owerre11Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada.

(Dated: November 9, 2018)

In the conventional ferromagnetic systems, topological magnon bands and thermal Hall effectare due to the Dzyaloshinskii-Moriya interaction (DMI). In principle, however, the DMI is eithernegligible or it is not allowed by symmetry in some quantum magnets. Therefore, we expect thattopological magnon features will not be present in those systems. In addition, quantum magnets onthe triangular-lattice are not expected to possess topological features as the DMI or spin-chiralitycancels out due to equal and opposite contributions from adjacent triangles. Here, however, wepredict that the isomorphic frustrated honeycomb-lattice and bilayer triangular-lattice antiferro-magnetic system will exhibit topological magnon bands and topological thermal Hall effect in theabsence of an intrinsic DMI. These unconventional topological magnon features are present as aresult of magnetic-field-induced non-coplanar spin configurations with nonzero scalar spin chirality.The relevance of the results to realistic bilayer triangular antiferromagnetic materials are discussed.

I. INTRODUCTION

In recent years, the concept of topological band theoryhas been extended to bosonic systems. Consequently,topological magnon bands and the associated thermalHall effect in insulating quantum ferromagnets have gar-nered considerable attention. The experimental realiza-tions of these phenomena in the quasi-two-dimensional(2D) kagomé ferromagnets Cu(1-3, bdc)1,2 have furtherrekindled much interest in this area. Thermal Halleffect of magnons was previously realized experimen-tally in different 3D pyrochlore ferromagnets A2B2O7

3,4.These experimental studies follow from different theoret-ical proposals5–13. An extension to unfrustrated honey-comb magnets has been recently proposed by differentauthors14–19.

Generally speaking, it is believed that the topologicalmagnon phenomena in quantum ferromagnets1–4 resultfrom the DMI20,21, which plays the same role as spin-orbit coupling (SOC) in electronic systems22,23. To date,there is no experimental evidence of a counter-examplewhere topological magnon features originate from an al-ternative source other than the DMI in any magneticallyordered system. Therefore, the conception is that theDMI is mandatory for topological magnon properties toexist in magnetically ordered systems.

The frustrated magnets provide a platform to ex-plore this possibility as we have previously shown on thekagomé-type lattices24,25. But the kagomé-type latticesnaturally allow an intrinsic DMI, which is capable of in-ducing and stabilizing the coplanar spin structure26. Incontrast, most non-kagomé-type frustrated magnets donot allow an intrinsic DMI due to symmetry. Therefore,we wish to extend our analyses to those systems. On thehoneycomb lattice, geometric frustration is present whena next-nearest-neighbour (NNN) antiferromagnetic inter-action J2 competes with a nearest-neighbour (NN) an-tiferromagnetic interaction J1. This system is known asthe frustrated J1–J2 honeycomb-lattice Heisenberg anti-

ferromagnet (HLHAF). It possesses interesting phase dia-gram for J2/J1 127–40. The opposite limit J2/J1 1is unexplored. In this regime the geometric frustrationinduced by J2 yields a decoupled 120 coplanar order.The model is now isomorphic to a 120 coplanar orderon the stacked bilayer triangular-lattice Heisenberg anti-ferromagnet (TLHAF) with intraplane coupling J2 andsmall interplane coupling J141. Therefore, one can cap-ture the physics of bilayer TLHAF by studying the dom-inant J2 limit of the frustrated HLHAF.

In this paper, we predict that topological magnonbands and unconventional topological thermal Hall ef-fect will exist on the isomorphic honeycomb and bilayertriangular lattice J1–J2 model without any DMI. Weshow that the system possesses topological magnon bandswith nonzero Chern number C± = ±sgn(sinφ), wheresinφ is related to the field-induced scalar spin chiralityχ =

∑Si · (Sj × Sk), and i, j, k label sites on a unit

triangle; φ is the angle subtended by three non-coplanar(umbrella) spins. The corresponding thermal Hall con-ductivity κxy is tunable by the external magnetic field asit requires no DMI. Interestingly, topological propertiesare not expected to be present on the triangular latticeas the DMI (spin-chirality) cancels out due to equal andopposite contributions from adjacent triangles5. The cur-rent result is a counter-example where the field-inducedscalar-chirality of the non-coplanar (umbrella) spin struc-ture does not cancel as it is coupled to the magnetizationof the non-coplanar spins42.

These results are particularly interesting especially forthe stacked triangular-lattice antiferromagnetic materi-als with no intrinsic DMI. They include Ba3XSb2O9 (X≡Mn, Co, and Ni)43–49 and VX2 (X ≡ Cl, Br, and I)50,51and others52. The effects of topological magnons are alsomanifested by the measurement of nonzero thermal Hallconductivity κxy at various external magnetic fields alongthe z-axis1,2. We note that the quasi-2D bilayer metal-lic triangular-lattice magnet PdCrO2 with 120 copla-nar order also shows a finite anomalous Hall effect in

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a perpendicular-to-plane external magnetic field53. Wetherefore expect that an analogous thermal Hall effect inmagnetic insulators with charge-neutral excitations suchas magnons will be present and of great importance.

II. MODEL

The frustrated isomorphic J1–J2 Heisenberg model onthe honeycomb- and bilayer triangular-lattice in an ex-ternal magnetic field is given by

H =∑ij

JijSi · Sj −H∑i

Szi , (1)

where Jij = J1(J2) are nearest (next-nearest) neighbourantiferromagnetic interactions andH is the external mag-netic field along the z-axis perpendicular to the latticeplane. The Hamiltonian (1) has been extensively studiedon the honeycomb lattice in the context of ground state(thermodynamic) properties28–40. In the classical limitat zero magnetic field28,29, a collinear Néel order existsfor J2/J1 < 1/6. For J2/J1 > 1/6 it has a family ofdegenerate spiral order with incommensurate wave vec-tors. It was shown that spin wave fluctuations at leadingorder lift this accidental degeneracy in favour of specificwave vectors. In particular, the 120 coplanar order withordering wave vector K = Q =

(±2π/3

√3, 2π/3

)is ex-

pected to emerge for J2/J1 1. In this limit the Hamil-tonian (1) is isomorphic to the bilayer TLHAF with in-traplane coupling J2 and small interplane coupling J1.The two sublattices of the honeycomb lattice A,B areequivalent to the top and bottom sublattices of the bi-layer triangular lattice as shown in figure (1). Therefore,we study both systems simultaneously via equation (1).

III. RESULTS

A. Magnon band structures

It is advantageous to introduce the standard Holstein-Primakoff bosonization. The calculation is tedious butdoable as shown in Appendix (A). We have checked thatfor J2/J1 < 1/6 or equivalently J2/J1 → 0 the Hamil-tonian recovers the magnon band structures of collinear(canted) Néel antiferromagnet at H = 0 (H < Hs)as well as collinear ferromagnet at H = Hs, whereHs = 3(2J1 + 3J2) is the saturation field. These lim-iting cases require the DMI for topological features toexist as previously shown14–19.

We are interested in the dominant J2 limit correspond-ing to the isomorphic HLHAF and bilayer TLHAF witha stable 120 coplanar order. In this regime the magneticfield induces a non-coplanar (umbrella) chiral spin tex-ture with nonzero scalar spin chirality χ. We have shownthe magnon bands at zero magnetic field H = 0 in figure(2) (i.e., conventional 120 spin structure). They have

A

B

FIG. 1: Color online. (a) Schematics of the isomorphic J1–J2

honeycomb-lattice and bilayer triangular-lattice. (b) The firstBrillouin zone of the system with indicated paths.

0

2

4

6

8

E/J

1S

(a) J2/J1 = 1.1

Γ K M Γ

(b) J2/J1 = 1.3

K M Γ

FIG. 2: Color online. Dirac magnon bands of the conventional120 coplanar order at zero magnetic field H = 0. The bandslinearly touching at K and form a Dirac point. Inset showsthe circled points

Dirac point nodes at K. The Dirac nodes remain intacteven in the presence of (out-of-plane) DMI as it plays astability role in certain frustrated magnets rather than atopological role. In figure (3) we have shown the magnonbands for J2/J1 = 1.3 at two values of nonzero magneticfields. We see that the Dirac magnon nodes are gappedand the magnon bands become topological due to thepresence of nonzero scalar spin chirality χ. Notice thatthere is a roton-minimum near the ordering wave vectorof the coplanar spin structure at K = Q, which becomesgapless for large J2/J1. These features can be repro-duced in the bilayer triangular-lattice antiferromagneticsystems as shown explicitly in Appendix (B).

0

2

4

6

8

E/J

1S

(a) H = 0.2Hs

Γ K M Γ

(b) H = 0.25Hs

K M Γ

FIG. 3: Color online. Topological magnon bands of the non-coplanar (umbrella) spin structure for J2/J1 = 1.3 at twomagnetic field values. Inset shows the circled points.

3

0 0.5 1

ka/2π

0

2

4

6

8E(k

x)/J1S

(a) H = 0

0 0.5 1ka/2π

(b) H = 0.2Hs

FIG. 4: Color online. Magnon chiral edge modes for a stripgeometry periodic in x, but with open boundaries along y forJ2/J1 = 1.3. The black region corresponds to the magnonbulk bands. The chiral edge modes are shown in colors. (a)Dirac cones connected by flat chiral edge modes at zero mag-netic field H = 0. (b) Gapped Dirac cones with gapless chiraledge modes for small magnetic field H = 0.2Hs as a result ofthe emergent scalar spin chirality χ. Insets show the magni-fied chiral edge modes.

B. Berry curvature and Chern number

Most importantly, the magnon bands with nonzeroscalar spin chirality χ now acquire a nonzero Berry cur-vature, given by

Ωij;ks = −∑s6=s′

2Im[〈Pks|vi|Pks′〉 〈Pks′ |vj |Pks〉](Eks − Eks′)

2 , (2)

where vi = ∂(ηHk)/∂ki defines the velocity operatorsand η = diag(IN×N ,−IN×N ) is the diagonal of N × Nidentity matrix and s labels the bands. Here Pks is theparaunitary operator that diagonalizes ηHk. The Chernnumbers are given by

Cs =1

2π

∫BZ

dkidkj Ωij;ks. (3)

We have computed the Chern numbers numerically andestablished that for the two positive magnon bands, says = ±, C± = ±sgn(sinφ), where sinφ is related to thefield-induced scalar spin chirality χ and φ is the anglesubtended by three non-coplanar (umbrella) spins. Itchanges sign by reversing the sign of the magnetic field orthe scalar spin chirality, i.e., sinφ→ − sinφ as ϑ→ ϑ+π.

The existence of chiral magnon edge modes is anotheraspect of topological character of nontrivial magnons ininsulating quantum magnets. At zero magnetic field theDirac magnon bulk bands are connected by a flat chi-ral edge mode as shown in figure (4)(a). As the mag-netic field is turned on the flat edge modes are lifted duethe presence of scalar spin chirality χ as shown in fig-ure (4)(b). The chiral edge modes are now topologicallyprotected by the Chern numbers C± = ±sgn(sinφ).

0 0.5 1 1.5

T (K)

0

0.05

0.1

0.15

0.2(b)

H = 0.005H = 0.008

-0.01 0 0.01

H(T )

-0.2

-0.1

0

0.1

0.2

κxy(W

K−1m

−1) (a)

T = 1.2T = 1.25

FIG. 5: Color online. (a) Tunable thermal Hall conductivityκxy as a function of magnetic field H for two temperature val-ues. (b) Tunable thermal Hall conductivity κxy as a functionof temperature T for two magnetic field values. The couplingis set to J2/J1 = 1.3.

C. Topological thermal Hall effect

Conventionally, thermal Hall effect is induced by theDMI as reported in unfrustrated magnets1–19 as well asfrustrated magnets54–56. For frustrated magnets withnoncollinear and coplanar spin structure thermal Hallconductivity can be nonzero in the absence of the DMI.This is possible because an external magnetic field caninduce non-coplanar spin configurations with nonzeroscalar spin chirality χ. The interesting feature in thecurrent study is that the coplanar spin structure can bepresent in bilayer TLHAF without an intrinsic DMI43–49.The thermal Hall conductivity κxy can be derived fromlinear response theory10. We have shown the trends ofκxy as functions of the magnetic field and temperaturein figures (5) (a) and (b) respectively for a specific valueof J2/J1. Evidently, we capture a sign change in κxy asthe scalar spin chirality is reversed by reversing the signof the magnetic field. Interestingly, the trend of κxy issynthetic and tunable by the magnetic field as it requiresno intrinsic DMI. As we discuss below certain (honey-comb) triangular-lattice antiferromagnetic materials donot have an intrinsic DMI, so we do expect that κxy canbe tuned by an external magnetic field.

IV. CONCLUSION

We have predicted that topological magnon bands andunconventional topological thermal Hall effect will existin the isomorphic J1–J2 honeycomb-lattice and bilayertriangular-lattice antiferromagnets. These interestingfeatures originate from the magnetic-field-induced non-coplanar (umbrella) spin structure with nonzero scalarspin chirality and require no DMI in contrast to previ-ously studied unfrustrated magnets1–19 and frustratedmagnets54–57. Therefore they are unconventional andsynthetic as they can be tuned by the external magneticfield.

Most importantly, topological magnon features havenot been previously predicted on the triangular lattice

4

for the reasons we mentioned above. In realistic materialsthe bilayer triangular antiferromagnetic systems do notusually allow an intrinsic DM interaction, but an easy-plane (axis) anisotropy can be present. This is the case inthe bilayer triangular-lattice quantum antiferromagnetsBa3XSb2O9 (X ≡ Mn and Co) with a stable 120 copla-nar order and no intrinsic DM interaction43–49. They arealso quasi-two-dimensional (quasi-2D) with dominant in-traplane (J2) coupling and small interplane coupling (J1)and easy-axis anisotropy (∆).

The compounds VX2 (X ≡ Cl, Br, and I) also formquasi-2D layered triangular-lattice quantum antiferro-magnets with a stable 120 coplanar order and no in-trinsic DMI50–52. The net magnetic-field-induced scalarspin chirality in the non-coplanar regime will be nonzero.Therefore, the current predictions can be tested in thesematerials. Indeed, a finite κxy at various external mag-netic fields signifies that the magnetic excitations aretopologically nontrivial. As mentioned above, we havepreviously shown slightly similar results on the kagoméand star lattices24,25, but these lattice geometries nat-urally allow a DMI which stabilizes the coplanar spinstructure26. In contrast, the current results are differentin that the bilayer triangular antiferromagnetic materi-als mentioned above do not have an intrinsic DMI and anon-vanishing spin-chirality can be induced by applyingan external magnetic field in the plane perpendicular tothe magnets.

The bilayer honeycomb-lattice quantum antiferro-magnets Bi3X4O12(NO3) (X ≡ Mn, V, and Cr) arealso promising candidates; however it is believed thatJ2/J1 1 for X ≡ Mn, but an external magneticfield induces a transition to a 3D collinear Néel order atH ∼ 6T 27. In the collinear regime the DMI is mandatoryfor topological magnons to exist, and it can be allowed inthis compound58. There is also a possibility to synthe-size different honeycomb materials with dominant J259.We also expect the spontaneously-induced spin chiralityin the chiral spin liquid to have the same topological ef-fects on the underlying magnetic excitations. Hence, thescalar-chirality mechanism can help explain the recentlyobserved thermal Hall conductivity in a spin liquid ma-terial at nonzero magnetic field60.

Acknowledgements

Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and bythe Province of Ontario through the Ministry of Researchand Innovation.

Appendix A: Dominant J2 limit

We are interested in the dominant J2 limit of equation(1), corresponding to the isomorphic honeycomb-latticeand bilayer triangular-lattice antiferromagnets with a

stable 120 coplanar order. At zero field we take thespins to lie on the plane of the honeycomb (bilayer) tri-angle lattice taken as the xy plane. Then, we performa rotation about the z-axis on the sublattices by thespin oriented angles θi. As the external magnetic fieldis turned on, the spins will cant towards the direction ofthe field and form a non-coplanar configuration. Thus,we have to align them along the new quantization axisby performing a rotation about the y-axis by the fieldcanting angle ϑ. The total transformation of the spins is

Si = Rz(θi) · Ry(ϑ) · S ′i, (A1)

where

Rz(θi) · Ry(ϑ) =

cos θi cosϑ − sin θi cos θi sinϑsin θi cosϑ cos θi sin θi sinϑ− sinϑ 0 cosϑ

.

(A2)

Next, we plug the spin transformation (A1) into theHamiltonian (1). There are numerous terms but we re-tain only the terms that contribute to the free magnonmodel, given by

H =∑ij

Jij[

cos θijS ′i · S′j + sin θij cosϑz ·

(S ′i × S ′j

)(A3)

+ 2 sin2

(θij2

)(sin2 ϑS ′xi S ′xj + cos2 ϑS ′zi S ′zj

) ]−H cosϑ

∑i

S ′zi ,

where θij = θi − θj . We note that for Jij = J1,sin θij = 0, therefore the field-induced scalar spin chi-rality of the non-coplanar (umbrella) spin configurationsdefined as χ =

∑S ′i ·

(S ′j × S ′k

)is induced only within

the triangular plaquettes of the NNN bonds on the hon-eycomb lattice or the triangular plaquettes of the NN bi-layer triangular lattice. Here sin θij = νij | sin θij |, whereνij = ±1 denotes the sign of the magnon hopping alongthe triangular plaquettes of the honeycomb (bilayer tri-angular) lattice.

Usually, the net chirality vanishes on the triangularlattice because neighbouring triangular plaquettes con-tribute equal and opposite chirality5. But the field-induced spin chirality of the non-coplanar (umbrella) spinstructure will be finite as it is coupled to the magnetiza-tion of the non-coplanar spin configuration. The sign ofthe scalar-chirality is determined by the magnetic fieldand it has the same sign on each honeycomb (bilayer)triangle for H > 0, whereas for H < 0 the spins on eachhoneycomb (bilayer) triangle flip, now ϑ→ π+ϑ on eachtriangle. In this case the net scalar-chirality is nonzero42.The origin of the spin chirality can also be inferred fromgeometric frustration of the lattice, which can allow a chi-ral spin liquid phase. In this case, the scalar spin chiralitycan be spontaneously developed. Because of the scalar

5

spin chirality the system has already acquired a real spaceBerry curvature from the chiral magnetic spin structure.We therefore expect the spontaneously-induced and thefield-induced spin chirality to have the same topologicaleffects on the underlying magnetic excitations.

In the present case, it is advantageous to introduce theHolstein-Primakoff bosonization62: Szi = S−a†iai, S

+i ≈√

2Sai =(S−i)†, where S±i = Sxi ±iS

yi and a†i (ai) are the

bosonic creation (annihilation) operators. The magnontight binding Hamiltonian is given by

HJ1= S

∑〈i,j〉

[t1,z(a

†iai + a†jaj) + t1,r(a

†iaj + h.c.) (A4)

+ t1,o(a†ia†j + h.c.)

],

HJ2= S

∑〈〈i,j〉〉

[t2,z(a

†iai + a†jaj) + t2(e−iφija†iaj + h.c.)

(A5)

+ t2,o(a†ia†j + h.c.)

]; HH = Hϑ

∑i

a†iai,

where 〈i, j〉 and 〈〈i, j〉〉 denote the summations over theNN and NNN sites respectively.

t1,z = J1[1− 2 cos2 ϑ)], t1,r = −J1[1− sin2 ϑ

], (A6)

t1,o = J1 sin2 ϑ, t2,z =J22

[1− 3 cos2 ϑ], (A7)

t2 =√

(t2,r)2 + (t2,m)2, t2,r = −J22

(1− 3 sin2 ϑ/2),

(A8)

t2,m =

√3J22

cosϑ, t2,o =3J2

4sin2 ϑ, (A9)

and Hϑ = H cosϑ. The angle ϑ is determined from themean-field energy, given by

E0 = −3J12

(1− 2 cos2 ϑ

)− 3

2J2(1− 3 cos2 ϑ

)−H cosϑ,

(A10)

where E0 = EMF /NS2 and N is the total number of

sites on the honeycomb lattice. The magnetic field isrescaled in unit of S. Minimizing this energy yields thecanting angle cosϑ = H/Hs, where Hs = 3(2J1 +3J2) isthe saturation field. The solid angle subtended by threenon-coplanar spins is given by φij = νijφ, where φ =tan−1[t2,m/t2,r]. In Fourier space the Hamiltonian canbe written as H = 1

2S∑

k Ψ†kHkΨk+const., where Ψk =

(ψ†k, ψ−k), with ψ†k = (a†k,A, a†k,B).

Hk =

Ik −mk t1,rf∗k t2,oλ

∗k t1,of

∗k

t1,rfk Ik +mk t1,ofk t2,oλ∗k

t2,oλk t1,of∗k Ik +mk t1,rf

∗k

t1,ofk t2,oλk t1,rfk Ik −mk

,

(A11)

-1.5 0 1.5

kx/π

-1.5

0

1.5

ky/π

RΓ

N

Q K

MP

FIG. 6: Color online. The first Brillouin zone of the triangularlattice and the corresponding paths that will be adopted inthis section.

where fk = 1 + e−ika + e−i(ka+kb), λk = 2[cos ka +cos kb + cos(ka + kb)]; mk = 2t2 sinφ[sin ka + sin kb −sin(ka + kb)]; Ik = 3t1,z + 6t2,z + t2 cosφλk + H cosϑ =

3(J1 + J2) + t2 cosφλk. The vectors are a =√

3x andb = −

√3x/2+3y/2 with ka = k·a and kb = k·b. We diag-

onalize the Hamiltonian numerically via the generalizedBogoluibov transformation61. For J2/J1 < 1/6 or equiv-alently J2/J1 → 0 the Hamiltonian recovers collinear(canted) Néel antiferromagnet at H = 0 (H < Hs) aswell as collinear ferromagnet at H = Hs. These limitingcases require the DM interaction for topological featuresto exist as previously shown14–19. The dominant J2 limitis different and requires no DM interaction for topologicalfeatures to exist.

Appendix B: Bilayer triangular-latticeantiferromagnets

The dominant J2 limit of frustrated honeycomb latticeis isomorphic to the bilayer triangular-lattice Heisenbergantiferromagnets. The conventional Hamiltonian for bi-layer triangular-lattice Heisenberg antiferromagnets ap-plicable to real materials is given by

H = J2∑〈i,j〉,τ

[S⊥iτ · S

⊥jτ + ∆SziτSzjτ

]−H

∑i,τ

Szi,τ (B1)

+ J1∑〈i,j〉,ττ ′

[S⊥iτ · S

⊥jτ ′ + ∆SziτSzjτ ′

],

where τ labels the top and bottom layers and S⊥i =(Sxi ,S

yi ). Note that all the interactions are now nearest-

neighbour (NN). The easy-plane anisotropy lies in therange 0 ≤ ∆ ≤ 1. The NN intraplane coupling is J2 > 0and the NN interplane coupling is J1 > 0 with J1 J2,i.e., quasi-2D limit. After the rotation in the spin space

6

0

1

2

3

4

5

6

7

8E/J

1S

R M Γ N Q K M P

FIG. 7: Color online. Dirac magnon bands (at N and K) ofthe bilayer XXZ triangular-lattice antiferromagnet with theconventional 120 coplanar order at zero magnetic field H =0. The plot is generated with ∆ = 0.7 and J2/J1 = 1.3.

(A1) and (A2) we have

H =∑ij

Jij[

cos θijS ′i · S′j + sin θij cosϑz ·

(S ′i × S ′j

)(B2)

+ (∆− cos θij)(sin2 ϑS ′xi S ′xj + cos2 ϑS ′zi S ′zj

) ]−H cosϑ

∑i

S ′zi ,

where we have retained the free magnon model. Jij =J1(J2). Because of the antiferromagnetic interplane cou-pling the spins on the top layer are orientated in theopposite direction to those on the bottom layer, hencesin θij = 0 for Jij = J1 and the scalar-chirality van-ishes. However, each layer form a 120 coplanar orderand sin θij = νij sin(120) for Jij = J2, where νij = ±for magnon hopping on the top and bottom layers respec-tively. The scalar-chirality of the non-coplanar structureon both layers are along the positive z-axis for H > 0,whereas for H < 0 the spins on each triangular-layer flipby 180. Now ϑ → π + ϑ on each layer. Therefore thenet scalar-chirality is nonzero in both cases42.

We adopt the one-sublattice structure63,64 on eachlayer of the triangular lattice and label them A and B.From figure 1 (a) in the main text, we see that thestacking of the bilayer triangle is such that there aresix nearest-neighbours on each layer and three nearestneighbours between the layers. The parameters of thecorresponding tight-binding model are

0

1

2

3

4

5

6

7

8

E/J

1S

R M Γ N Q K M P

FIG. 8: Color online. Topological magnon bands of the bi-layer XXZ triangular-lattice antiferromagnet with the non-coplanar (umbrella) spin configuration at nonzero magneticfield H = 0.2Hs. The plot is generated with ∆ = 0.7 andJ2/J1 = 1.3.

t1,z = J1[1− (1 + ∆) cos2 ϑ)], t1,r = −J1[1− 1 + ∆

2sin2 ϑ

],

(B3)

t1,o =J1(1 + ∆)

2sin2 ϑ, t2,z =

J22

[1− (2∆ + 1) cos2 ϑ],

(B4)

t2 =√

(t2,r)2 + (t2,m)2, t2,r = −J22

(1− (2∆ + 1)

2sin2 ϑ

),

(B5)

t2,m =

√3J22

cosϑ, t2,o =J2(2∆ + 1)

4sin2 ϑ, (B6)

with cosϑ = H/Hs and Hs = 3J1(1 + ∆) + 3J2(2∆ + 1).In the basis Ψ†k = (a†k,A, a−k,A, a

†k,B , a−k,B), the mo-

mentum space Hamiltonian is given by

Hk =

Ik −mk t2,oλk t1,rf∗k t1,of

∗k

t2,oλ∗k Ik +mk t1,ofk t1,rfk

t1,rfk t1,of∗k Ik +mk t2,oλk

t1,ofk t1,rf∗k t2,oλ

∗k Ik −mk

, (B7)

where fk = 1+e−ik1 +e−i(k1+k2), λk = 2[cos k1+cos k2+cos(k1 +k2)]; mk = 2t2 sinφ[sin k1 +sin k2−sin(k1 +k2)];Ik = 3t1,z + 6t2,z + t2 cosφλk + H cosϑ = 3J1 + 3J2 +t2 cosφλk. Here ki = k · ei and the primitive vectors ofthe triangular lattice are e1 = x and e2 = −x/2+

√3y/2.

The Brillouin zone paths are depicted in figure (6).It is evident that when the interplane coupling van-

ishes, i.e., J1 = 0 = t1,r = t1,o, the Hamiltonian reducesto two decoupled triangular-lattice XXZ antiferromag-nets. For J1 6= 0 we have shown the magnon bandsof the bilayer triangular-lattice XXZ antiferromagnets at

7

H = 0 andH = 0.2Hs respectively in figures. (7) and (8).We see that the magnon bands have the same structureas the frustrated honeycomb lattice shown in the maintext. Topological magnon bands in figure (8) directly

imply the existence of nonzero Chern numbers, magnonedge modes, and thermal Hall conductivity. We note that∆ = 1 has the same topological features as expected.

∗ Electronic address: sowerre@perimeterinstitute.ca1 M. Hirschberger et al., Phys. Rev. Lett. 115, 106603(2015).

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