hosho katsura, naoto nagaosa and patrick a lee- theory of the thermal hall effect in quantum magnets

8/3/2019 Hosho Katsura, Naoto Nagaosa and Patrick A Lee- Theory of the Thermal Hall Effect in Quantum Magnets

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arXiv:0904.3427v3

[cond-mat.str-

el]28Jul2010

Theory of the Thermal Hall Effect in Quantum Magnets

Hosho Katsura1, Naoto Nagaosa1,2, Patrick A Lee31Cross Correlated Materials Research Group, Frontier Research System,

Riken,2-1 Hirosawa, Wako, Saitama 351-0198, Japan2Department of Applied Physics, The University of Tokyo,

7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan3Department of Physics, Massachusetts Institute of Technology,

77 Massachusetts Avenue, Cambridge, MA 02139

We present a theory of the thermal Hall effect in insulating quantum magnets, where the heatcurrent is totally carried by charge-neutral objects such as magnons and spinons. Two distinct typesof thermal Hall responses are identified. For ordered magnets, the intrinsic thermal Hall effect formagnons arises when certain conditions are satisfied for the lattice geometry and the underlyingmagnetic order. The other type is allowed in a spin liquid which is a novel quantum state sincethere is no order even at zero temperature. For this case, the deconfined spinons contribute to thethermal Hall response due to Lorentz force. These results offer a clear experimental method to provethe existence of the deconfined spinons via a thermal transport phenomenon.

PACS numbers: 71.10.Hf, 72.20.-i, 75. 47.-m

The ground state and low energy excitations of corre-

lated electronic systems are the subject of recent inten-sive interests, and especially the possible quantum liquidstates are the focus both theoretically and experimen-tally [15]. For the quantum magnets, magnetic suscep-tibility, neutron scattering and specific heat are the ex-perimental tools to study this issue. In the conductingsystems, on the other hand, charge transport propertiesalso offer important clues to the novel electronic statessuch as the non-Fermi liquid or the quantum Hall liq-uid. Therefore, a natural question is whether there areany transport properties in insulating quantum magnetswhich provide insight into the ground state. To answerthis question, we study in this Letter the thermal Hall

effect theoretically and find several different mechanismsleading to the classification of the quantum magnets.

For a finite Hall response, time-reversal symmetrymust be broken due to magnetic field and/or magneticordering. The Hall effect in itinerant magnets, wherethe spin structure and conduction electron motion arecoupled, has been studied extensively. In this case, inaddition to the usual Lorentz force, the scalar (spin) chi-

rality defined for three spins as Si (Sj Sk) plays animportant role [68]. The scalar chirality acts as a ficti-tious magnetic flux for the conduction electrons and givesrise to a non-trivial topology of the Bloch wave functions,

leading to the Hall effect. It is natural to expect that asimilar effect occurs even in the localized spin systemsfor e.g. the spin current [9]. Another important toolto detect charge-neutral modes is the thermal transportmeasurement. In low-dimensional magnets, the ballisticthermal transport property was predicted from the in-tegrability of the one-dimensional Heisenberg model [10]and has been experimentally observed in Sr2CuO3 [11].In -(ET)2Cu2(CN)3, one of possible candidates for two-dimensional quantum spin liquids [1], the thermal trans-

port measurement was used as a probe to unveil the na-

ture of low-energy spin excitations [12]. The measure-ments have been limited to the longitudinal thermal con-ductivity so far. In this paper we predict a non-zero ther-mal Hall conductivity, i.e., the Righi-Leduc effect whichwill provide important information as described below.

First, we need to consider the influence of the externalmagnetic field on localized spin systems. In addition tothe Zeeman coupling, we have the ring exchange processleading to the coupling between the scalar chirality andexternal magnetic fields. This coupling is derived fromthe t/U expansion for the Hubbard model at half fillingwith on-site Coulomb interaction U and complex hoppingtij = te

iAij [13, 14] and its explicit form is given by

Hring = 24t3

U2sin Si (Sj Sk), (1)

where is the magnetic flux through the triangle formedby the sites i, j, and k in a counterclockwise way. Sincethe coefficient is proportional to t3/U2, it is expected tobe small. In the vicinity of the Mott transition, however,this coupling is not negligible. We first examine the effectof Hring within the spin-wave approximation. Then wefind that if the lattice geometry and the magnetic ordersatisfy certain conditions, the magnons can experiencethe fictitious magnetic field and there occurs the intrin-sic thermal Hall effect, i.e., the thermal Hall conductivity

xy due to the anomalous velocity of the magnons. In thiscase, xy is independent of the lifetime of magnons (),whereas the longitudinal one xx depends on [15, 16].It can be regarded as a bosonic analogue of the quan-tum Hall effect with zero net flux [17]. We also derivea TKNN-type formula [18] of the thermal Hall conduc-tivity for a general free-bosonic Hamiltonian. It shouldbe possible to apply this formula to the recently foundphonon thermal Hall effect [19]. Finally, we consider theeffect of Hring in quantum spin liquids. Since there is no
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(a)

i

j k

i

(b) (c)

i

j k

i

FIG. 1: (a) A portion of the triangular lattice with ferromag-netic order. The blue arrows indicate the directions of spins.The counterclockwise rotations (red arrows) indicate the or-der of sites in Hring. We use this convention throughout thisLetter. (b) Antiferromagnetic model on a square lattice. Theinteraction along the diagonal edge is smaller than that alongthe vertical or horizontal edge. (c) A portion of the triangularlattice with the 120 order.

magnetic order in such a system, it has been proposedthat deconfined fermionic spinons exist. In contrast tothe magnons, the spinons, which are the gauge depen-dent object, can feel the vector potential A just as inthe case of electrons, leading to the Landau level forma-tion [14]. We propose a novel way to detect the spinondeconfinement via the thermal Hall effect measurement ina candidate of quantum spin liquid, -(ET)2Cu2(CN)3.

NO-GO theorem for the coupling to magnetic flux. Let us first consider the spin-wave expansion of Eq. (1)to find a system in which the intrinsic thermal Hall ef-fect occurs. We consider the collinear ground state spinconfigurations. The fluctuation of the scalar chirality upto the second order in S S S is written as

Si( Sj Sk)+Sj( Sk Si)+Sk( Si Sj), (2)

where Sa denotes the a-component (a = x, y, or z) ofthe ordered moment. Note here that the linear orderterms in S vanish since Si Sj = 0. As an exam-ple, we consider the ferromagnetic Heisenberg model ona triangular lattice shown in Fig. 1(a) with an orderedmoment S0 along y. In this case, the quadratic termsin S from Eq. (1) always cancel. To explain this, letus focus on the edge jk. From the upper triangle, thisedge gives S0( Sj Sk)y. On the other hand, from thelower triangle, it gives S0( Sk

Sj)y which cancels out

the former one. Since such a cancellation occurs on anyedge, Hring in Eq. (1) does not contribute to the spin-wave Hamiltonian to quadratic order. This observationleads us to conclude that such a cancellation occurs forany ferromagnetic model where each edge is shared bythe equivalent cells such as plaquettes and triangles. Asimilar cancellation occurs for certain antiferromagneticsystems. An example is shown in Fig. 1(b). In this ex-ample, there are several different types of ring exchangeprocesses, but again the cancellation between the cells

sharing a link occurs for the collinear antiferromagneticconfiguration. Finally, for noncollinear spin structures,we considered the 120 magnetic order on a triangularlattice (Fig. 1(c)) and conclude that the cancellation

again occurs since ordered components of Si and that ofSi are the same as shown in Fig. 1(c).

Intrinsic thermal Hall effect in the spin-wave approx-

imation. Once we understand the principles of the

cancellation, it is rather easy to find an example whereit does not occur, namely, when the link is shared by in-equivalent cells. An example is the ferromagnetic modelon the kagome lattice. In this case, the spin wave Hamil-tonian is influenced by the magnetic flux . We willdevelop below a theoretical formalism to calculate thethermal Hall conductivity in terms of the Kubo formula.For this purpose, we consider a general Hamiltonian fornoninteracting bosons which can be regarded as a spin-wave Hamiltonian within quadratic order in S:

H = j,

h( Rj), h( Rj) =1

2

tbRj+

bRj + h.c.,

where bRj annihilates a boson at the lattice point of the

-th site in the j-th unit cell and are vectors connect-ing Rj and its neighboring sites. The hopping t is ingeneral complex. In momentum space, the Hamiltonianis written as H =

k b

(

k)H,(k)b(k), where b(k)is a Fourier transform of bRj and repeated indices are

summed over. Using the local energy operator h( Rj),the average energy current density is defined by

jE i[H,j

Rjh( Rj)]/V, (3)

where V is the total volume [20]. Using the

Fourier transform of

h(Rj) defined by h(q) =

j, eiqRjh( Rj), the energy current density is rewrit-

ten as jE = q[h(0), h(q)]|q=0/V. Using this fact, thefollowing convenient expression for jE is obtained:

jE =1

2V

k

b(k)

kH(k)2

b(k), (4)

where the differential operator k acts only on H(k)2. Weintroduce the spin wave basis |u(k) which diagonalizesH(k) with eigenvalues (k). It is important to note thateven in this basis jE is not diagonal. In addition to theexpected diagonal term d

dk(k), there are off-diagonal

terms which can be thought of as arising from anomalousvelocities. As we see below, these terms are responsiblefor xy, just as in the case of the intrinsic anomalous Halleffect in metals [21].

Starting from the Kubo formula, the following expres-sion analogous to the TKNN formula [18] can be obtained


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FIG. 2: (a) The unit cell of the kagome lattice (the areaenclosed by the dotted line). The vectors ai (i = 1, 2, and 3)correspond to green, red, and blue arrows, respectively. Thearrows on the edges indicate the sign of the phase factor ei/3

(see main text). The fictitious fluxes through triangle andhexagon are and2, respectively. (b) Dispersions of bandsin the case of = /3 with JS = 1. The right figure showsthe enlargement of dispersion around = 0 where the lowest

band has a quadratic dispersion, i.e., 1(k) JS(k2x + k

2y).

for the thermal Hall conductivity xy:

xy = 12T

Im

BZ

d2k

(2)2n(k)

kxu(k)(H(k) + (k))2kyu(k) ,(5)

for the non-interacting spin waves (free bosons) where

the integral is over the Brillouin zone (BZ), and n(k) =

(e(k)1)1 is the Bose distribution function. We can

show that the integrand in Eq. (5) is independent of a k-dependent phase change of the spin wave basis functions|u(k) in a similar way as for the orbital magnetiza-tion [22].

Thermal Hall effect in kagome ferromagnet. Wenow apply the above formula to the ferromagnetic modelon the kagome lattice. The Hamiltonian is given by H =

, h + h with

h/ = J( Si Sj + Sj Sk + Sk Si) KSSi (Sj Sk),

where J is the ferromagnetic exchange coupling, Kis proportional to sin according to Eq. (1), and thesum is taken over all the triangles in the kagome lat-tice (see Fig. 2(a)). Again, note that the sites i,

j, and k form a triangle ( or ) in a counterclock-wise way. Using the Holstein-Primakoff transformation[S+j = (2S nj)1/2bj , Sj = bj(2S nj)1/2, Szj = S njwith nj = b

jbj], we obtain the spin-wave Hamiltonian as

HSW = 4JSj

nj S

J2 + K2j,k

(ei/3bjbk + h.c.),

where tan(/3) = K/J and the sum is taken over allthe nearest neighbor bonds. This Hamiltonian con-tains a non-trivial phase factor and is quite similar tothe electronic model considered by Ohgushi et al. [8].The Fourier transform of the Hamiltonian is given byH(k) = 4JS 2JS(cos(/3))1(k, ) with

(k, ) = 0 cos k1ei/3 cos k3ei/3

cos k1ei/3 0 cos k2ei/3cos k3e

i/3 cos k2ei/3 0

,

where ki = k ai with a1 = (1/2,

3/2), a2 = (1, 0),and a3 = (1/2,

3/2) as shown in Fig. 2(a). If

the magnetic field is absent, i.e., = = 0, the up-per band is dispersionless and the other two bands areparticle-hole symmetric around = 3JS. The disper-sions of three bands (0 1(k) 2(k) 3(k))for = /3 are shown in Fig. 2(b). In the limitof low temperature and weak magnetic field, the dom-inant contribution to the integral in Eq. (5) comes from

= 1 (lowest band) and small |k| due to the Bose fac-tor n(k). By an explicit calculation [23], we find that

kxu1(k)|kyu1(k) i|k|2/(27

3) around k = 0 andobtain

xy (6JS)2

2T

0

dk

2

k

eJSk2 1k2

27

3

=

36

3T,

(6)where we have replaced the integration over the BZ withthat over all k. In this way, a non-zero thermal Hall con-ductivity is indeed realized by the coupling between thescalar chirality and the magnetic field in the ferromag-netic kagome lattice. The obtained linear dependence of

xy

on T is due to the dimensionality (D = 2) and thequadratic dispersion around (k) = 0.

Thermal Hall effect in quantum spin liquids. Asdiscussed above, the spin Hamiltonian contains the mag-netic flux , and the spin waves or magnons are influ-enced by the magnetic field only through the off-diagonalmatrix elements of the thermal currents, correspond-ing to the intrinsic anomalous Hall effect. This is insharp contrast to the case of electrons which is cou-pled to the vector potential A, and the usual Hall ef-fect due to the Lorentz force occurs there. In this re-spect, it is interesting to note that the spin operator Sican be represented by the fermion operators fi and fi

(called spinons with spin 1/2) as Si =

, fifi/2( = (x, y, z): Pauli matrices, , =, ) with theconstraint

f

ifi = 1. In this representation, the ex-

change interaction JSi Sj can be written as Jijij/2with ij =

f

ifj . This ij is called the order pa-

rameter of the resonating valence bond (RVB) which de-

scribes the singlet formation between the two spins Siand Sj . In the mean field approximation for ij , the free

fermion model for fi, fi emerges [24]. The phase aij


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of the order parameter ij = |ij|eiaij and the La-grange multiplier a0 to impose the constraint above con-stitutes the gauge field, which is coupled to the spinons.In the confining phase of this gauge field, two spinonsare bound to form a magnon. On the other hand, inthe deconfining phase, the spinons behave as nearly freequasi-particles. The latter case is realized in some ofthe quantum spin liquid states [24]. Similar state has

been obtained also for the Hubbard model [25], whichcontains the gapped charge excitations. This charge de-grees of freedom is represented by the U(1) phase factoreii, i.e., the electron operator ci is decomposed intothe product fie

ii , which is coupled to aij Aij whereAij is the vector potential (Peierls phase) correspondingto the magnetic flux [25]. Then, the Maxwell termLg = (1/g)

dr

(F F)2 (g: coupling constant,F = a a, F = A A, ) is generatedby integrating over the charge degrees of freedom. Tosummarize, the spinons are described by the Lagrangian

L=

j,

fj(

ia0j

)fj

j,k

tfeiajkfjfk + Lg, (7)

Following the previous works [14, 25], we take the spinonmetal with a Fermi surface as a candidate for the 2Dquantum spin liquid realized in -(ET)2Cu2(CN)3[12].In a magnetic field Fxy = Bz, the average of the gaugeflux Fxy = cFxy is induced with c a constant of theorder of unity because of the coupling between Fxy andFxy in Lg [14, 24]. Therefore, the spinons are subjectto the effective magnetic field Fxy and to the Lorentzforce.

Let us first estimate the spinon lifetime from therecent thermal transport measurements in that material.The longitudinal thermal conductivity is obtained fromthe Wiedemann-Franz law by assuming a Fermi liquid ofspinons:

xxsp = 22

3

F

k2BT

h 1

d, (8)

where F is the Fermi energy and d 16A is the inter-layer distance. After a subtraction of the phonon con-tribution, xxsp is estimated to be 0.02 WK1m1 atT = 0.3 K [12]. We obtain F / = 56.5 and with

F = J 250 K, we estimate 1.72 1012 s. Nextwe examine xysp . As has been shown in [14], the gaugeflux for spinons is comparable to the applied magneticflux and hence xysp (c)xxsp , where c = eB/mc isthe cyclotron frequency with the effective mass of spinonmc. Estimating mc 2/(Ja2) with assuming the lat-tice spacing a 10A, we obtain c 0.086 B withB being measured in Tesla. Therefore, the thermal Hallangle xysp /

xxsp c becomes of the order of 0.1, which

is easily measurable, with a weak magnetic field B 1T

such that the spin-liquid ground state is not disturbed.Also note that compared with the intrinsic thermal Halleffect discussed above, the magnitude of this Lorentz-force driven thermal Hall conductivity is much larger bythe factor of (F/)2. Therefore, the observation ofthe thermal Hall effect is a clear signature of such de-confined spinons in the spin liquid, and experiments on-(ET)2Cu2(CN)3 is highly desirable.

Another important difference between the spinon con-tribution and the intrinsic term is that the spinons arediffusive and see the field A. Thus in a small sampleone can expect mesoscopic effects such as universal con-ductance fluctuations of the thermal conductivity as afunction of B. Using the Wiedemann-Franz law, we ex-pect the relative fluctuation in xx and xy to be of order/(F) for each coherent volume with dimension

in

where = vF and in = vFin and in is some inelasticscattering time much longer than at low temperatures.The fluctuation is reduced by

N if the sample contains

N coherent volumes. We estimate the elastic mean free

path to be 400A, so that at low temperatures this effectmay be observable in micron-scale samples.

In conclusion, we have studied theoretically the ther-mal Hall effect in the quantum spin systems induced bythe external magnetic field. There are three cases, i.e.,(i) no thermal Hall effect, (ii) intrinsic thermal Hall ef-fect by the magnons, and (iii) large thermal Hall effectdue to the Lorentz force. (i) corresponds to the mostof the conventional (anti)ferromagnets on triangular andsquare and cubic lattices, while (ii) to the magnets on aparticular lattice structure such as kagome, and (iii) tothe spin liquid with deconfined spinons. Therefore, thethermal Hall effect offers a unique experimental method

to gain an important insight on the ground state/low en-ergy excitations of the quantum magnets.

The authors are grateful to N. P. Ong, T. Senthil,Y. Taguchi, Y. Tokura, and S. Yamashita for their valu-able comments and discussions. This work is partly sup-ported in part by Grant-in-Aids (No. 17105002, No.19048015, No. 19048008) from the Ministry of Educa-tion, Culture, Sports, Science and Technology of Japan.PAL acknowledges support by NSF DMR-0804040.

[1] Y. Shimizu et al., Phys. Rev. Lett. 91, 107001 (2003).[2] S. Yamashita et al., Nature Phys. 4, 459 (2008).[3] Y. Okamoto et al., Phys. Rev. Lett. 99, 137207 (2007).[4] J. S. Helton et al., Phys. Rev. Lett. 98, 107204 (2007).[5] O. I. Motrunich, Phys. Rev. B 72, 045105 (2005).[6] N. Nagaosa, J. Phys. Soc. Jpn. 75, 042001 (2006).[7] Y. Taguchi et al., Science 291, 2573 (2001).[8] K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev.

B 62, R6065 (2000).[9] S. Fujimoto, Phys. Rev. Lett. 103, 047203 (2009).

[10] X. Zotos, F. Naef, and P. Prelovsek, Phys. Rev. B 55,


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11029 (1997).[11] A. V. Sologubenko et al., Phys. Rev. B 62, R6108 (2000).[12] M. Yamashita et al., Nature Phys. 5, 44 (2009).[13] D. Sen and R. Chitra, Phys. Rev. B 51, 1922 (1995).[14] O. I. Motrunich, Phys. Rev. B 73, 155115 (2006).[15] N. A. Sinitsyn, J. Phys.: Condens. Matter 20, 023201

(2008).[16] Also the extrinsic contribution to the thermal Hall con-

ductivity from the skew scattering [15] is expected. How-

ever, at the scattering events with the low energy limit,the s-wave scattering is dominant and the skew scatteringeffect is expected to be very small.

[17] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).[18] D. J. Thouless et al., Phys. Rev. Lett. 49, 405 (1982).

[19] C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phys. Rev.Lett. 95, 155901 (2005).

[20] G. D. Mahan, Many-Particle Physics (Plenum, NewYork, 1990), p.32.

[21] F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004).[22] D. Ceresoli et al, Phys. Rev. B 74, 024408 (2006).[23] Here we consider the perturbative expansion of Eq. (5)

and keep only the first order terms in . See supplemen-tary items for a detailed derivation.

[24] P.A.Lee, N. Nagaosa, and X.G. Wen, Rev. Mod. Phys.78, 17 (2006).

[25] S-S. Lee and P. A. Lee, Phys. Rev. Lett. 95, 036403(2005).

Supplementary Items for Theory of Thermal Hall Effect in Quantum Magnets

Off-diagonal terms in energy current density

In this section, we give an explicit expression for the energy current density jE in the basis where the Hamiltonianis diagonal. As we have explained in the main text, jE is not diagonal in this basis. We start with the non-interactingHamiltonian in the Fourier space:

H =k,,

b(k)H,(k)b(k), (9)

where b(k) denotes the Fourier transform of bRj and repeated indices are summed over. The single-particle

Hamiltonian H(k) can be diagonalized by the unitary matrix g(k) as g(k)H(k)g(k) = (k), where (k) =diag(1(k),...,n(k)). Using g(k) we define a new basis of bosons as

i(k) =j

g

(k)ijbj(k). (10)

In this basis, H is written as H =

k(k)

(

k)(k).

Let us now see the off-diagonal structure of jE in the basis of (k). This is clarified by introducing the Berry

connection (or Maurer-Cartan 1-form) defined by (k) = g(k)kg(k). Using , the energy current density (Eq. (4)

in the main text) is written as

jE =1

2V

k

(k)

k(k)2 + [(k), (k)2]

(k). (11)

Here we have used the relation (kg(k))g(k) = g(k)(kg(k)). The first term in the bracket is a diagonal one

corresponding to (k)k (k). On the other hand, the second term represented by the commutation relation givesthe off-diagonal elements of the energy current density. Similarly to the case of the intrinsic anomalous Hall effect,this off-diagonal term can be regarded as arising from anomalous velocities. The thermal conductivity tensor can beobtained by substituting the above expression for jE into the Kubo formula:

xy =V

T

0

dt

0

djxE(i)jyE(t)th, (12)

where = 1/T is the inverse temperature and ...th denotes thermal average. Along the same lines as the derivationof the TKNN formula [18], one can obtain Eq. (5).


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A detailed derivation of the thermal Hall conductivity in the Kagome ferromagnet

Eigenvalue problem

In the main text, we have studied the thermal Hall effect in the kagome ferromagnet. To obtain the spin-wavespectrum and eigen-modes, we need to solve the eigenvalue problem associated with the following matrix:

(k, ) = 0 cos(

k

a1)ei/

3 cos(k

a3)ei/

3

cos(k a1)ei/3 0 cos(k a2)ei/3cos(k a3)ei/3 cos(k a2)ei/3 0

. (13)Before study this matrix, we consider a more general matrix

M =

0 a ca 0 b

c b 0

(14)

and its secular equation given by

3 = (|a|2 + |b|2 + |c|2) + (abc + abc). (15)Now we apply Vietes solution of cubic equation. To do so, we compare Eq. (15) with the following identity:

cos3

3

=

3

4cos

3

+

1

4cos . (16)

Then, we obtain one of the solutions 1 = 2A cos(/3) with cos = B/(2A) (0 ), where A and B are definedthrough the following relations:

3A2 = |a|2 + |b|2 + |c|2, A2B = abc + abc. (17)The other two solutions can also be written in terms as 2 or 3 = 2A cos[( 2)/3]. The normalized eigenvectorcorresponding to n (n = 1, 2, 3) is explicitly given by W

1n (

2n |b|2, na + bc,nc + ab)T where W2n = 3[2A22n +

(A2 2n)|b|2 + nA2B].Let us now apply the above general result to our specific problem of (k, ). We identify 3A2 with 1 + f(k), and

A2B with f(k) cos(), where f(k) = 23j=1 cos(

k aj). Then, the eigenvalues of (k, ) are given by

n(k, ) = 2

1 + f(k)

3cos

n(k)

3

, (n = 1, 2, 3) (18)

with

cos 1(k) =

27

4

(f(k)cos )2

(1 + f(k))3, 2(k) = 1(k) + 2, 3(k) = 1(k) 2. (19)

It is useful to note that 1 + f(k) =3

j=1 cos2(k aj). The normalized eigenvector corresponding to n(k, ) is written

as

|un(k) =1

Wn(k, ) n(

k, )2 cos2 k2ei/3

n(k, )cos k1 + e2i/3

cos k2 cos k3ei/3n(k, )cos k3 + e2i/3 cos k1 cos k2

, (20)

where Wn(k, )2 = 2(1 + f(k))n(k)

2 + (1 + f(k) 3n(k)2)cos2 k2 + 3n(k)f(k)cos . As for the ferromagneticKagome model, the Hamiltonian is defined by H(k) = 4JS 2JS(cos(/3))1(k, ) and hence the eigen-energiesare given by

n(k) = 4JS

1

1 + f(k)

3

cos(n(k)/3)

cos(/3)

. (21)


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Derivation of Eq. (6)

We now derive Eq. (6) in the main text from the general formula for xy (see Eq. (5)). In the limit of lowtemperature and weak magnetic field, the contribution from the lowest band ( = 1) dominates in RHS of Eq. (5)

due to the Bose factor n(k). Therefore, we approximate Eq. (5) as

xy 12T

Im

BZd2k

(2)2n1(k)

kxu1(k)(H(k) + 1(k))2

kyu1(k). (22)

Inserting the resolution of the identity at each k, i.e., 1 =3

=1 |u(k)u(k)|, we can rewrite the above expressionas

xy 12T

=2,3

BZ

d2k

(2)2n1(k)(1(k) + (k))

2 Im

kxu1(k)u(k)u(k)kyu1(k) . (23)

Note that we have omitted = 1 from the summation above since it can be shown thatkxu1(k)|u1(k)u1(k)|kyu1(k) is real using the fact |u1(k) is normalized, i.e., ku1(k)|u1(k) + u1(k)|ku1(k) =0. In the low temperature limit, the dominant contribution to the integral in Eq. (23) comes from small |k| due tothe Bose factor. In the vicinity ofk = 0, one can expand cos 1(k) in Eq. (19) and 1(k, ) in Eq. (18) with respect

to k as

cos 1(k) cos (1 + O(k4

)), 1(k, ) 2 cos(/3)1 163

j=1k2j + O(k

4

)

. (24)

Therefore, in the vicinity ofk = 0, 1(k) JS|k|2. Furthermore, using Eq. (19) and (24), we find 1(k) + 2(k) 2JS(3 3 tan(/3) + O(k2)) and 1(k) + 3(k) 2JS(3 +

3tan(/3) + O(k2)). Then, we approximate 1 + 2

and 1 + 3 by 6JS and rewrite Eq. (23) as

xy 12T

BZ

d2k

(2)2n1(k)(6JS)

2=2,3

Im

kxu1(k)u(k)u(k)kyu1(k)

= (6JS)2

2T

BZ

d2k

(2)2n1(k)

3=1

Im

kxu1(k)u(k)u(k)kyu1(k)

= (6JS)2

2TBZ

d2k

(2)2 n1(k) Im

kxu1(k)kyu1(k)

. (25)

Here, we have again used the fact that kxu1(k)|u1(k)u1(k)|kyu1(k) is real. Note that if we take into account the- and k-dependencies of 1 + i (i = 2, 3), they just give higher order terms in and T. (As we will see in the next

paragraph, Im[kxu1(k)|kyu1(k)] is already first order in . )For our purpose to obtain the thermal Hall conductivity, we need to calculate the imaginary part of

kxu1(k)|kyu1(k). Now we consider the perturbative expansion of |u1(k) with respect to and keep up to thefirst order terms in . The zeroth and the first order wavefunctions are given by

|u(0)1 (k) =1

W1(k, 0)

1(

k, 0)2 cos2 k21(k, 0) cos k1 + cos k2 cos k31(k, 0) cos k3 + cos k1 cos k2

(26)

and

|u(1)1 (k) =i(/3)

W1(k, 0)

01(k, 0) cos k1 2cos k2 cos k3

1(k, 0)cos k3 + 2 cos k1 cos k2

, (27)

respectively. Then, from the Taylor expansion around k = 0, we obtain

Im[kxu1(k)|kyu1(k)] Im[kxu(0)1 (k)|kyu(1)1 (k) + kxu(1)1 (k)|kyu(0)1 (k)]=

27

3(k2x + k

2y). (28)


8/8

8

Therefore, for the first order term in , the Berry curvature around the point k = 0 is quadratic in |k|. SubstitutingEq. (28) into Eq. (25), we obtain Eq. (6) shown in the main text.

hosho katsura, naoto nagaosa and patrick a lee- theory of the thermal hall effect in quantum magnets

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