Floquet Topological Magnons

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

We introduce the concept of Floquet topological magnons — a mechanism by which a synthetictunable Dzyaloshinskii-Moriya interaction (DMI) can be generated in quantum magnets using cir-cularly polarized electric (laser) field. The resulting effect is that Dirac magnons and nodal-linemagnons in two-dimensional (2D) and three-dimensional (3D) quantum magnets can be tuned tomagnon Chern insulators and Weyl magnons respectively under circularly polarized laser field. TheFloquet formalism also yields a tunable intrinsic DMI in insulating quantum magnets without aninversion center. We demonstrate that the Floquet topological magnons possess a finite thermalHall conductivity tunable by the laser field.

Quantum magnets without an inversion center allow anintrinsic Dzyaloshinskii-Moriya interaction (DMI) [1, 2],which is a consequence of spin-orbit coupling (SOC) andit is usually fixed in different magnets. The associatedmagnon bands in the magnetically ordered systems havea nontrivial topology with Chern number protected chi-ral magnon edge modes in 2D systems and magnon sur-face states in 3D systems. They are dubbed topologicalmagnon Chern insulators [3–9] and Weyl magnons [10–13] respectively. They are the analogs of electronic topo-logical (Chern) insulators [14–16] and Weyl semimetals[17–19]. However, due to the charge-neutral property ofmagnons, topological magnonic materials are believed tobe potential candidates for low-dissipation magnon trans-ports in insulating quantum magnets and they are appli-cable to magnon spintronics and magnetic data storage[20]. Topological magnonic materials (i.e. magnon Cherninsulators andWeyl magnons) also possess a thermal Halleffect as predicted theoretically [21–25] and observed ex-perimentally [26–28]. To date, topological magnon bandshave been realized only in a quasi-2D kagomé ferromag-net [29].

Generally, every quantum ferromagnetic material doesnot have a strong intrinsic DMI necessary for topologi-cal magnons to exist. For instance, the single crystals ofthe ferromagnetic honeycomb compounds CrX3 (X ≡ Br,Cl, and I) show no evidence of DMI [30–35], and kagoméhaydeeite also does not have a finite (topological) energygap in the observed spin-wave spectra [36], suggestingthat the DMI does not play a significant role in hay-deeite. These 2D ferromagnetic materials with negligibleDMI are candidates for Dirac magnons [37], and the 3Dferromagnetic counterparts are candidates for nodal-linemagnons [13, 38]. By applying a circularly polarized laserfield in these “topologically trivial” systems, one can gen-erate topological magnons (i.e., magnon Chern insulatorsand Weyl magnons) via a tunable synthetic laser-inducedDMI. This is particularly important as it offers a way totune the DMI in magnetic materials.

The Floquet theory [39] of laser-driven systems pro-vides a theoretical as well as experimental method toengineer such topologically nontrivial systems. This for-malism is mostly dominated by electronic systems [40–59]and also optical bosonic systems [60–65]. Moreover, an

applied laser field provides a means for coherent controlof the magnetization in 1D quantum magnets [66–70].

In this Letter, we introduce the Floquet formalism to2D and 3D ferromagnetic quantum magnets in the pres-ence of a circularly polarized electric (laser) field. Weshow that the underlying charge-neutral magnons acquirea time-dependent Aharonov-Casher phase [71], whichgenerates a tunable synthetic DMI in magnetic systemswith negligible intrinsic DMI. This leads to emergentnontrivial magnon Chern insulators and Weyl magnonswith finite thermal Hall conductivity, which can be ma-nipulated by the laser field. Our results provide a novelplatform to engineer topological magnons in topologicallytrivial insulating quantum magnets. We hope that theseresults will extend the experimental search for topolog-ical magnons to a broader class of insulating quantummagnetic materials even without an intrinsic DMI.

Let us consider 2D and 3D ferromagnetic spin systemsdescribed by the pristine Hamiltonian

H0 = −J∑

〈α,β〉Sα · Sβ , (1)

where 〈α, β〉 denotes summation over nearest-neighbour(NN) sites and Sα are the magnetic spin vectors at thelattice sites α located at rα and J is the ferromagneticinteraction. For now we take the intrinsic DMI to benegligible, which is the case in most ferromagnetic com-pounds. Therefore, the goal is to generate a syntheticDMI by periodic modulation of the lattice using a time-dependent electric field of a laser light.

We are interested in the underlying magnon excita-tions of the quantum spin Hamiltonian (1) as describedby the Holstein-Primakoff transformation: Szα = S −a†αaα, S

+α ≈

√2Saα = (S−α )†, where a†α(aα) are the

bosonic creation (annihilation) operators, and S±α =Sxα± iSyα denote the spin creation and annihilation oper-ators which correspond to the hopping terms. However,we will retain the spin operators in order to show theexplicit form of the laser-induced DMI. As magnons arecharge-neutral bosonic quasi-particles they do not inter-act with electromagnetic field except through their mag-netic dipole moment, which we assume to be in the z-direction µ = gµB z, where µB is the Bohr magneton andg the Landé g-factor. Now suppose that a circularly po-

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larized laser (electromagnetic) field with dominant oscil-lating electric field components E(t) is irradiated perpen-dicular to the magnetic material lying on the x-y plane.The magnetic dipole moment of magnon quasi-particleshopping in such an electric field background will acquirethe time-dependent Aharonov-Casher phase [71]

θαβ(t) =gµB~c2

∫ rβ

rα

(E(t)× z

)· d`, (2)

where ~ is the reduced Planck’s constant and c is thespeed of light. The oscillating electric field obeys the re-lationE(t) = −∂A(t)/∂t, whereA(t) is the time-periodicvector potential. We choose an oscillating electric fieldsuch that

E(t)× z = E0(sinωt,− cosωt, 0), (3)

where E0 is the amplitude of the electric field. The cor-responding time-dependent Hamiltonian is given by

H(t) = −∑

〈α,β〉

[J

2

(S−α S

+β e

iθαβ(t) + h.c.)

+ JSzαSzβ

],

(4)

Noticing that the direction of the vector pointing fromrα to rβ defines a relative angle φαβ , we get θαβ(t) =λ sin(ωt− φαβ) where

λ =gµBE0a

~c2, (5)

is the dimensionless parameter characterizing the lightintensity in the magnonic Floquet formalism and a isthe lattice constant. Notice the difference between thedimensionless parameter in Eq. (5) and that of electronicFloquet formalism [41, 43].

The Floquet theory is a standard mechanism to studydriven quantum systems in the likes of Eq. 4 [40–43]. Itenables one to transform a time-dependent model intoa static effective model governed by what is called theFloquet Hamiltonian. Now, we proceed with this formal-ism. The static time-independent effective Hamiltoniancan be expanded in the power series of ω−1 and it isgiven by Heff =

∑i≥0 ω

−iH(i)eff . We calculate the series

expansion of the effective Hamiltonian using the discreteFourier component of the time-dependent HamiltonianHn = 1

T

∫ T0dte−inωtH(t) with period T = 2π/ω, where

Hn = −∑

〈αβ〉

[Jn,⊥2

(S−α S

+β e−inφαβ + h.c.

)+ Jδn,0S

zαS

zβ

],

(6)

with Jn,⊥ = JJn(λ), and Jn(λ) is the Bessel functionof order n ∈ Z. δn,` = 1 for n = ` and zero other-wise. We have used the standard relation e[iz sin(x)] =∑∞m=−∞ Jm(z)eimx. In the high frequency limit ω J

the leading order contribution is the zeroth order effectiveHamiltonian given by H(0)

eff = H0, where

H(0)eff = −

∑

〈αβ〉

[J0,⊥(SxαS

xβ + SyαS

yβ) + JSzαS

zβ

], (7)

and J0,⊥ = JJ0(λ). Thus, the zeroth order term yieldsan XXZ ferromagnetic Hamiltonian. For quantum spin-1/2 ferromagnetic systems, Eq. 7 is equivalent to theBose-Einstein condensation of hardcore bosons studiedby Matsubara and Matsuda [72], but in this case J ≥|J0,⊥|. By lowering the frequency the first order contri-bution to the effective Hamiltonian is non-negligible. Itis given by

H(1)eff =

∞∑

n=1

1

n

[Hn,H−n

]. (8)

The commutator of the spin operators results in a prod-uct of three spins reminiscent of the scalar spin chiral-ity. In order to see this we note that the z-componentof the spins vanishes for n ≥ 1, therefore the com-mutation relation in the first order term involves only[S+α S−β , S

+ρ S−γ

]= 2

(δβρS

zβS

+α S−γ − δαγSzαS+

ρ S−β

). We

use this relation together with the identity J−n(z) =(−)nJn(z) and obtain

H(1)eff =

∑

∆/∇J

(1)αβ Sγ · (Sα × Sβ), (9)

where

J(1)αβ = −

∞∑

n=1

2(−)nJ2n,⊥ sin (nΦαβ) , (10)

with Φαβ = φαγ−φβγ . Sγ = Szγ z and γ is the intermedi-ate lattice site between α and β [21] and the sum is overthe triangular plaquettes of the chosen lattice geometry.

The main result of this Letter is the induced syntheticDMI by circularly polarized laser field as given in Eq. (9).The DMI points along the z-direction perpendicular tothe x-y plane. On the kagomé and pyrochlore latticesthe synthetic DMI lies within the bonds of the NN sites,whereas on the honeycomb lattice it lies within the bondsof the next-nearest neighbour (NNN) sites. The DMI isthe primary source of magnon Chern insulators and Weylmagnons in insulating quantum ferromagnets. Conse-quently, Dirac magnons in 2D ferromagnetic systems andnodal-line magnons in 3D ferromagnetic systems withnegligible intrinsic DMI will be driven to magnon Cherninsulators and Weyl magnons respectively by the circu-larly polarized laser field.

Now, we exemplify this method by utilizing the 2Dhoneycomb ferromagnets for simplicity (see Fig. 1). Inthis case the relative angle is given by φαγ−φβγ = 2π

3 ναβ[57, 58], where ναβ = +/− for hopping within ∆/∇ re-spectively. Hence J (1)

αβ ≈√

3ναβJ21,⊥. The total effective

3

FIG. 1. The schematic of the honeycomb lattice with laser-field-induced DMI. The blue dotted and crossed circles denotethe direction of the synthetic DM vectors pointing in and outof the lattice plane on the NNN bonds. Here, δi are thevectors connecting the NN sites and bi connect the NNN sites.The different colours of the solid circles red(green) are usedto denote the two sublattices A(B).

FIG. 2. Floquet magnon band structure of circularly laser-driven honeycomb quantum ferromagnetic spin system forJ = 1 meV λ = 1.5 in units of gµBa/~c2 and ~ω = 2.5.(i) Floquet Dirac magnon in the zeroth order approximationDF = 0. (ii) Floquet topological magnon in the first orderapproximation DF ' 0.014J .

Floquet Hamiltonian up to first order is given by

Heff = −∑

〈αβ〉

[J0,⊥(SxαS

xβ + SyαS

yβ) + JSzαS

zβ

](11)

+DF

∑

∆/∇ναβSγ · (Sα × Sβ),

where DF =√

3J21,⊥/ω. We note that |J0,⊥| ≤ J , so that

the spins would prefer to order magnetically along thez-direction. The synthetic DMI does not affect the ferro-magnetic ordering of the spins. We study the underlyingmagnon excitations of Eq. 11 by implementing the lin-ear order Holstein-Primakoff transformation mentionedabove on the two sublattices A and B of the honeycomblattice. In the bosonic representation the synthetic DMIis imaginary. It generates a synthetic magnetic gaugeflux within the NNN sites. The momentum space Hamil-tonian is given by

Heff(k) = h0I2×2 + h(k) · σ, (12)

0 0.5 1

λ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

-κF xy(λ,T

)

(i)

T = 0.3JT = 0.5JT = 0.75J

0 0.5 1

T/J

0

0.005

0.01

0.015

0.02(ii)

λ = 0.1λ = 0.15λ = 0.2

FIG. 3. Floquet tunable thermal Hall conductivity. (i) κFxy

vs. λ for different temperatures. (ii) κFxy vs. T for different

amplitudes. Here, J = 1 meV and ~ω = 2.5J .

where σ = (σx, σy, σz) are Pauli matrices and I2×2 isan identity 2 × 2 matrix. h0 = 3JS, and h(k) =(hx(k), hy(k), hz(k)), with hx(k) = −J0,⊥S

∑j cosk ·

δj , hy(k) = −J0,⊥S∑j sink · δj , and hz(k) =

2DFS2∑j sink · bj . The Floquet magnon bands are

given by

εFηk = h0 + η√

h(k) · h(k), (13)

where η = ± labels the top and the bottom bands respec-tively. Henceforth we set S = 1/2. The Floquet magnonbands are depicted in Fig. 2 for zeroth order (i) and firstorder (ii) contributions to the effective Hamiltonian. Inthe former (i), the system exhibits Dirac magnon points,but they are different from the Dirac magnon pointsin the undriven systems [37] because the Floquet Diracmagnons can be manipulated by anisotropic laser fieldamplitude, i.e., A0 = (Ax, Ay, 0). In this case the Diracmagnon points can move away from the high symmetrypoints of the Brillouin zone at ±K = (±4π/3

√3, 0) and

can also be gapped out by fine-tuning the anisotropicamplitudes. However, the system will still remain topo-logically trivial at the zeroth order as time-reversal sym-metry is preserved.

A nontrivial band topology arises by lowering the fre-quency and the first order correction becomes impor-tant, which breaks time-reversal symmetry. This leadsto a synthetic DMI which induces a gap (∆gap ∼ 2|DF |)at the Dirac points as shown in Fig. 2(ii). The Flo-quet magnon bands now acquire a nonzero Berry cur-vature which has the form ΩFηk =

(∇×AF

ηk

)zwhere

AFηk = i 〈ψFηk|∇|ψFηk〉 is the Berry connection and ψFηk

are the Floquet eigenvectors of Heff(k). The FloquetChern numbers are given by the integration of the Berrycurvatures over the Brillouin zone,

CFη =1

2π

∫

BZ

d2k ΩFηk. (14)

4

It is easily shown that the Floquet Chern numbers areproportional to the synthetic DMI: CFη ∼ ηsgn[DF ],which can be controlled by the laser field.

The nontrivial topology of the Floquet magnon bandshas an important transport consequences. It can lead toFloquet thermal magnon Hall effect. This refers to thegeneration of a transverse heat current upon the applica-tion of a longitudinal temperature gradient. In this case,the synthetic laser-field-induced Berry curvature due tothe DMI appear in the equations of motion of a magnonwave packet in the same mathematical structure as amagnetic field in the Lorentz force. In other words, theBerry curvature due to the DMI acts like an effectivemagnetic field in momentum space on the magnons. Theresulting effect is the production of a transverse thermalHall conductivity, which can derived from linear responsetheory [22]. The Floquet thermal Hall conductivity isgiven by

κFxy(λ, T ) = − k2BT

(2π)2~∑

η=±

∫

BZ

d2kc2 (nη) ΩFηk, (15)

where nη ≡ nB(εηk) = (eεηk/kBT − 1)−1 is the Bosefunction close to thermal equilibrium, c2(x) = (1 +

x)(ln 1+x

x

)2−(lnx)2−2Li2(−x), and Lin(x) is a polylog-arithm. In Fig. 3 we have shown the plots of κFxy(λ, T )as functions of the parameters in units of kB/~. For thetwo-band honeycomb ferromagnets the Floquet thermalHall conductivity is negative and it is tunable by theamplitude (frequency) of the laser field. In particular,κFxy(λ, T ) can be tuned off at the zeros of Jn(λ). This isusually not possible in materials with a strong intrinsicDMI in the absence of a laser field.

However, applying circularly polarized laser field inmagnetic materials with a strong intrinsic DMI can in-duce a tunable DMI. Using the simple honeycomb fer-romagnetic model as an example, the suitable DMI dueto inversion symmetry breaking of the lattice is of theform [7] HDMI = D0

∑〈〈αβ〉〉 ναβ z · Sα × Sβ , where

the summation is taken over the triangular plaquettesof the NNN sites. In the presence of a circularly polar-ized laser field with sufficiently high frequency, the ze-roth order correction to the effective Hamiltonian givesHeffDMI = D0J0(λ)

∑〈〈αβ〉〉 ναβ z ·Sα×Sβ . Now the value

of the intrinsic DMI can be manipulated by varying λ.For λ < λc ≈ 2.4048 a gap exists at the Dirac pointsand it closes near the first zero of the Bessel functionλ ≈ λc and reopens for λ > λc. Consequently, the sign ofthe Chern numbers (Berry curvatures) of the two bandschanges as shown in Fig. 4. Hence, a sign change emergesin κFxy(λ, T ) (not shown). We note that a sign change isnot possible in the undriven system on the honeycomblattice [8, 9]. The complete topological phase transi-tion including a trivial insulating regime with zero Chernnumber can be found by including NNN Heisenberg in-teraction as we have shown in the supplemental material[73].

It is important to note that the circularly polarized

0 0.5 1

kx/(2π/√3)

ε(kx)F/J

λ < λc

C+ = −1

C− = 1

0 0.5 1

kx/(2π/√3)

λ ≈ λc

0 0.5 1

kx/(2π/√3)

λ > λc

C+ = 1

C− = −1

FIG. 4. Evolution of sign change in the Floquet Chernnumbers of honeycomb ferromagnets with tunable intrin-sic DMI for different values of λ in units of gµBa/~c2 andD0/J = 0.15. The black lines are the magnon bulk bandsand blue lines denote the chiral magnon edge modes gaplessat ε(kx = π/

√3) = h0 as indicated by the dashed line.

laser field is necessary to induce a tunable synthetic DMIin quantum ferromagnets with negligible intrinsic DMI.However, in frustrated magnets with a coplanar magneticorder, a static magnetic field applied perpendicular tothe plane of the magnets can induce a tunable syntheticscalar spin chirality due to non-coplanar spin configura-tions. In this case tunable topological magnons can beproduced even in the absence of an intrinsic DMI [74–76].

In conclusion, we have shown that the time-dependentAharonov-Casher phase acquired by charge-neutralmagnons in a time varying circularly polarized electric(laser) field induces a synthetic DMI, leading to non-trivial topological magnons in the absence of an intrinsicDMI. Therefore, Dirac magnons and nodal-line magnonsbecome magnon Chern insulators and Weyl magnons re-spectively under radiation. Our results also showed thatthe intrinsic DMI in insulating quantum magnets withoutan inversion symmetry can be manipulated with circu-larly polarized laser field. Hence, the emergent thermalHall effect can be turned. It is experimentally feasibleto investigate the proposed phenomena in magnetic in-sulators using ultrafast terahertz spectroscopy [77]. Thisformalism may find application in the switching of mag-netization and magnon spin current by a laser field asan important step towards magnon spintronics and mag-netic data storage [78, 79]. We believe that these resultswill extend the study of topological magnons to differ-ent magnetic materials without any restrictions. In aninhomogeneous static electric field a time-independentAharonov-Casher phase can be acquired by magnons andleads to magnonic Landau levels in insulating magnets[80].

5

Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and by

the Province of Ontario through the Ministry of Researchand Innovation.

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Floquet Topological Magnons: Supplemental Material

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

I. TOPOLOGICAL MAGNON PHASEDIAGRAM

For a fixed λ such that Jn(λ) 6= 0 the honeycomb Flo-quet topological magnons maps to the static XXZ ferro-magnetic quantum spin Hamiltonian with DM interac-tion

H = −∑

〈α,β〉[J⊥(SxαS

xβ + SyαS

yβ) + JzS

zαS

zβ ] (1)

+D∑

〈〈αβ〉〉ναβ z · Sα × Sβ − J2

∑

〈〈α,β〉〉Sα · Sβ .

We have added an isotropic next-nearest-neighbour(NNN) ferromagnetic exchange coupling J2 and Jz > J⊥as before. In the bosonic representation we obtain

H = −t⊥∑

〈αβ〉(a†αaβ + h.c.)− t2

∑

〈〈αβ〉〉(eiφαβa†αaβ + h.c.)

(2)

+ tA∑

α∈Aa†αaα + tB

∑

α∈Ba†αaα.

The mean-field energy has been dropped. t⊥ = J⊥S, t2 =√t′22 + t2D, t

′2(tD) = J2S(DS), tA(B) = 3Jz,A(B)S+6J2S.

The phase factor is φαβ = ναβφ, where φ = arctan(tD/t′2)

is a magnetic flux generated by the DM interaction onthe NNN sites.

Next, we Fourier transform the magnon hoppingHamiltonian (2) into momentum space and the resultingHamiltonian can be written as H =

∑k ψ†k · H(k) · ψk,

with ψ†k = (a†kA, a†kB). The momentum space Hamilto-

nian is given by

H(k) = d0(k)I + d(k) · σ, (3)

where I is an identity 2 × 2 matrix and σ = (σx, σy, σz)are Pauli matrices and d = (dx, dy, dz) are given by

d0(k) = ε0 − 2t2 cosφ∑

j

cosk · bj , (4)

dx(k) = −t⊥∑

j

cosk · δj , (5)

dy(k) = −t⊥∑

j

sink · δj , (6)

dz(k) = M − 2t2 sinφ∑

j

sink · bj , (7)

with ε0 = (tA + tB)/2 + 6J2S and M = (tA− tB)/2. Themagnon bands are given by

εs(k) = d0 ±√d · d = d0 + sε(k), (8)

-π 0 π

φ

0

M

t2C± = ∓1C± = ±1

C± = 0

C± = 0

3

√3

-3

√3

FIG. 1. The complete phase diagram of the Haldane magnoninsulator. In the topological magnon insulator regime C± =±1 and in the trivial magnon insulator C± = 0.

where s = ± for the upper and lower magnon bandsrespectively. The the eigenvectors are given by

|ψsk〉 =1√2

(√1 + s

dz(k)

ε(k), − se−iϕk

√1− sdz(k)

ε(k)

)T,

(9)where

tanϕk =dy(k)

dx(k). (10)

A. Magnon phase transition

The model above is termed a Haldane magnon in-sulator. It has only been studied in the topologicalphase, which is the only phase one would get at theisotropic point J⊥ = Jz [1, 2](see also [3]). There-fore, the full topological phase diagram of the Haldanemagnon insulator has not been mapped out. In this sec-tion, we will present the complete topological phase di-agram of Haldane magnon insulator and also show thatit differs from the electronic counterpart [4] in terms oftransport properties. For φ 6= nπ, the magnon bandshave nonzero Berry curvature Ωsk = (∇×Ask)z whereAsk = i 〈ψsk|∇|ψsk〉 is the Berry connection. The Chernnumber is given by

arX

iv:1

705.

0469

4v5

[co

nd-m

at.s

tr-e

l] 2

8 Se

p 20

18

2

Cs =1

2π

∫d2kΩsk. (11)

The maximum contribution to the Berry curvature comesfrom the states near the corners of the Brillouin zoneKτ = (τ4π/3

√3, 0) with τ = ±. Therefore, we expand

the Hamiltonian near this point and the differential oper-ator can be expressed in polar coordinates (k,φk) wherek = |k|. The Berry curvature takes the form

Ωτks =1

k

(∂(kAτsϕk)

∂k− ∂Aτsk

∂ϕk

), (12)

where Aτsϕk = 1k i 〈ψτks|∂ϕkψτsk〉 and Aτsk = i 〈ψτks|∂kψτks〉.

We note that ϕk appears as a U(1) phase in the eigen-vectors, therefore Aτsϕk and Aτsk are independent of ϕk,thus the second term in Eq. 12 vanishes and the resultingexpression becomes

Ωτks =1

k

∂

∂kPτs (k); Pτs (k) = i 〈ψτsk|∂φkψτsk〉 . (13)

By integrating the Berry curvature we obtain

Cs =∑

τ=±[Pτs (∞)− Pτs (0)], (14)

= −s2

∑

τ=±τsgn[M + 3

√3τt2 sinφ]. (15)

A topological magnon phase transition occurs at Mc =3√

3t2 sinφ. The regime M < Mc corresponds to a topo-logical magnon Chern insulator with C± = ∓1 and theregime M > Mc corresponds to a trivial magnon insu-lator with C± = 0.The phase diagram of the Haldanemagnon insulator is depicted in Fig. 1. From bulk-edgecorrespondence we expect chiral magnon edge modes inthe vicinity of the bulk gap, which are protected bythe Chern numbers. As shown in Fig. 2, the topologi-cal regime possesses gapless chiral magnon edge modeswhereas the chiral magnon edge modes in the trivialmagnon insulator regime contribute to the bulk magnonbands.

B. Magnon quantum anomalous Hall effect

In electronic systems, quantum anomalous Hall effectrefers to the generation of a transverse quantized Hallconductivity without an applied magnetic field. Basi-cally, an electric field E induces a charge current J e andthe Berry curvature acts as an effective magnetic field inmomentum space. For insulating quantum ferromagnetsthe magnon excitations are charge-neutral and do notexperience a Lorentz force. Instead, a temperature gra-dient −∇T induces a heat current JQ, and the Berrycurvature due to the DMI appears in the equations ofmotion of a magnon wave packet in the same mathemat-ical structure as a magnetic field in the Lorentz force.

0 0.5 1

kx/(2π/√

3)

0

0.5

1

1.5

2

2.5

3

ε(kx)

(i)

0 0.5 1

kx/(2π/√

3)

(ii)

FIG. 2. Topological and trivial magnon insulator phases onthe honeycomb quantum ferromagnets for (t′2 = tD = t2 =0.2t⊥) or φ = π/4. The parameters are chosen such that thephase transition occurs at (M/t⊥)c ≈ 0.73 where the magnonbulk gap closes (not shown). (i) Topological Chern magnoninsulator with gapless magnon edge modes (blue lines) forM/t⊥ = .1. (ii) Trivial magnon insulator with bulk gap andno gapless magnon edge modes (blue lines) for M/t⊥ = 1.

In ferromagnets, this does not require an applied mag-netic field analogous to the electronic counterparts. Thethermal Hall conductivity is given by

κxy = − k2BT

(2π)2

∑

s=±

∫

BZ

d2kc2 (ns) Ωsk, (16)

where ns ≡ nB [εsk] = [eεsk/kBT − 1]−1 is the Bose func-tion, c2(x) = (1 + x)

(ln 1+x

x

)2 − (lnx)2 − 2Li2(−x), andLin(x) is a polylogarithm.

The major difference here is that magnon transportsrequire a finite temperature due to the Bose function. Inother words, magnonic systems do not have conductionand valence bands and no bands are fully occupied orcompletely empty. Instead, the magnons can be ther-mally populated at low temperatures near the k = 0mode. In fact, the maximum contribution of κxy comesfrom the lowest magnon band near the k = 0 mode atlow temperatures.

Unlike electronic systems, the topological and the triv-ial insulating regimes in Fig. 2 will possess a finite κxydue to the population of the band near the k = 0 modeat low temperatures. In this limit the Berry curvature ofthe lowest band in the topological and trivial insulatingphases has the form

Ω−(k→ 0) ' c1kxk2y + c2k

2xk

2y

c3, (17)

3

where c1 = −27(Mt2)/8, c2 = 81√

3t2 sinφ/8 and c3 =4(M2 + 9t22)3/2. In the low-temperature limit κxy can bewell-approximated as

κxy '1

T

∫d2k

(2π)2nB [ε−(k)][ε−(k) + ε+(k)]2Ω−(k).

(18)

Near the k = 0 mode, ε−(k) + ε+(k) ' 2(tA + tB) = t0and ε−(k) ' a + bk2, where a = (tA + tB)/2 − 3t⊥ andb = 3t⊥/4. Now, we can parameterize the momentum

in the polar coordinate (kx = k cos θ, ky = k sin θ) andperform the angular part of the integration,

κxy 'c2c3

t2016πT

∫ ∞

0

k dkk4

e(a+bk2)/T − 1,

=T 2

16πb3c2c3

Li3[

exp(− aT

)]. (19)

This is the low-temperature thermal Hall conductivity inthe topological and trivial insulating phases.

[1] S. A. Owerre, J. Phys.: Condens. Matter 28, 386001(2016).

[2] S. A. Owerre, J. Appl. Phys. 120, 043903 (2016).[3] S. K. Kim et.al., Y. Tserkovnyak, Phys. Rev. Lett. 117,

227201 (2016).[4] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).