Einstein-de Haas Effect of Topological Magnons

Jun Li,1 Trinanjan Datta,2, 1, ∗ and Dao-Xin Yao1, †

1State Key Laboratory of Optoelectronic Materials and Technologies,School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

2Department of Chemistry and Physics, Augusta University, 1120 15th Street, Augusta, Georgia 30912, USA

We predict the existence of Einstein-de Haas effect in topological magnon insulators. Temperature variationof angular momentum in the topological state shows a sign change behavior, akin to the low temperature thermalHall conductance response. This manifests itself as a macroscopic mechanical rotation of the material hostingtopological magnons. We show that an experimentally observable Einstein-de Haas effect can be measured in thesquare-octagon, the kagome, and the honeycomb lattices. Albeit, the effect is the strongest in the square-octagonlattice. We treat both the low and the high temperature phases using spin wave and Schwinger boson theory,respectively. We propose an experimental set up to detect our theoretical predictions. We suggest candidatesquare-octagon materials where our theory can be tested.

The Einstein-de Haas (EdH) effect, predicted in 1908 bythe Nobel Laureate Owen W. Richardson, was the test of avery fundamental concept - did circulating electrons give riseto magnetic moments? [1]. Seven years later, Albert Einsteinand Wander J. de Haas experimentally demonstrated the pos-sibility of transferring magnetic to mechanical angular mo-mentum [2]. Subsequently, their experimental initiative wasperfected by Stewart [3]. The transfer of momentum betweenelectron spins and lattice degrees of freedom in a freely sus-pended magnetized body is a direct outcome of the angularmomentum conservation principle. The rotation can be in-duced by an externally applied transient magnetic field or tem-perature. The reverse effect, generation of a magnetic momentby mechanical rotation, called the Barnett effect has also beenrealized [4]. The very first gyromagnetic ratio measurementswere performed using the EdH effect. At present, g-factormeasurements performed using the EdH technique provide amore accurate value compared to electron-spin resonance orferromagnetic resonance. In addition, the gyromagnetic ratiohas important consequences in the fields of quantum electro-dynamics [5], Bose-Einstein condensates [6], molecular mag-netism, nano-magneto-mechanics, spintronics, and ultrafastmagnetism [7, 8].

The realization of topology, an abstract mathematical con-cept, in correlated electronic and magnetic systems has revo-lutionized the field of condensed matter physics and materialsscience [9]. In a topological state, the bulk and the boundarymay have entirely different properties. For example, in a quan-tum spin Hall state the bulk is insulating, whereas the edgecan support helical transport modes. These novel topologicalstates are robust against external perturbation such as an exter-nal magnetic field. In the bulk of the quantum Hall state, thetopological number that governs the integer Hall conductance(called Chern number) provides a criterion to classify the dif-ferent topological phases which otherwise cannot be relatedby a continuous mapping [10]. A topological band theory hasbeen developed to explain bulk-boundary correspondence be-tween the edge and the bulk states [11].

∗ Corresponding author:tdatta@augusta.edu† Corresponding author:yaodaox@mail.sysu.edu.cn

Spin-wave excitations (magnons) can be topologically pro-tected. Recently, it has been demonstrated that topologicalmagnons could be transported along the sample edge in athermal Hall experiment (chiral edge state) [12]. Differentfrom the electrical Hall effect, in its thermal Hall counter-part, current is produced by a temperature gradient rather thanan electric field. Thus a heat, instead of a charge, currentis detected. The presence of Dzyaloshinskii-Moriya (DM)interaction mimics the role of time-reversal breaking mag-netic field [12]. Current prototypical examples of topologicalmagnon materials include Lu2V2O7 [12], Cu-(1,3 bdc) [13],and CrI3 [14]. Dirac magnons have been proposed in a hon-eycomb ferromagnet [15]. A chiral topological magnon insu-lator in three dimensions has also been studied [16]. ThermalHall effect has been found in various other magnetic systemssuch as spin liquids [17, 18], multiferroics [19], antiferromag-nets [20–23], and the pseudogap phase of a cuprate supercon-ductor [24]. Other neutral quasi-particles such as phonons canalso exhibit thermal Hall effect [25, 26]. Magnons, topolog-ical [27] or otherwise [28], play an important role in devicephysics, too.

In this article, we marry the century old EdH experimen-tal technique with the contemporary concept of topologicalmagnons. We explicitly demonstrate that an EdH effect oftopological magnon excitations can be experimentally ob-served, at least, in three different frustrated magnetic lattices– square-octagon, kagome, and honeycomb. The EdH behav-ior of thermal Hall effect in both the low and high temper-ature phases is elucidated using a combination of spin wavetheory and Schwinger boson mean-field theory formalism, re-spectively. A temperature evolution map of the angular mo-mentum sign change and the corresponding thermal Hall con-ductance behavior κxy for realistic experimental conditionsshow the validity of our proposal. We provide an estimateof the EdH effect, within a Stewart experiment like set-up, forthe square-octagon, the kagome, and the honeycomb lattices.We find that the square-octagon geometry has the largest lowtemperature EdH response within a similar parameter setting.Additionally, we use the non-equilibrium Green’s function(NEGF) method to track the evolution of propagating edgeheat current contributions that can flip their orientation. Thisfurther confirms the presence of the sign change behavior. Fi-

arX

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nd-m

at.s

tr-e

l] 2

0 M

ay 2

020

2

J1

J2

D1

D2

FM

non-FM

non-FM

A+0(+1,-1,-1,+1)

A-0

(-1,-1,+1,+1)

B0+

(-1,-1,+1,+1)

B0-(+1,+1,-1,-1)

-1 -0.5 0 0.5 1-0.5

-0.25

0

0.25

0.5

D1/J1

D2/J1

a b

Fig. 1. The square-octagon lattice and the topological phase diagram. a Lattice and magnetic interactions. J1 and J2 are two types ofnearest neighbor ferromagnetic interactions within cell and between cells, respectively. D1 and D2 denote the nearest and next-nearest neighborDzyaloshinskii-Moriya interactions allowed by symmetry, according to the Moriya’s rules. b The topological phase diagram is reported for ageneric Chern number set C1,C2,C3,C4. The index of the Chern number is the same as the energy band order E1 < E2 < E3 < E4. Thesubscripted A and B symbols classify the various ferromagnetic topological phases. A phase diagram exploring a comprehensive parameterregion is reported in the Supplementary Information.

nally, we propose candidate square-octagon materials whereour theory can be tested.

ResultsSquare-octagon spin model. We consider a spin-1/2 Heisen-berg system on a square-octagon lattice, see Fig. 1. As shown,the geometry consists of four two-dimensional square sub-lattices obtained by replacing each site of a square lattice bya tilted 45 square plaquette. The side of every octagon andthe square is set to (

√2 − 1)a, where a is the edge length

of the square unit cells, see red dashed square in the figure.The model Hamiltonian contains nearest-neighbor (nn) ferro-magnetic (FM) Heisenberg exchange interaction, the nn andnext-nearest-neighbor (nnn) Dzyaloshinskii-Moriya (DM) in-teraction terms (see Fig. 1), and an external Zeeman magneticfield term. The total Hamiltonian is given by

H = −∑〈mn〉

JmnSm · Sn − h∑

m

S zm

+∑〈mn〉

Dmn · (Sm × Sn) +∑〈〈mn〉〉

Dmn · (Sm × Sn), (1)

where Sm denotes the spin angular momentum at site m,h = µBgH is the magnetic field interaction, g is the g-factor,µB is the Bohr magneton, and H is the external magnetic field.The ferromagnetic exchange coupling is chosen to be posi-tive, Jmn > 0. We set J1 and J2 equal to unity and take thespin to be S = 1/2. The DM interaction Dmn is over the nnand the nnn neighbors. As shown in Fig. 1b, the DM inter-action is tuned over a range of parameters to study the EdH

effect in the square-octagon model. The nonzero DM termscombine with the exchange interaction to generate a complexphase factor. Hence, the presence of DM coupling imparts anon-trivial topological character to the bands.

In order to study topological magnon excitations, we con-sider a linear spin-wave theory approach in the low tem-perature regime where magnon excitations are well-defined.Higher order interaction effects will not change the main con-clusions of our article. The energy bands were obtained by di-agonalizing the Fourier transformed Hamiltonian as outlinedin the Methods and Supplementary Information. While themagnon picture is effective at low temperatures and offers ex-planation of experimental data [12], it fails at higher tempera-tures where the ground state is not purely ferromagnetic. Theapproximation

√2S − b†b →

√2S becomes invalid. Con-

sequently, we utilize the Schwinger boson mean-field theory(SBMFT) [29] formalism for the thermally disordered quan-tum paramagnetic regime [30]. The Schwinger boson ap-proach is capable of describing experimental results [31] andcan additionally give rise to the spin Nernst effect [30]. Calcu-lation details on the SBMFT application are provided in Sup-plementary Information.

Topological energy bands. The topological identity of en-ergy bands can be characterized by non-zero Chern numbersof bulk bands or edge states in a finite-size system. Withinour model, considering a strip sample, we draw edge states asshown in Fig. 2a. In the figure, the dense overlapped states arebulk states, which are also the projection of energy bands in

3

0 π 2π

3

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1

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1 2 3 4

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10ε/J1

kx

y

x x x

D2/J1

D1/J1-1 0 1

-0.5

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0.5

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

1 mK 30 mK

60 mK 200 mK

1 mK 30 mK

60 mK 200 mK

a b c

d

ε = 0.637

ε = 2.36

ε = 1.5

Fig. 2. Heat current orientation reversal with temperature in a square-octagon lattice. a Energy band diagram corresponding to topo-logical edge states for the A+0 phase (D1,2/J1 = 0.6, 0 with J1 = 1K), see Fig. 1. The dashed lines indicate the choice of energy levels forthe non-equilibrium green function (NEGF) calculation. b NEGF heat current plots displaying thermal Hall current direction reversal. Energyselection of the left, middle, right plots are 0.637J1, 1.5J1, and 2.36J1 respectively. The temperature of the left and right leads are set toTL = 2.5J1 and TR = 1.5J1. The opacity of color represents the magnitude of local magnon density of states. c Edge angular momentum evolu-tion with temperature. Positive and negative regions of circulation are colored orange and blue, respectively. d Corresponding κxy temperatureevolution. Panels c and d track the sign change evolution at four different temperatures ranging from 1 mK to 200 mK.

the ky direction. The states lying in the gaps and connectingtwo bulk bands are edge states. In the absence of DM inter-action, the first (second) band touches the third (fourth) bandat the M(π, π) (Γ(0, 0)) point. The intermediate energy bandsare flat at those high symmetry points. With a nonzero DMterm, we can open gaps at those degenerate points to turn thesystem into a magnon insulator [32]. The gap expressions are±2(D1 + 2D2)S or ±2|D1 − 2D2|S at the Γ and the M points,respectively. See Supplementary Information for more de-tail. Note, Chern numbers are well-defined in band insulators,only.

Phase diagram. The topological magnetic phase diagram fora set of model parameters is shown in Fig. 1. The FM and non-FM regions are highlighted in the (D2/J1,D1/J1) plane. Thezero temperature phase diagram was constructed via energycomparison between non-ferromagnetic versus ferromagneticstates (which have the lowest energy). The topological phases,characterized by Chern numbers [9], are shown in Fig. 1. Thebulk-boundary correspondence can be seen in Fig. 2. Accord-ing to Fig. 1, the Chern number combination is 1,−1,−1, 1.Since the winding numbers are w1 = −w3 = 1, w2 = 0, thereis only one loop of edge state in the first and the third gapwith opposite orientation. Hence, there is no edge state in thesecond gap. The group velocity of the edge states show a chi-rally moving magnon current localized at the edges, whosedirection is consistent with the total winding number in thatgap. A thorough phase diagram calculation is reported in theSupplementary Information.

Thermal Hall Effect. We studied the topological magnonHall effect using the non-equilibrium Green’s function(NEGF) formalism, see Fig. 2. Calculation details are pro-vided in the Supplementary Information. A temperature gra-dient was introduced to a finite sample to study the magnonheat current along the x -direction. The generation of Berrycurvature (due to the presence of DM interaction) leads to acurrent deflection in the y -direction. This results in chiraledge current transport. The chiral current is localized. For aselected energy level in the gap, the orientation of the chiralcurrent corresponds well with the sign of the winding numberfor the edge states as shown in Fig. 2a. With a temperaturegradient along the x-direction, there is a net thermal Hall con-ductance between the top and the bottom edge [33]. Thereis also a net magnon current transport from the hotter to thecolder part of the sample [34]. Within our parameter setting,each sub panel of Fig. 2b represents the κxy current corre-sponding to the dashed energy levels of 0.637J1, 1.5J1, and2.36J1 in the gap, respectively. As the figure shows, there is anet non-zero current for the left and the right sub panels, butthere is no net edge state current in the second gap.

Theoretical [31, 35–37] and experimental [13] studies haveshown that thermal Hall conductance can change sign as tem-perature or magnetic field is varied. In Fig. 2c and Fig. 2d,we show the sign evolution of angular momentum togetherwith the variation of thermal Hall conductance with temper-ature. See Methods and Supplementary Information for de-tails on how these plots were obtained. The shape change ofthe boundary implies that previously identified positive zones

4

0 0.1 0.2 0.3

0.2

0.1

0

-0.1

-0.2

-0.3

γm/γe

T/J1

a b

Total

Edge Current

Self-Rotation

Measurement System

Magnon Sample

θ

h T

h = 0h = 0.1J1

Fig. 3. Temperature dependence of the topological gyromagneticratio γm in a square-octagon lattice. The gyromagnetic ratio contri-bution, compared to the electronic value, for individual topologicaledge current, self-rotation of the magnon wave packet, and the totalangular momentum contribution to γm are shown, both in zero andnon-zero external field h. J1 is the nearest neighbor ferromagneticinteraction. Parameters choices were D1,2/J1 = 0.6, 0.1.

have now been converted into negative areas (the blue colorcomes orange). We can see from Fig. 2c and Fig. 2d that thereis a pinch point at around 60 mK where the transition hap-pens. Currently, several competing theoretical proposals haveattempted to explain the thermal Hall sign change behavior.The theoretical constructs range from the high temperaturelimiting value of thermal conductance to the Chern numberof bands [36], propagation direction of edge modes [35], non-uniform Berry curvature over the Brillouin zone [31], and sen-sitivity of weight function in the thermal Hall formula [31].The importance of the weight function is manifested in thesign change behavior of Nernst conductivity [38]. Irrespec-tive of the conceptual reason for sign variation, we show thata measurable EdH mechanism can be observed in a host offrustrated lattices.

Einstein-de Haas Effect. The EdH effect is the ultimatemacroscopic manifestation of a very subtle microscopic ex-change of angular momentum originating from the electronspin (an inherently quantum object). The quantum to classicaltransfer is succinctly summed up in the equation

∆Lsample + ∆Linner = 0. (2)

Magnons carry angular momentum in units of ~. However,in a topological magnon insulator the magnon carries an ad-ditional angular momentum originating from Berry curva-ture [39].

The EdH effect of topological magnons occur when the sys-tem is exposed to a rapidly changing magnetic field or tem-perature. The sudden change causes the magnetization of thesystem to react. Thus, an angular momentum deviation, re-lated to the magnetization change by the gyromagnetic ratioγe, is inflicted. Concurrently, topological magnons, will expe-rience an additional angular momentum change resulting fromthe Berry curvature. This consists of the self rotation and the

edge current of magnons. So the key to estimating the mag-nitude of the EdH effect of a topological magnon insulator isthe gyromagnetic ratio of magnons. We can define the gyro-magnetic ratio of magnons as the angular momentum dividedby the magnetic moment of magnons, which is related to themagnetization change of the system. The magnetic momentof the magnons is expressed as

∆M = −~

γeN

∑nk

nB(εnk), (3)

where the gyromagnetic ratio of our spin system is γe =

2me/(ge); g is the Lande factor, e and me are the charge andmass of the electron, respectively. Within the low temperatureapproximation, the mass of the magnon can be approximatedas the effective mass at the bottom of the first band. Thus, thegyromagnetic ratio of topological magnons is given by

γm =Ltot

∆M, (4)

where Ltot represents the total sum of the edge and self-rotation angular momentum. The result of our calculation isshown in Fig. 3. We find two competing angular momentumcomponents which oppose each other. The self-rotation partis larger in magnitude, but with an opposing sign. The to-tal gyromagnetic contribution has a nonzero value in the zerotemperature limit. Note, the additional amplitude of the EdHeffect of topological magnons, compared to the pure two di-mensional ferromagnetic system, is appreciable as long as theDM interaction is not too small. To further analyze the physi-cal content of Fig. 3, we compare and contrast the topologicalgyromagnetic ratio with respect to the kagome and the honey-comb lattice systems.

In Fig. 4 we show the results of our γm/γe calculation. Wenotice that in the very low temperature limit (almost zero), thesquare-octagon lattice has a non-zero response. This sets itapart from the other usual lattice candidates such as kagomeand honeycomb, that have been experimentally explored tillnow. The EdH response of these lattices is at least two tofour orders of magnitude smaller. In Fig. 4a we show thetemperature variation of the topological gyromagnetic ratiocompared to the electronic value. The curve is truncated ata point where the magnetic ordering begins to enter a hightemperature disordered paramagnetic phase. The disorderingtemperature for the various lattices studied in this article arelisted in the Supplementary Information. While initially thesystem is sensitive to temperature changes, at higher temper-atures the response hits a plateau. This could potentially bedue to magnon excitations not being well-defined anymore aswe encroach the thermal disordering temperature. Although,we have demonstrated the existence of the EdH effect in onlythese three lattices, our formalism, analysis approach, andeventual conclusions will hold for a wider variety of frustratedferromagnetic systems.

In contrast to the topological gyromagnetic ratio response,the EdH caused by electrons is determined by the electron it-self. Considering this, we define a differential gyromagnetic

5

—— S-O: J2, D1, 2/J1 = 1, -0.6, 0

—— S-O: J2, D1, 2/J1 = 1, 0, 0.3

—— Kagome: D1,2/J1 = -0.6, 0

—— Kagome: D1, 2/J1 = 0, 0.3

—— Honeycomb: D2/J1 = 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6

0.3

0.1

0.2

0.05

0.15

0.25

T/J1

γm/γe

—— S-O: J2, D1, 2/J1 = 1, -0.6, 0

—— S-O: J2, D1, 2/J1 = 1, 0, 0.3

—— Kagome: D1,2/J1 = -0.6, 0

—— Kagome: D1, 2/J1 = 0, 0.3

—— Honeycomb: D2/J1 = 0.3

0 0.1 0.2 0.3 0.4 0.5 0.6

0.3

0.1

0.2

0.05

0.15

0.25

T/J1

γm*/γe

a b

Fig. 4. Comparison of Einstein de-Haas effect response in square-octagon, kagome, and honeycomb lattices. a The bare topological toelectronic gyromagnetic ratio variation with temperature is shown. b The differential gyromagnetic ratio response. In both panels the plots aretruncated at the thermal disordering temperature above which an ordered ferromagnetic ground state ceases to exist.

ratio response γ∗m of the topological magnons as

γ∗m =

(∂Ltot/∂T∂∆M/∂T

)h. (5)

For electrons γ∗e is equal to γe. However, the situation is en-tirely different for topological magnons which are excitationquasiparticles. For these excitation modes the gyromagneticratio is renormalized from the bare γm response. Each magnonmode has its own gyromagnetic response. Additionally, themagnons can be excited or annihilated. Thus, experimentallythe gyromagnetic ratio response cannot simply be measuredfrom γm itself, but from a response to a temperature change.Evaluating the above equation shows that the system doeshave a maximal response as seen in Fig. 4b, before dippingoff. A magnetic field can enhance the EdH, just as in Fig. 3b.Thus, there is an optimal temperature at which the magnon in-sulator will have the best possible response. Thus, from an ex-perimental point of view, this is the optimal temperature zonein which our theory can be tested. For the square-octagon lat-tice this value is around T/J1 = 0.1. For the kagome and thehoneycomb the ratio is about 0.24.

High temperature phase. The magnon picture is valid in thelow temperature limit, below the thermally disordered quan-tum paramagnetic regime. The spinon formalism, consideredwithin the Schwinger Boson Mean Field theory (SBMFT) ap-proach, is more appropriate in the high temperature phase. Toanalyze the disordered high temperature phase of the square-octagon lattice, we define ten mean fields – the symmet-ric part of J1-type coupling ζ1↑, ζ1↓ and the anti-symmetricpart ξ1↑, ξ1↓, also for D2-type coupling we have ζ3↑, ζ3↓, ξ3↑and ξ3↓, the remaining two mean fields are for the J2-typecoupling η2, and we have a Lagrangian multiplier λ for theconstraint

∑s 〈b

†msbms〉 = 2S . Among the ten mean fields,

four anti-symmetric parts are not important. The result isdisplayed in the inset of Fig. 5. Please see SupplementaryInformation for detailed derivations. From the figure, we

can see that all mean fields vanish above Tc ≈ 0.81J1, theboundary of the ferromagnetic-paramagnetic phase transition.In the zero temperature limit and with an external non-zerofield h > 0, the mean fields tend to the following limitsλ → 2J1S + J2S + h/2, ζ1↑ = ζ3↑ → 2S , η2 → S andζ1↓ = ζ3↓ = ξ1↑ = ξ1↓ = ξ3↑ = ξ3↓ = 0 in the ferromag-netic phase region. Thus, the Hamiltonian of the spin downis the same as the magnon H↓ = Hmag [30]. However, theHamiltonian of the spin up corresponds to a topologically triv-ial H↑ = Hmag(D1 = D2 = 0). The focus of the present work ison the fully magnetized regime of the sample. At low temper-ature, where the spinons condense into magnons, the resultsfrom the magnon and the spinon calculations agree. This isa check on the validity of our calculation from two separateformalism. We do anticipate spinons to exhibit EdH effect athigher temperatures.

Discussion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.2

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0.5

1

1.5

T/J1

Mean Fields/J1D1,2/J1 = 0.6, 0.1

h/J1 = 0.1

T/J1

κ/10-3Wm-1K

-1

D1,2/J1 = 0.4, 0.15h/J1 = 0.1

Spinon: ↑

Spinon: ↓

Spinon: ↑ + ↓

Magnon

ζ3

ζ3

ζ1

ζ1

η2

λ

Fig. 5. Thermal Hall κxy response of spinon and magnon fields.Inset picture shows converged values of spinon mean field parame-ters. The magnon curve is terminated at the critical thermal disorder-ing temperature.

6

We have shown that the square-octagon, the kagome, and thehoneycomb lattices can exhibit observable EdH effect. The in-fluence of a non-zero Berry curvature originating from DM in-teractions and its underlying topological identity is preservedeven though the lattice structure changes. Thus, similar con-clusions, not qualitatively differently, can be obtained fromother lattices which can support a non-zero Berry curvature.Our conclusions should also hold for other recently studiedsystems such as the kagome lattice [35, 40], the Lieb lat-tice [37], or the star lattice [41]. It is also interesting to con-sider a model with two kinds of DM interaction, competingwith each other on the square-octagon lattice. More precisely,the system should possess a broken up-down symmetry as in aferro- or ferri- magnetic system. In antiferromagnets, the DMinteraction can only result in the spin Nernst effect [38].

It is challenging to find an appropriate material that haslarge enough DM interactions. Here we discuss some pos-sible real materials where our theory could be tested. Recentstudies have predicted two-dimensional square-octagon com-pounds in the C or the group-V materials [42–46]. Monolayeror multi-layer (V-III bonded) Haeckelite compounds are otherpotential examples [47–49]. The square-octagon XY2 com-pound, with two sublattices, has been recently synthesized,too [50]. If magnetic dopants could be introduced into the X-sublattice, then the bilayer structure is another potential can-didate. Besides the above examples, quasi two-dimensionalmaterials can possess intrinsic ferro- or antiferromagnetismin a nearly square-octagon structure, also [51, 52]. In thesecompounds, Fe or Mn transition metal ions form an interpen-etrating square-octagon layer with the rare earth element re-siding in an almost square-octagon geometry. If we removethe central atom of the rare-earth sublattice to introduce a va-cancy, or dope all the rare earths by nonmagnetic atoms, asquare-octagon structure as proposed in our model could beachieved.

0 0.1 0.2 0.3

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T/J1

a b

Total

Edge Current

Self-Rotation

Measurement System

TopologicalSample

ΔTh

Magnon

θ

Fig. 6. Proposed experimental set up of the Einstein-de Haaseffect of topological magnons. The disk shaped sample is exposedto an external heat bath with a temperature gradient and also to anexternal field h, if required. The mirror mounted on the axle of thedisk will rotate and deflect the incident light. The deflection angle isproportional to the Einstein de-Haas effect.

In Stewart’s original experiment [3], he observed the EdHeffect in iron and nickel. In his set up, the sample was sus-pended inside a solenoid by a quartz fiber. A changing currentin the solenoid circuit caused an angular momentum impulse.This generated a mechanical rotation of the suspended sys-tem. The rotation angle was measured as the deflection of aspot of light that reflected from a mirror attached to the freelysuspended system. The deflection was proportional to the an-gular momentum change originating from the EdH effect. Therotational motion of the system was dictated both by materialproperties and the amount of the transient angular momentumchange.

To test our theoretical prediction, we propose an experi-mental set-up similar to that of Stewart as shown in Fig. 6.The difference is that our system is two-dimensional. Thus,the sample is disc shaped. Immediately below the samplethere is a heater or an electric plug to adjust the temperature ormagnetic field. Note, a strong magnetic field can also polarizethe ground state to a ferromagnetic phase. The sample is con-nected to a suspension wire or equipment suitable enough toprovide the torque. The rotational force can be measured bya detecting device such as a mirror attached to the wire witha light beam and a light sensor, or some form of atomic forcedetector. In an actual experiment, the temperature needs to beappropriately adjusted so that it is neither too high nor too low,see Fig. 4b for guidance. Note, in Stewart’s paper, his differ-ential equation response was modeled as a damped harmonicoscillator characterized by a damping coefficient and a mo-ment of inertia. The initial angular momentum impulse, whichappeared as an initial condition, was determined by measur-ing the first amplitude of oscillation. If the EdH of magnonsis ultrafast, which implies the time scale of the inner angularmomentum transferred to the whole sample is much shorterthan those of the relaxation time in the experiment, then ourmodel is applicable. If not, we need to consider a new micro-scopic mechanism of spin-to-rotation transfer.

MethodsSpin wave theory. The spin Hamiltonian is transformedinto a magnon Hamiltonian following the usual Holstein-Primakoff transformations: S +

m =(2S − b†mbm

)1/2bm, S −m =

b†m(2S − b†mbm

)1/2, S z

m = S − b†mbm. Only bilinear termsare retained. The Hamiltonian has the form H = E0 −∑

mn

[(Jmn + iνmnDmn) S b†mbn + H.c.

]+ h

∑m b†mbm, where E0

is ground state energy. νmn = −νnm designates the direction ofDM interaction. Subsequently, energy bands are obtained viadiagonalization of the Hamiltonian matrix. The boundaries ofthe topological phase diagram was determined based on track-ing gap closures and opening. [10, 53]. The edge states werecalculated for a strip sample in open boundary condition. InFig. 2, we considered W = 120, which is large enough to fixthe edge states [10]. Different edge conditions will give riseto different edge states. But, the topology features remain un-changed. Our selection of two edges is similar to Fig. 1. Forfurther details on spin wave theory see Supplementary Infor-mation.

Schwinger boson theory. In contrast to spin-wave theory,

7

SBMFT considers the following transformations – Sm =

b†ms(σ)stbmt/2, where σ are Pauli matrices, s and t are spin in-dices. The bosonic commutation relations [bms, b

†nt] = δmnδst

keep the S U(2) algebra invariant. As an approximation,ten mean-fields λi are defined to describe the ground state.The spinon excitation was taken as a perturbation by usingthe Hartree-Fock decoupling scheme where AB 7→ 〈A〉 B +

A 〈B〉 − 〈A〉 〈B〉. This reduces the quartic interaction termsof the Hamiltonian to bilinear contributions. The value ofthe mean-fields were determined by minimizing the free en-ergy F = F0(λ) + kBT

∑kµs ln

(1 − exp

(−Ekµs/kBT

)). Ten

self-consistency equations ∂F/∂λi = 0 (i = 1, . . . , 10) werenumerically solved to obtain the results. For further details,please see Supplementary Information.

Chern number. The thermal Hall effect can be considered asthe bosonic version of the integer quantum Hall effect. Thussome properties of topological insulators can also be found inour system, such as topological numbers, and conducting edgestates. The topological invariant of our topological magnonstate is the Chern number [9], which is the integral of theBerry curvature over the Brillouin zone. The Berry curvatureof a band λ is given by

Ωλk = i∑µ,λ

〈λ|∇kH(k)|µ〉 × 〈µ|∇kH(k)|λ〉(Eλ − Eµ)2 .

The Chern number is given by the integral

Cλ =1

2π

∫BZ

d2kΩλk. (6)

The topological phases characterized by the Chern numbersare shown in Fig. 1. See Supplementary Information for cal-culation details and a complete phase diagram based on theChern numbers.

Thermal Hall conductance. The electronic Hall conduc-tance, originating from the Kubo formula, is the integral ofBerry curvature with an appropriate weight, which relates theanomalous velocity of the magnons, multiplied by the Fermidistribution function over all energies [53, 54]. With this de-scription, the topological phase is still included within theframework of Bloch’s wave function and band theory, that is,topological band theory. Similarly, the thermal Hall conduc-tance can be expressed as a weighted summation of the Berrycurvature, according to linear response theory as [33, 39]

κxy = −k2

BT~a

1N

∑λk

c2(nB(ελk))Ωλk, (7)

where the weight function is given by c2(x) = (1 + x)(ln(1 +

1/x))2 − (ln x)2 − 2Li2(−x), where Lis(z) =∑∞

n=1 zn/ns is thepolylogarithm function. c2 is a decreasing function similarto the Bose distribution. The lattice constant is chosen asa = 0.1nm in this article. See Supplementary Informationfor thermal Hall calculation details.Non-equilibrium Green’s function (NEGF). To perform theNEGF calculation, the square-octagon lattice sample was di-vided into three parts – the left lead, the central part, and

the right lead. The lead temperatures were kept at unequalTL (left) and TR (right) values to ensure a thermally non-equilibrium state. The magnon transport of the central partis what we are concerned with [34]. The Hamiltonian for thispart can be written as

H =∑

α,β=L,C,R

b†αHαβbβ. (8)

The block diagonal form of the Hamiltonian matrix is thenused to compute the self energy, the lesser, the retarded, andthe advanced Green’s function of the central part. Further-more, quantities such as the local density of magnons and thelocal current were determined from the Green’s functions as

ρm =i~(G<

CC)mm

πa(9)

jmn =ε

2πRe[(G<

CC)mnHnm − (G<CC)nmHmn]. (10)

The two leads and the central part are supposed to be largeenough to eliminate the influence of boundaries. But, in prac-tice, for a schematic calculation a 4×10 lattice site discretiza-tion suffices to demonstrate the basic EdH effect, see Fig. 2.For further calculation details, please refer to the Supplemen-tary Information.

Data availabilityAll relevant data are available from the authors upon request.

References1. Richardson, O. W. A mechanical effect accompanying magneti-

zation. Phys. Rev. (Series I) 26, 248–253 (1908).2. Einstein, A. Experimenteller nachweis der ampereschen

molekularstrome. Naturwissenschaften 3, 237–238 (1915).3. Stewart, J. Q. The moment of momentum accompanying mag-

netic moment in iron and nickel. Physical Review 11, 343–363(1918).

4. Barnett, S. J. Magnetization by rotation. Phys. Rev. 6, 239–270(1915).

5. Schwinger, J. On quantum-electrodynamics and the magneticmoment of the electron. Phys. Rev. 73, 416–417 (1948).

6. Kawaguchi, Y., Saito, H. & Ueda, M. Einstein-de Haas effect indipolar Bose-Einstein condensates. Phys. Rev. Lett. 96, 080405(2006).

7. Ganzhorn, M., Klyatskaya, S., Ruben, M. & Wernsdorfer, W.Quantum Einstein-de Haas effect. Nature Communications 7,11443 (2016).

8. Mentink, J. H., Katsnelson, M. I. & Lemeshko, M. Quantummany-body dynamics of the Einstein-de Haas effect. Phys. Rev.B 99, 064428 (2019).

9. Hasan, M. & Kane, C. Colloquium: Topological insulators. Re-views of Modern Physics 82, 3045–3067 (2010).

10. Hatsugai, Y. Edge states in the integer quantum Hall effect andthe Riemann surface of the Bloch function. Physical Review B48, 11851 (1993).

11. Bansil, A., Lin, H. & Das, T. Colloquium: Topological bandtheory. Rev. Mod. Phys. 88, 021004 (2016).

12. Onose, Y. et al. Observation of the magnon Hall effect. Science329, 297–299 (2010).

8

13. Hirschberger, M., Chisnell, R., Lee, Y. S. & Ong, N. P. ThermalHall effect of spin excitations in a Kagome magnet. Physicalreview letters 115, 106603 (2015).

14. Chen, L. et al. Topological spin excitations in honeycomb ferro-magnet cri3. Phys. Rev. X 8, 041028 (2018).

15. Pershoguba, S. S. et al. Dirac magnons in honeycomb ferromag-nets. Phys. Rev. X 8, 011010 (2018).

16. Li, B. & Kovalev, A. A. Chiral topological insulator of magnons.Phys. Rev. B 97, 174413 (2018).

17. Watanabe, D. et al. Emergence of nontrivial magnetic excitationsin a spin-liquid state of kagome volborthite. Proceedings of theNational Academy of Sciences 113, 8653–8657 (2016).

18. Kasahara, Y. et al. Unusual thermal Hall effect in a Kitaev spinliquid candidate α-RuCl3. Physical review letters 120, 217205(2018).

19. Ideue, T., Kurumaji, T., Ishiwata, S. & Tokura, Y. Giant thermalHall effect in multiferroics. Nature materials 16, 797 (2017).

20. Doki, H. et al. Spin thermal Hall conductivity of a Kagomeantiferromagnet. Physical review letters 121, 097203 (2018).

21. Kawano, M. & Hotta, C. Thermal hall effect and topologicaledge states in a square-lattice antiferromagnet. Phys. Rev. B 99,054422 (2019).

22. Nakata, K., Kim, S. K., Klinovaja, J. & Loss, D. Magnonic topo-logical insulators in antiferromagnets. Phys. Rev. B 96, 224414(2017).

23. Samajdar, R. et al. Enhanced thermal hall effect in the square-lattice neel state. Nature Physics 15, 1290–1294 (2019).

24. Grissonnanche, G. et al. Giant thermal Hall conductivity in thepseudogap phase of cuprate superconductors. Nature 571, 376–380 (2019).

25. Strohm, C., Rikken, G. & Wyder, P. Phenomenological evidencefor the phonon Hall effect. Physical review letters 95, 155901(2005).

26. Sugii, K. et al. Thermal Hall effect in a phonon-glassBa3CuSb2O9. Physical review letters 118, 145902 (2017).

27. Thingstad, E., Kamra, A., Brataas, A. & Sudbø, A. Chiralphonon transport induced by topological magnons. Phys. Rev.Lett. 122, 107201 (2019).

28. Xiao, J., Bauer, G. E. W., Uchida, K.-c., Saitoh, E. & Maekawa,S. Theory of magnon-driven spin seebeck effect. Phys. Rev. B81, 214418 (2010).

29. Sarker, S., Jayaprakash, C., Krishnamurthy, H. & Ma, M.Bosonic mean-field theory of quantum Heisenberg spin systems:Bose condensation and magnetic order. Physical Review B 40,5028 (1989).

30. Kim, S. K., Ochoa, H., Zarzuela, R. & Tserkovnyak, Y. Realiza-tion of the Haldane-Kane-Mele model in a system of localizedspins. Physical review letters 117, 227201 (2016).

31. Lee, H., Han, J. H. & Lee, P. A. Thermal Hall effect of spins ina paramagnet. Physical Review B 91, 125413 (2015).

32. Owerre, S. A first theoretical realization of honeycomb topolog-ical magnon insulator. Journal of Physics: Condensed Matter28, 386001 (2016).

33. Matsumoto, R. & Murakami, S. Theoretical prediction of a ro-tating magnon wave packet in ferromagnets. Physical ReviewLetters 106, 197202 (2011).

34. Zhang, L., Ren, J., Wang, J.-S. & Li, B. Topological magnon in-sulator in insulating ferromagnet. Physical Review B 87, 144101(2013).

35. Mook, A., Henk, J. & Mertig, I. Edge states in topologicalmagnon insulators. Phys.Rev.B 90, 024412 (2014).

36. Mook, A., Henk, J. & Mertig, I. Magnon Hall effect and topol-ogy in Kagome lattices: A theoretical investigation. PhysicalReview B 89, 134409 (2014).

37. Cao, X., Chen, K. & He, D. Magnon Hall effect on the Lieb lat-tice. Journal of Physics: Condensed Matter 27, 166003 (2015).

38. Cheng, R., Okamoto, S. & Xiao, D. Spin Nernst effect ofmagnons in collinear antiferromagnets. Physical review letters117, 217202 (2016).

39. Matsumoto, R. & Murakami, S. Rotational motion of magnonsand the thermal Hall effect. Physical Review B 84, 184406(2011).

40. Seshadri, R. & Sen, D. Topological magnons in a Kagome-latticespin system with x x z and Dzyaloshinskii-Moriya interactions.Physical Review B 97, 134411 (2018).

41. Owerre, S. Magnon Hall effect without Dzyaloshinskii-Moriyainteraction. Journal of Physics: Condensed Matter 29, 03LT01(2016).

42. Sheng, X.-L. et al. Octagraphene as a versatile carbon atomicsheet for novel nanotubes, unconventional fullerenes, and hy-drogen storage. Journal of Applied Physics 112, 074315 (2012).

43. Guan, J., Zhu, Z. & Tomanek, D. Tiling phosphorene. ACS nano8, 12763–12768 (2014).

44. Zhang, Y., Lee, J., Wang, W.-L. & Yao, D.-X. Two-dimensionaloctagon-structure monolayer of nitrogen group elements and therelated nano-structures. Computational Materials Science 110,109–114 (2015).

45. Kou, L. et al. Tetragonal bismuth bilayer: A stable and robustquantum spin Hall insulator. 2D Materials 2, 045010 (2015).

46. Ersan, F., Akturk, E. & Ciraci, S. Stable single-layer structure ofgroup-V elements. Physical Review B 94, 245417 (2016).

47. Camacho-Mojica, D. C. & Lopez-Urıas, F. GaN haeckelitesingle-layered nanostructures: Monolayer and nanotubes. Sci-entific reports 5, 17902 (2015).

48. Shahrokhi, M., Mortazavi, B. & Berdiyorov, G. R. New two-dimensional boron nitride allotropes with attractive electronicand optical properties. Solid State Communications 253, 51–56(2017).

49. Gurbuz, E., Cahangirov, S., Durgun, E. & Ciraci, S. Single layersand multilayers of GaN and AlN in square-octagon structure:Stability, electronic properties, and functionalization. PhysicalReview B 96, 205427 (2017).

50. Li, W., Guo, M., Zhang, G. & Zhang, Y.-W. Gapless MoS2

allotrope possessing both massless Dirac and heavy fermions.Physical Review B 89, 205402 (2014).

51. Ma, X., Whalen, J. B., Cao, H. & Latturner, S. E. Competingphases, complex structure, and complementary diffraction stud-ies of R–δFeAl4−xMgxTt2 intermetallics (R = Y, Dy, Er, Yb;Tt= Si or Ge; x < 0.5). Chemistry of Materials 25, 3363–3372(2013).

52. Calta, N. P. & Kanatzidis, M. G. Quaternary aluminum sili-cides grown in Al flux: RE5Mn4Al23−xSix (RE = Ho, Er, Yb) andEr44Mn55(AlSi)237. Inorganic chemistry 52, 9931–9940 (2013).

53. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs,M. Quantized Hall conductance in a two-dimensional periodicpotential. Physical Review Letters 49, 405 (1982).

54. Kohmoto, M. Topological invariant and the quantization of theHall conductance. Annals of Physics 160, 343–354 (1985).

AcknowledgementsT. D. acknowledges funding support from Sun Yat-SenUniversity Grant Nos. OEMT-2017-KF-06 and OEMT-2019-KF-04. J. L. and D. X. Y. are supported byNKRDPC-2017YFA0206203, NKRDPC-2018YFA0306001,NSFC-11974432, GBABRF-2019A1515011337, and Lead-ing Talent Program of Guangdong Special Projects.

9

Author contributions

T. D and D. X. Y conceived and designed the project. J. Lperformed the calculation. T. D. and D. X. Y checked the cal-culations. T. .D. proposed the Einstein-de Haas experimentalinterpretation. All authors contributed to the discussion, anal-ysis, and writing of the paper.

COMPETING INTERESTS

The authors declare no competing interests.

ADDITIONAL INFORMATION

Supplementary Information for the paper will be availablevia the journal’s weblink after publication.

Correspondence and requests for materials should be ad-dressed to T. Datta or D. X. Yao.