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Magnon Hall effect on the Lieb lattice

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 166003 (7pp)

doi:10.1088/0953-8984/27/16/166003

Magnon Hall effect on the Lieb lattice Xiaodong Cao, Kai Chen and Dahai He Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of China E-mail: [email protected] Received 21 January 2015, revised 27 February 2015 Accepted for publication 3 March 2015 Published 30 March 2015 Abstract

Ferromagnetic insulators without inversion symmetry may show magnon Hall effect (MHE) in the presence of a temperature gradient due to the existence of Dzyaloshinskii–Moriya interaction (DMI). In this theoretical study, we investigate MHE on a lattice with inversion symmetry, namely the Lieb lattice, where the DMI is introduced by adding an external electric field. We show the nontrivial topology of this model by examining the existence of edge states and computing the topological phase diagram characterized by the Chern numbers of different bands. Together with the topological phase diagram, we can further determine the sign and magnitude of the transverse thermal conductivity. The impact of the flat band possessed by this model on the thermal conductivity is discussed by computing the Berry curvature analytically. Keywords: magnon Hall effect, edge states, flat band (Some figures may appear in colour only in the online journal)

of magnon opens new possibilities to achieve dissipationless transport and flexible heat control in insulating magnets [9–14]. The previous studies on MHE mainly focus on the twodimensional kagome lattice which can be layered to form the pyrochlore Lu2 V2 O7 , host of the experimental observation of MHE [7]. It is well known that the DMI [15, 16] which is responsible for MHE, is absent in a lattice with inversion symmetry. However, as discussed in [17, 18], the DMI may be induced by an external electric field. This raises the question whether MHE may also exist in a system which holds the inversion symmetry? It can be shown that the inversion symmetry can simplify the problem of the evaluation of the Z2 invariant greatly in the study of TI [19]. As a paradigm of inversion symmetry, the Lieb lattice, a linecentered square lattice, shows inversion symmetry and has specific properties that come from its topology. Intense investigations of the Lieb lattice have been conducted both theoretically and experimentally in electronic systems [20– 23], photonic systems [24] and cold atoms [25]. The main focus of these studies is basically on the flat band and single Dirac cone located in the Brillouin zone, which may have a novel physical impact. Here, we present another possible host of MHE, the Lieb lattice with complex next-nearest neighbor(NNN) magnon hopping. Our main results can be summarized as follows. We discuss the tight-binding Hamiltonian which can support nontrivial topology and its corresponding band structure. We

1. Introduction

Intensive study on topological insulator (TI), which is characterized by the possession of edge states or surface states connecting the gapped bulk valence band and conduction band, has provided us with a deep understanding of the topological aspect of matter [1, 2]. It also stimulates attempts to achieve the application of dissipationless spin-based electronics [3]. However, for the reason that spin transport in topological insulator is carried by electron, the accompanying Joule heating makes dissipationless transport hard to achieve. A promising candidate for dissipationless transport is ferromagnetic insulator possessing MHE, which is the bosonic analogy of TI. MHE describes the existence of a transverse thermal conductivity with a temperature gradient along the longitudinal direction and it has been predicted and observed in magnetic insulators [4–7]. These investigations extend the study on the topological aspect of electrons to that of magnons which are the collective excitation of localized spins. The authors in [4] found that if the link between two spins is not shared by two equivalent cells, then the intrinsic thermal Hall effect may occur, i.e. there exists a transverse thermal conductivity. Further studies [5, 6] show that the thermal Hall effect can be understood by the semiclassical picture that the magnon wave pocket undergoes a self-rotational motion and a rotational motion along the edge of the sample material [5, 6] and the topological nature of MHE has been discovered [8]. In addition to its theoretical significance, the charge-free nature 0953-8984/15/166003+07$33.00

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X Cao et al

then discuss the topological phase diagram characterized by the Chern numbers and explore its relation with the transverse thermal conductivity. As recently discussed in [26], the transverse thermal conductivity can change sign by adjusting temperature in specific parameter regimes and this property exhibits in the model we studied. Together with the topological phase diagram, we can determine the sign of the transverse thermal conductivity in the high temperature limit. In fact, the sign of the transverse thermal conductivity corresponding to three out of four topological phases can be determined in this way, leaving one specific topological phase whose sign depends on details of the band structure. This specific topological phase exhibits high Chern number and shows the property of sign changing in dependence on temperature. It means that we can control the direction of the transverse heat flow by the tuning of temperature, which makes it promising for device design. Finally, we pay specific attention to the case that the NNN magnon hopping is purely imaginary, which leads to the existence of a flat band. We analyze the impact of the flat band on thermal conductivity by computing the Berry curvature analytically. The flat band has a vanishing contribution to the transverse and longitudinal thermal conductivity due to the vanishing of both the Berry curvature and group velocity. This paper is organized as follows. In section 2, we present our model and discuss its band structure; in section 3, we present the topological aspects of this model; in section 4, we explore the parameter dependence of the transverse thermal conductivity and show its relation with the topological phase diagram; in section 5, we pay specific attention to the case which holds the flat band; Finally, in section 6, a summary of this paper is presented and the possible experimental realization is discussed.

Figure 1. (a) The structure of the Lieb lattice. Each cell contains three different sites. a1 = (2a, 0) and a2 = (0, 2a) are the lattice vectors and a = 1 is the nearest site distance. The coupling of two sites along the arrow corresponds to (J2 + iD)S, while the coupling of the opposite direction corresponds to (J2 − iD)S. (b) The first Brillouin zone of the Lieb lattice.

induced by the external electric field with Dmn = Dez ∝ E × emn . ez is the unit vector in the z direction which is chosen as the quantization axis and emn is the unit vector connecting the localized spins on site m and n. The last term is the Zeeman term caused by coupling with an external magnetic field along the z direction, with h = gH µB where g is the g factor of electron and µB is the Bohr’s magneton and H is the strength of the external magnetic field. This Zeeman term, as adapted in the experimental observation of MHE in Lu2 V2 O7 [7], can reduce the dimension of the model from three to two because only the components of the DM vectors along the external magnetic field can contribute to the transverse thermal conductivity in the xy plane. In the following, we set h = 0+ to align the direction of the localized spins. Sˆ + = Sˆ x + iSˆ y and Sˆ − = Sˆ x − iSˆ y are the spin ladder operators. The Lieb lattice is illustrated in figure 1. There are three sites in each unit cell, a1 = (2a, 0) and a2 = (0, 2a) are the lattice vectors and a = 1 is the nearest site distance. The nearest coupling exists between site 1 and site 2, site 1 and site 3. The next-nearest coupling exists between site 2 and site 3. When the coupling is along the arrow, i.e. from site 3 to site 2, the strength is (J2 + iD)S, while the coupling along the opposite direction corresponds to the strength (J2 − iD)S. For a ferromagnetic insulator described by (1), one can perform the standard Holstein–Primakoff [27] transformation,  ˆ Sˆ + = 2S − nˆ b,  † − ˆ Sˆ = bˆ 2S − n, ˆSz = S − n, ˆ (2)

2. Model and band structure

The MHE is due to the coupling of the spin current with an effective gauge potential. In the lattice without inversion symmetry, the DMI provides the effective vector  potential. The conventional DMI is given by HDM = m,n Dmn ·   Sˆ m × Sˆ n [15, 16], where D is the DM vector and Sˆ m is the localized spin on site m, together with the Heisenberg interaction in the xy plane, the total Hamiltonian has the form of H = Hxy + HDM . In the lattice with inversion symmetry, the conventional DMI is absent [4]. However, as studied in [17, 18], an external electric field may serve as a gauge field coupling with the spin current. Therefore, the Hamiltonian can be written as  H = −J1 Sˆ m · Sˆ n

where nˆ = bˆ † bˆ is the magnon density operator and S = 1/2 for spin one-half system. The total Hamiltonian has two parts if we ignore the magnon–magnon interaction,

m,n

   J2 + iD J2 − iD ˆ + ˆ − + ˆ− z ˆz ˆ ˆ − Sm Sn + Sn Sm + J2 Sm Sn 2 2 m,n  (1) Sˆ zn , −h

H = H0 + HSW ,   HSW = −J1 S bˆ †m bˆ n + bˆ †n bˆ m − (nˆ m + nˆ n )

n

−S

where J1 denotes the nearest-neighbor(NN) coupling, J2 denotes the NNN coupling and D is the strength of DMI





[(J2 + iD)bˆ †m bˆ n + (J2 − iD)bˆ †n bˆ m − J2 (nˆ m + nˆ n )],

m,n

2

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Figure 2. Magnon band structure of the Lieb lattice with different J2 and D. J1 = 1 and S = 0.5 in all cases. (a) With J2 = D = 0, three bands touch each other at the Dirac cone in the first Brillouin zone. (b) With J2 = 0.2, D = 0, a gap is opened between the lower two bands by the NNN coupling and the formal flat band becomes dispersive. (c) With J2 = 0.2, D = 0.2, a gap is opened between the upper two bands and a Dirac cone forms between the two lower bands. (d) With J2 = 0.2, D = 0.4, two gaps have been opened by the NNN coupling and DMI. All three bands are now isolated from each other.

H0 = −h



S − J1

 m,n

m

S 2 − J2



S2,

 4J1 S ∓2J1 S cos2 (kx ) + cos2 (ky ). In the first Brillouin zone, three bands touch each other at the Dirac cone as shown in figure 2(a). If the NNN coupling is included but DMI is turned off, a gap will be opened between the two lower bands and the formal flat band becomes dispersive as depicted in figure 2(b). Noting that, for the existence of NNN coupling, the diagonal elements of hk are different, which is equivalent to the case where an on-site energy is added to the sites with four neighbors or conversely the inequivalent sites with two neighbors as discussed in the electronic system [20] and cold atoms [28]. Furthermore, as depicted in figure 2(c), when the DMI is turned on and J2 = D, a new gap is opened between the upper two bands while the gap between the lower two bands is closed by the touching at the single Dirac cone. When J2 = D, all three bands are isolated and two gaps are opened.

(3)

m,n

where H0 is the classical ground energy and HSW represents the spin-wave excitation contribution. In the following, we will just consider the spin-wave part of the Hamiltonian. The Fourier transformation of the magnon creation and annihilation reads 1  −ik·(Rl +rm ) ˆ bm (k), e bˆ Rl +rm = Nµ k 1  ik·(Rl +rm ) ˆ † bm (k), e (4) bˆ †Rl +rm = Nµ k where Nµ is the number of unit cells. Rl is the position of the lth unit cell and rm is the relative position of the mth site in the unit cell with m = 1, 2, 3. The spin-wave part of the Hamiltonian now has the matrix form of  † bˆ k hk bˆ k , HSW = (5)

3. Topological aspect

To show the nontrivial topology of this model, we now examine the existence of edge state and classify the topological phase diagram by Chern numbers of three bands. First, we consider a quasi-one-dimensional periodic ribbon with finite layers Ny in y direction, while it retains the translational symmetry in x direction. By partial Fourier transformation in x direction, the Hamiltonian has the structure of, for example Ny = 4,   A C 0 0  B A C 0   Hribbon =  (6)  0 B A C , 0 0 B A

k

with hk (1, 1) = 4J1 S, hk (2, 2) = hk (3, 3) = 4J1 S + 4J2 S, hk (1, 2) = hk (2, 1) = −2 cos (kx )J1 S, hk (1, 3) = hk (3, 1) = −2 cos (ky )J1 S, hk (2, 3) = hk∗ (3, 2) = −4J2 S cos (kx ) cos (ky ) − 4iDS sin (kx ) sin (ky ) and bˆ †k = (bˆ †1 (k), bˆ †2 (k), bˆ †3 (k)). By using the periodic boundary condition in both x and y direction, we can obtain the dispersions of the three bands after the diagonalization of the 3 × 3 matrix. When J2 = D = 0, the spectrum of hk consists of one degenerate flat band E2 = 4J1 S and two dispersive bands E1,3 (k) = 3

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Figure 3. The dispersion relations of the quasi-one-dimensional Lieb ribbon with Ny = 100 and J1 = 1. (a) With J2 = D = 0, two bands touch the flat band at the Dirac cone. (b) With J2 = 0.2, D = 0, a gap is opened between the lower two bands by the NNN coupling, while the other band is still isolated. (c) With J2 = 0.2, D = 0.1, the DMI induces nontrivial topology of the dispersion relation with a pair of edge states between the upper two bands. (d) With J2 = D = 0.2, the two lower formal gapped bands touch each other at the newly formed Dirac cone. (e) With J2 = 0.2,D = 0.4, two pairs of edge states emerge and all bulk bands are gapped. (f ) The special case for the existence of a flat band with a pure imaginary NNN magnon hopping(J2 = 0,D = 0.2).

with A(1, 1) = 4J1 S, A(2, 2) = A(3, 3) = 4J1 S + 4J2 S, A(1, 2) = A(2, 1) = −2J1 S cos (kx ), A(1, 3) = A(3, 1) = −J1 S, A(2, 3) = A(3, 2) = −2J2 S cos (kx ) + 2DS sin (kx ) and   0 0 −J1 S C =  0 0 − (2J2 S cos (kx ) + 2DS sin (kx ))  , (7) 0 0 0

and Bknx ky

=i

HSW HSW ϕn ϕn† ∂ ∂k ϕn − (kx ↔ ky )  ϕn† ∂ ∂k x y

(En − En )2

n =n

,

(9)

with En and ϕn the eigenvalue and eigenvector of the nth band, respectively. There are four different topological phases characterized by the Chern numbers (C 1 , C 2 , C 3 ) of the three bands as shown in figure 4. Although the parameter range presented here is [0, 2] for J2 and D, we have computed the Chern numbers of the three bands in a wider parameter range and find that no new topological phase emerges. The tuning of J2 and D indeed drive the topological phase transition. This topological phase diagram can help us to determine the sign and magnitude of the transverse thermal conductivity as discussed in the following.

and B = C T . As shown in figure 3, when there is no NNN coupling and DMI, there is one Dirac cone in the first Brillouin zone; and when the NNN coupling is turned on, a gap is opened between the two lower bands, while the upper two bands remains touching; when the DMI is turned on and J2 < D, all bulk bands are gapped and a pair of edge states emerges between the upper two bands; the lower two bands, which are formally gapped, touch each other at the newly formed Dirac cone when J2 = D; when J2 > D two pairs of edge states emerge and the bulk bands are all isolated. The special case of existence of flat band is depicted in figure 3(f ). The above discussions suggest that topological phase transition may happen with the tuning of D and J2 . To gain a better understanding of the existence of topological phase transition, we further compute the Chern numbers of three bands as the invariant of topological phase. The Chern number is given by the integration of the Berry curvature over the first Brillouin zone [1, 2, 29],  1 n dkx dky Bknx ky , (8) C = 2π

4. Transverse thermal conductivity

In the following, we study the transverse thermal conductivity and its relation with the topological phase diagram. The intrinsic contribution to the transverse thermal conductivity can be formulated by the semiclassical equation of motion of the magnon wave pocket which includes the anomalous velocity in terms of the Berry curvature [5, 6],  kB2 T  xy c2 (ρi )Bki x ky dkx dky , (10) κ = (2π )2 h ¯ i BZ

with the sum running over the three bands and the integral is over the first Brillouin zone. ρi = 1/ (exp (¯hEi /kB T ) − 1) is

BZ

4

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Figure 4. Topological phase diagram of our model. The Chern numbers of the three bands are depicted in (a)–(c), respectively. (d) The topological phase diagram characterized by the Chern numbers of the three bands. There are four different topological phases in the parameter range of D/J1 ∈ [0, 2] and J2 /J1 ∈ [0, 2].

the Bose distribution with Ei as the ith eigenvalue. c2 is given by   1 + ρi 2 − (ln ρi )2 − 2Li2 (−ρi ), (11) c2 (ρi ) = (1 + ρi ) ln ρi x where Li2 (x) = − 0 du(ln (1 − u)/u), x ∈ (−∞, 1) is the dilogarithm of the Spence function. c2 (ρi ) as the function of energy for different temperatures is depicted in figure 5. The high temperature limit is π 2 /3. The high temperature limit of the transverse thermal conductivity can be obtained as follows [26],  kB  xy Ei Bki x ky dkx dky . (12) κlim = lim κ xy = − T →∞ ¯ i (2π)2 h BZ

This high temperature limit can be used to estimate the sign and maximum magnitude of the transverse thermal conductivity, even though it can never be reached in the ferromagnetic phase. To build a relation between the sign of the transverse thermal conductivity and the topological phase, we assume a flat band of this model and replace the k-dependent band energy in (12) by its average energy E i . Then we have kB  i xy C Ei . (13) κlim ∝ − (2π )2 h ¯ i

Figure 5. c2 (ρi ) as a function of energy E for different temperatures.

The high temperature limit is π 2 /3. Here we set h ¯ = kB = 1.

determined as  E 2 − E 3 < 0, Phase(0, −1, 1);    E 1 − E 3 < 0, Phase(−1, 0, 1); xy κlim ∝  E + E 3 − 2E 2 , Phase(−1, 2, −1);   1 E 3 − E 2 > 0, Phase(0, 1, −1).

(14)

The sign of κ xy can be determined in all topological phases except for Phase(−1, 2, −1). As depicted in figure 6(a),

The signs of the transverse thermal conductivity corresponding to the four topological phases as depicted in figure 4 can be 5

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X Cao et al

hk can  be found as E2 = 2J1 and E1,3 = 2J1 ∓ R with R = R12 + R22 + R32 . Noting that, all three bands are gapped for R3 = 0. The eigenvectors can be obtained with a U(1) gauge symmetry. As discussed in [30], this U(1) symmetry disappears in the Berry curvature hence the Chern number is independent of a specific choice of eigenstate. The eigenstates are found as 1 |2  = (iR3 , R2 , −R1 ), R 1 |1,3  =  R 2(R 2 − R12 ) ×(∓RR2 + iR1 R3 , R1 R2 ∓ iRR3 , R 2 − R12 ).

(16)

Then the Berry curvatures for the three bands are given by Figure 6. The transverse thermal conductivity as function of

R · (∂kx R × ∂ky R), R3 and the Chern numbers are given by π π 1 2 1,3 dkx dky C = 0, C = ∓1 = ∓ 2π =∓ Bk2x ky = 0, Bk1,3 x ky

temperature for different D with J1 = 1. (a) With J2 = 0.2, all κ xy < 0. (b) With J2 = 0.6, κ xy can change sign by varying D and temperature. The relation between k xy and the topological phase diagram is explained in the main text. Here we set h ¯ = kB = 1.

when J2 = 0.2, our system is either in Phase(0, −1, 1) or Phase(−1, 0, 1) in dependence of D, so the transverse thermal conductivity always has κ xy < 0 according to (14). Furthermore, for |E 1 − E 3 | > |E 2 − E 3 |, κ xy in Phase(−1, 0, 1) magnitude is larger than κ xy in Phase(0, −1, 1). This agrees with the observation of the increasing of magnitude of κ xy in figure 6(a) with the increasing of D. For J2 = 0.6, as depicted in figure 6(b), our system will experience a topological phase transition from Phase(0, 1, −1) to Phase(−1, 2, −1) with the increasing of D, so the sign of κ xy will change from positive which is the consequence of in Phase(0, 1, −1) to values depending on the details of the band structure. We can see that when J2 = D = 0.6, κ xy can change sign with the increase of temperature after reaching its minimum.

(17)

0 0 2 2 2 2 2 2J D(cos ky sin kx + cos kx sin ky + sin2 kx sin2 ky ) . × 1 2 (J1 cos2 kx + J12 cos2 ky + 4D 2 sin2 kx sin2 ky )3/2

(18) This analytical result agrees with our numerical result as shown in figure 4 when J2 = 0. Now, from the semiclassical equation of motion of the magnon wave pocket [5], 1 ∂En (k) ˙ (19) − k × Bknx ky , h ¯ ∂k where r is the center of the magnon wave pocket, it is apparent that the flat band does not contribute to the transverse and longitudinal thermal conductivity for both the vanishing of the group velocity ∂E2 (k)/∂k and anomalous velocity caused by the Berry curvature. Although the flat band makes no contribution to the transverse thermal conductivity, the maximum magnitude of κ xy is achieved due to the contribution of another two bands when J2 = 0, as shown in figure 7. We can see a good agreement of the sign of κ xy between our classification in (14) and the numerical result. r˙ n =

5. Flat band

Finally, we discuss the impact of flat band, which is a specific property of the Lieb lattice, on the thermal conductivity. Generally, it is difficult to calculate the Chern number analytically. But in this specific case, i.e. J2 = 0, we can obtain the Berry curvature and Chern number analytically [30]. When J2 = 0, the DMI provides a purely imaginary NNN hopping amplitude for magnon and we can write the Hamiltonian as hk = R · L, where R = (R1 , R2 , R3 ) with R1 = −J1 cos kx , R2 = −J1 cos ky and R3 = 2D sin kx sin ky . L = (L1 , L2 , L3 ) is a vector of the following Gell-mann matrices     0 1 0 0 0 1 L1 =  1 0 0  , L2 =  0 0 0  , 0 0 0 1 0 0   0 0 0 L3 =  0 0 −i  , (15) i 0 0 which form the spin-1representation of the Lie algebra SU(2) which satisfy Li , Lj = iij k Lk . The three eigenvalues of

6. Summary

In summary, we explored the magnon Hall effect on the Lieb lattice which has inversion symmetry. Although the conventional DMI may be absent, an external electric field can induce the DMI. By showing the existence of the edge state and nontrivial topology invariant, we confirm the existence of the magnon Hall effect in this model. Furthermore, following the semiclassical equation of motion of the magnon wave pocket, we discussed the sign of the transverse thermal conductivity in the high temperature limit and its relation with the topological phase diagram characterized by Chern numbers of the three bands. We found the transverse thermal 6

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Acknowledgments

The authors thank J Zheng and X Lu for helpful discussions and the Xiamen Supercomputer Center for the use of its computing facilities. This work was financially supported by NSFC No. 11047185 and No. 11105112 and FRFCU (Grant No. 2010121009). References [1] [2] [3] [4] [5] [6] [7]

Figure 7. The contour plot of the transverse thermal conductivity as

function of D/J1 and J2 /J1 at temperature T = 5. The dashed line depicts the contour κ xy = 0. The maximum magnitude of κ xy is achieved at J2 = 0. Here we set h ¯ = kB = 1.

[8] [9] [10] [11]

conductivity can change sign in dependence on temperature in one of the four different topological phases, while the signs of the others are fixed. Finally, we paid special attention to the flat band and we found the flat band does not contribute to both transverse and longitudinal thermal conductivity. This is contrary to our expectation that the flat band may contribute remarkably to the transverse thermal conductivity which will reach the maximum magnitude when the flat band appears. As for possible experimental realizations, the essential points are finding real materials which have the structure of the Lieb lattice and the way to induce the DMI. Similar to the first experimental observation of MHE in Lu2 V2 O7 with a pyrochlore structure, for which MHE can be modeled by the kagome lattice in the presence of a perpendicular magnetic field, a straightforward generalization of the Lieb lattice from 2D into 3D is the perovskite structure, which exists in some materials. A prominent example is the high-Tc cuprate superconductors, such as YBa2 Cu3 O7 or Bi2 Sr 2 CaCu2 O8 , which are layered perovskite structure consisting of weakly coupled 2D CuO2 planes with Lieb lattice structure. For the pyrochlore structure, the inversion symmetry is broken and there exists nonzero DMI naturally. However, since the perovskite structure has inversion symmetry, we need the magnetoelectric effect to induce the DMI by an external electric field. Therefore, the possible candidates to find the experimental realization might be the complex perovskite oxides, which exhibit ferromagnetism at low temperature and the magnetoelectric effect, as discussed in previous studies [31].

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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Magnon Hall effect on the Lieb lattice.

Ferromagnetic insulators without inversion symmetry may show magnon Hall effect (MHE) in the presence of a temperature gradient due to the existence o...
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