Topological Magnons in Kitaev Magnets with Finite Dzyaloshinskii--Moriya Interaction at High Field

Kangkang Li(厉康康)$^{\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/2/027502

⇦Chinese Physics Letters, 2023, Vol. 40, No. 2, Article code 027502⇨ Topological Magnons in Kitaev Magnets with Finite Dzyaloshinskii–Moriya Interaction at High Field Kangkang Li (厉康康)* Affiliations Department of Physics, Zhejiang Normal University, Jinhua 321004, China Received 23 November 2022; accepted manuscript online 13 January 2023; published online 5 February 2023 ^*Corresponding author. Email: physeeks@163.com Citation Text: Li K K 2023 Chin. Phys. Lett. 40 027502 Abstract There have been intensive studies on Kitaev materials for the sake of realization of exotic states such as quantum spin liquid and topological orders. In realistic materials, the Kitaev interaction may coexist with the Dzyaloshinskii–Moriya interaction, and it is of challenge to distinguish their magnitudes separately. Here, we study the topological magnon excitations and related thermal Hall conductivity of kagome magnet exhibiting Heisenberg, Kitaev and Dzyaloshinskii–Moriya interactions exposed to a magnetic field. In a strong magnetic field perpendicular to the plane of the lattice ([111] direction) that brings the system into a fully polarized paramagnetic phase, we find that the magnon bands carry nontrivial Chern numbers in the full region of the phase diagram. Furthermore, there are phase transitions related to two topological phases with opposite Chern numbers, which lead to the sign changes of the thermal Hall conductivity. In the phase with negative thermal conductivity, the Kitaev interaction is relatively large and the width of the phase increases with the strength of Dzyaloshinskii–Moriya interaction. Hence, the present study will contribute to the understanding of related compounds.

DOI:10.1088/0256-307X/40/2/027502 © 2023 Chinese Physics Society Article Text In recent years, quantum materials with bond-dependent anisotropic Kitaev spin interactions have been the subjects of experimental and theoretical studies, because there are frustrations from the competing exchange couplings coexisting with the geometric frustration of the underlying lattices, which may lead to unusual magnetic orders as well as gapped and gapless spin liquids with fractional excitations.[1-8] However, the Kitaev interaction is often accompanied by the Dzyaloshinskii–Moriya interaction (DMI)[9,10] in realistic materials, and it is of challenge to distinguish their magnitudes separately.[11,12] The Kitaev interaction also shows its exotic properties in magnonic side. For instance, various lattices such as honeycomb lattice[12-15] and kagome lattice[16] with Kitaev interaction host topological magnon bands. However, the effect of the Kitaev interaction can be generally similar to that of the DMI, which can also induce topological magnons and has already been studied in lots of experimental and theoretical works.[17-25] In some cases, the spin wave spectrum of the Heisenberg–Kitaev model can even reduce to that of the Heisenberg model with DMI, and then hosts the same topological property.[15] Therefore, the same magnon bands and topological property can be generated either by Kitaev interaction, DMI, or their combination. Fortunately, it was shown that one can distinguish whether a system is Kitaev interaction or DMI dominated by further investigating the magnonic transport properties, for example, in honeycomb Kitaev magnet.[11,12] In this Letter, we study the topological magnon excitations and related thermal Hall conductivity of kagome magnet exhibiting Heisenberg, Kitaev and DM interactions exposed to a magnetic field. It is well known that the magnon bands of kagome ferromagnet with DMI carry nonzero Chern numbers.[20] On the other hand, the kagome magnet with Kitaev interaction, whose ground state is canted ferromagnetic order,[26,27] also hosts topological magnon excitations.[16] Hence, one may wonder what would happen when there are DMI and Kitaev interactions in kagome magnet simultaneously. Since the Kitaev interaction does not support ferromagnetic order,[26,27] we apply a magnetic field perpendicular to the plane of the lattice ([111] direction), which is strong enough to bring the system into fully polarized paramagnetic phase.[15] We find that the magnon bands carry nontrivial Chern numbers in the full region of the phase diagram. Furthermore, there are phase transitions related to two topological phases with opposite Chern numbers, which lead to the sign changes of the thermal Hall conductivity. In the phase with negative thermal conductivity, the Kitaev interaction is relatively large and the width of the phase increases with the strength of DMI. Thus the present study will contribute to understanding of related compounds. Model and Method. We consider interacting spins residing on the kagome lattice as shown in Fig. 1(a). The model is described by the spin Hamiltonian \begin{align} H=\,&J\sum_{\langle ij\rangle}\boldsymbol{S}_i\cdot\boldsymbol{S}_j+K\sum_{\langle ij\rangle}S_i^{\gamma_{ij}}S_j^{\gamma_{ij}}\notag\\ &+D\sum_{\langle ij\rangle}\nu_{ij}\hat{{\boldsymbol z}^\prime}\cdot(\boldsymbol{S}_i\times\boldsymbol{S}_j)-h\sum_i\hat{{\boldsymbol z}^\prime}\cdot\boldsymbol{S}_i.\tag {1} \end{align} Here, $\boldsymbol{S}_i$ and $\boldsymbol{S}_j$ are $S=1/2$ spins residing on the nearest-neighbor (NN) lattice sites, $J$ and $K$ denote the Heisenberg and Kitaev exchange couplings, respectively. The Cartesian components $\gamma_{ij}$ equal $x$, $y$, and $z$, depending on the bond type as shown in Fig. 1(a). We parameterize the Heisenberg and Kitaev terms by $J=\cos\psi$, $K=\sin\psi$, $\psi \in [0, 2\pi)$, with the energy unit $J^2+K^2=1$. The NN DMI is represented by the third term, where $\hat{{\boldsymbol z}^\prime}$ is the unit vector along [111] direction, and $\nu_{ij}=\pm1$ correspond to the anticlockwise and clockwise directions of the NN couplings in a triangle, respectively. In kagome magnet the Kitaev interaction does not support ferromagnetic order,[26,27] hence we add the final term of Zeeman coupling to a applied magnetic field along the [111] direction, which is strong enough to bring the system into fully polarized paramagnetic phase. We study a reasonable parameter range of $D\in[0, 1]$, and set $h=10$ (in units of $S$) all through the work to ensure the stability of the system. Further, the Holstein–Primakoff (HP) transformation[28] is employed to rewrite the Hamiltonian in terms of magnon creation and annihilation operators $a_i^†$ and $a_i$. Under the linear spin wave approximation, we keep only the quadratic terms of the magnon operators. Then the Fourier transformation is performed to rewrite the Hamiltonian in momentum space with basis $\varPsi_{\boldsymbol{k}^\prime}^†=(a_{1\boldsymbol{k}^\prime}^†, a_{2\boldsymbol{k}^\prime}^†, a_{3\boldsymbol{k}^\prime}^†, a_{1-\boldsymbol{k}^\prime}, a_{2-\boldsymbol{k}^\prime}, a_{3-\boldsymbol{k}^\prime})$, and we get the magnon Hamiltonian matrix $h(\boldsymbol{k}^{\prime})$ (see the Supplemental Material for details). Note that, for the present model with fully polarized phase, there are number-nonconserving terms $a^† a^†$ with their coefficients proportional to the Kitaev coupling $K$. Finally, the eigenvalues and eigenvectors are obtained by diagonalizing the dynamic matrix $I_-h(\boldsymbol{k}^{\prime})$ with $I_-=[(I,0),(0,-I)]$, $I$ the $3\times3$ identity matrix.

Fig. 1. (a) The structure of the kagome lattice. There are three spins residing in a primitive cell, which are denoted by red, green and blue sites. Red, green and blue bonds between NN sites $(i,j)$ carry three distinct Kitaev couplings $S_i^xS_j^x$, $S_i^yS_j^y$, and $S_i^xS_j^x$, respectively. Meanwhile, there are isotropic Heisenberg couplings in all the NN bonds. The arrows in a triangle denote the coupling directions whose DM vectors are in the [111] direction. The field is also applied in the [111] direction. The kagome lattice sits on the (111) plane, and we define a new 2D frame $x^\prime y^\prime$ on the kagome lattice with basis vectors $\boldsymbol{a}_1=(1, 0)$ and $\boldsymbol{a}_2=(1,\sqrt{3})/2$. (b) The first Brillouin zone of the kagome lattice.

Topological Magnons. We characterize the topological property of the model by the Chern number[29-31] of the $n$th magnon band as follows: \begin{align} C_n=\frac{1}{2\pi}\int_{_{\scriptstyle \rm BZ}}{d}k_x^{\prime}{d}k_y^{\prime}B_{k_x^{\prime}k_y^{\prime}}^n, \tag {2} \end{align} with the Berry curvature of the $n$th band being \begin{align} B_{k_x^{\prime}k_y^{\prime}}^n\!={i}\!\sum_{n^\prime\ne n}\!\frac{\langle\phi_n|\frac{\partial h(\boldsymbol{k}^{\prime})}{\partial k_x^{\prime}}|\phi_{n^\prime}\rangle\langle\phi_{n^\prime}|\frac{\partial h(\boldsymbol{k}^{\prime})}{\partial k_y^{\prime}}|\phi_n\rangle\!-\!(k_x^{\prime}\leftrightarrow k_y^{\prime})}{(E_n-E_{n^\prime})^2}, \tag {3} \end{align} where $E_n$ and $\phi_n$ are the eigenvalue and eigenvector of the $n$th band, respectively.

Fig. 2. (a) The topological phase diagram of our model as a function of $D$ and $\psi$, where $J=\cos\psi$ and $K=\sin\psi$. The topological phases are characterized by the Chern numbers of magnon bands, and the phase diagram is filled by two phases $(1,0,-1)$ and $(-1,0,1)$. The three red points on the dashed line with $D=0.7$ reside in phase $(-1,0,1)$ with $\psi=1.57\pi$, at phase boundary with $\psi=1.726\pi$ and in phase $(1,0,-1)$ with $\psi=2\pi$, respectively. (b) The Berry curvatures of the first and third magnon bands for $\psi=1.57\pi$ (top panel) and $\psi=2\pi$ (bottom panel) with $D=0.7$. The first Brillouin zone is indicated by a dashed hexagon.

The Heisenberg kagome magnet with only DMI or Kitaev interaction both host topological magnon excitations.[16,20] The similar effects on magnon topology of the DMI and Kitaev interaction stem from their common origin, the spin-orbit coupling. However, due to their different formalisms of spin operators, there should also be competing effects. Here we find that the magnon bands of our model carry nontrivial Chern numbers in the full region of the phase diagram with the tuning of $D$ and $\psi$, as shown in Fig. 2(a). We characterize the topological phase by $(C_1,C_2,C_3)$, and the phase diagram is filled by two phases $(1,0,-1)$ and $(-1,0,1)$. Note that we only show the parameter range $D\in[0.02,1]$ for the phase diagram, because there are complicated phases and phase transitions in the range $D\in[0,0.02]$ and the magnitude of the related thermal conductivity is nearly zero, and these results are deviated from the theme of this study. One may note that the two separated regions of the phase $(-1,0,1)$ have nearly the same shape, and they may be related to each other by $\psi+\pi$. However, it is not exactly the case. The mapping $\psi\rightarrow\psi+\pi$ takes $(J,K)$ to $(-J,-K)$, and the Zeeman coupling to the field only gives an energy shift of $h$ for the whole spectrum, then if we have the eigenvalue of the $n$th band $E_{n,\psi,D}(\boldsymbol{k}^\prime)=\widetilde{E}_{n}(\boldsymbol{k}^\prime)+h$, we must also have $E_{n,\psi+\pi,-D}(\boldsymbol{k}^\prime)=-\widetilde{E}_{n}(\boldsymbol{k}^\prime)+h$. Hence, we have the relation $[E_{n,\psi,D}(\boldsymbol{k}^\prime)+E_{n,\psi+\pi,-D}(\boldsymbol{k}^\prime)]/2=h$, which means that $E_{n,\psi,D}(\boldsymbol{k}^\prime)$ and $E_{n,\psi+\pi,-D}(\boldsymbol{k}^\prime)$ are mirror reflections of each other about the energy axis $E=h$. Thus the closing and re-opening of the band gap, which denotes a topological phase transition, is simultaneous for the bands $E_{n,\psi,D}(\boldsymbol{k}^\prime)$ and $E_{n,\psi+\pi,-D}(\boldsymbol{k}^\prime)$. Therefore, if we have $E_{n,\psi+\pi,-D}(\boldsymbol{k}^\prime)=E_{n,\psi+\pi,D}(\boldsymbol{k}^\prime)$, the two separated regions of the phase $(-1,0,1)$ can be related to each other by $\psi+\pi$ and have exactly the same shape. However, we have checked that the sign change of $D$ affects the band shape slightly and breaks the simultaneity of the related phase transitions. Consequently, we only have similar but not the same shape of the two regions of the phase $(-1,0,1)$. Now the same Chern numbers of the two regions can also be understood easily. The mirror reflection about a constant energy axis does not change the Chern numbers of the bands, however, it changes the order of the bands. If the bands $E_{n,\psi,D}(\boldsymbol{k}^\prime)$ carry Chern numbers $(-1,0,1)$, then the mirror-reflected bands $E_{n,\psi+\pi,-D}(\boldsymbol{k}^\prime)$ will carry Chern numbers $(1,0,-1)$. However, the sign change of $D$ changes the sign of the Chern number of every band. Thus the bands $E_{n,\psi+\pi,D}(\boldsymbol{k}^\prime)$ carry Chern numbers $(-1,0,1)$, which are the same as those of the bands $E_{n,\psi,D}(\boldsymbol{k}^\prime)$. To gain a deeper insight into the topological property, we calculate the Berry curvatures of the magnon bands for $\psi=1.57\pi$ and $\psi=2\pi$ both with $D=0.7$, which are in the phases $(-1,0,1)$ and $(1,0,-1)$, respectively, as shown in Fig. 2(b). As for $\psi=1.57\pi$ with $D=0.7$, the Berry curvature of the first band is negative all over the Brillouin zone, and the integral of it gives Chern number $-1$. While for the third band, its Berry curvature is always positive and gives Chern number $1$. For $\psi=2\pi$ with $D=0.7$, the situation is reversed. The sign of the Berry curvature of the second band is momentum dependent for both cases, and they are not shown here.

Fig. 3. Upper: magnon bands for (a) $\psi=1.57\pi$, (b) $\psi=1.726\pi$, and (c) $\psi=2\pi$, all with $D=0.7$, which are shown in Fig. 2(a) as red points. For the point (b) sitting at the phase boundary, the bands are gapless and there are Dirac points at the $K$ points. Note that the top band in (b) has small dispersion. Lower: the corresponding magnon bands in a strip geometry: (d) $\psi=1.57\pi$, (e) $\psi=1.726\pi$, and (f) $\psi=2\pi$, all with $D=0.7$. There are in-gap edge modes in (d) and (f), and nondispersive edge modes connecting the Dirac points in (e).

In Fig. 3 we show the bulk magnon bands and their corresponding band structures in a strip geometry for $\psi=1.57\pi$, $\psi=1.726\pi$, and $\psi=2\pi$, all with $D=0.7$, which are given in Fig. 2(a) as red points. For the point sitting at the phase boundary, the band structure is nearly the same as that of the Heisenberg kagome ferromagnet without DMI, the bands are gapless and there are Dirac points at the $K$ points,[20] except that here the top band has small dispersion. Due to the bulk-edge correspondence,[32] there are in-gap edge modes in the gapped band structures and nondispersive edge modes connecting the Dirac points. Transverse Thermal Hall Conductivity with Sign Change. The momentum-independent sign structure of the Berry curvatures will lead to the sign change of the magnon thermal Hall conductivity, as we will show in the following. The transverse thermal Hall conductivity of magnon can be calculated by[33-35] \begin{align} \kappa_{x^{\prime}y^{\prime}}=\frac{k_{\scriptscriptstyle{\rm B}}^2\,T}{(2\pi)^2\hbar}\sum_n\int_{_{\scriptstyle \rm BZ}}c_2(\rho_n)B_{k_x^{\prime}k_y^{\prime}}^n{d}k_x^{\prime}{d}k_y^{\prime}, \tag {4} \end{align} with the sum running over the three bands and the integral over the first Brillouin zone. Here $\rho_n=1/(\exp(E_n/k_{\scriptscriptstyle{\rm B}}T)-1)$ is the Bose distribution with $E_n$ as the $n$th eigenvalue; $c_2$ is given by \begin{align} c_2(\rho_n)=(1+\rho_n)\Big({\ln}\frac{1+\rho_n}{\rho_n}\Big)^2\!-\!({\ln}\rho_n)^2\!-\!2{\rm Li}_2(-\rho_n), \tag {5} \end{align} where ${\rm Li}_2(-\rho_n)$ is the polylogarithm function of order $2$.

Fig. 4. (a) The transverse thermal conductivity as a function of temperature for $\psi=1.57\pi$, $\psi=1.726\pi$, and $\psi=2\pi$, all with $D=0.7$, which are given in Fig. 2(a) as red points. The thermal conductivity are always negative and positive for the phases $(-1,0,1)$ and $(1,0,-1)$, respectively, and it is always zero for the phase boundary. (b) The transverse thermal conductivity as a function of $\psi$ with $D=0.7$ and $T=10$. There are sign changes at the phase boundaries. Here we set $\hbar=k_{\scriptscriptstyle{\rm B}}=1$.

As shown in Figs. 4(a) and 4(b), the thermal Hall conductivities are always negative and positive for the phases $(-1,0,1)$ and $(1,0,-1)$ respectively, with the sign changes happening at the phase boundaries, whose conductivities are always zero. The sign structure of the thermal conductivity stems from the sign structure of the Berry curvatures of the magnon bands. As for the phase $(-1,0,1)$, the negative and positive Berry curvatures of the first and third bands contribute to the Chern numbers $-1$ and 1, respectively. However, in the integral of the thermal conductivity, there is a weight $c_2(\rho_n)\ge0$, which decreases monotonically with energy,[25] thus the negative contribution from the first band is greater than the positive contribution from the third band, which leads to the negative conductivity. For the phase $(1,0,-1)$, the situation is reversed and thus its conductivity is positive. We have checked that the contributions from the second band are nearly zero for both phases. From the physical angle, the lowest magnon band is thermally occupied maximally at low temperatures, and then dominate the thermal conductivity. We note that the phase $(-1,0,1)$ with negative thermal conductivity resides around $\psi=0.6\pi$ and $\psi=1.6\pi$ in the phase diagram, where the Kitaev interaction is relatively large with $|K|\approx0.95$. Moreover, the width of the phase increases with the strength of DMI. These phenomena are helpful for distinguishing the relative strengths of the isotropic Heisenberg interaction, the Kitaev interaction, and the DMI. In summary, we have studied the topological magnon excitations and related thermal Hall conductivity in a kagome magnet exhibiting Heisenberg, Kitaev and DM interactions exposed to a magnetic field. We consider an enough strong field to bring the system into fully polarized paramagnetic phase. It is found that the magnon bands carry nontrivial Chern numbers in the full region of the phase diagram. Furthermore, there are phase transitions related to two topological phases with opposite Chern numbers, which lead to the sign changes of the thermal Hall conductivity. In the phase with negative thermal conductivity, the Kitaev interaction is relatively large and the width of the phase increases with the strength of DMI. Since the effects of magnon-magnon interactions are suppressed by the exchange scale divided by the applied field strength, the paramagnetic phase here is suitable for the exploration of related physics. Thus, we believe that the present study will contribute to understanding of related compounds. Acknowledgments. We thank Changle Liu for helpful discussion. This work was supported by the National Natural Science Foundation of China (Grant No. 12104407), and the Natural Science Foundation of Zhejiang Province (Grant No. LQ20A040004). References Phase Diagram and Quantum Order by Disorder in the Kitaev

K_{1} - K_{2}

Honeycomb MagnetHeisenberg-Kitaev model on the hyperhoneycomb latticeVaporization of Kitaev Spin LiquidsHyperhoneycomb Iridate

β − {Li}_{2} {IrO}_{3}

as a Platform for Kitaev MagnetismTopological spin liquids in the ruby lattice with anisotropic Kitaev interactionsInfinite projected entangled-pair state algorithm for ruby and triangle-honeycomb latticesGround state phase diagram of the Kitaev-Heisenberg model on a honeycomb-triangular latticeKitaev-Heisenberg models for iridates on the triangular, hyperkagome, kagome, fcc, and pyrochlore latticesA thermodynamic theory of “weak” ferromagnetism of antiferromagneticsAnisotropic Superexchange Interaction and Weak FerromagnetismThermal Hall signatures of non-Kitaev spin liquids in honeycomb Kitaev materialsInterplay of Dzyaloshinskii-Moriya and Kitaev interactions for magnonic properties of Heisenberg-Kitaev honeycomb ferromagnetsTopological excitations in the ferromagnetic Kitaev-Heisenberg modelSign Structure of Thermal Hall Conductivity and Topological Magnons for In-Plane Field Polarized Kitaev MagnetsTopological magnons in Kitaev magnets at high fieldsThermal Hall conductivity with sign change in the Heisenberg-Kitaev kagome magnetTheory of the Thermal Hall Effect in Quantum MagnetsObservation of the Magnon Hall EffectThermal Hall Effect of Spin Excitations in a Kagome MagnetTopological Magnon Bands in a Kagome Lattice FerromagnetA first theoretical realization of honeycomb topological magnon insulatorTopological magnon insulator in insulating ferromagnetDirac and Nodal Line Magnons in Three-Dimensional AntiferromagnetsWeyl and Nodal Ring Magnons in Three-Dimensional Honeycomb LatticesMagnon Hall effect on the Lieb latticeGround-state phase diagram of the Kitaev-Heisenberg model on a kagome latticeQuantum-classical crossover in the spin-

\frac{1}{2}

Heisenberg-Kitaev kagome magnetField Dependence of the Intrinsic Domain Magnetization of a FerromagnetColloquium : Topological insulatorsTopological insulators and superconductorsBerry phase effects on electronic propertiesChern number and edge states in the integer quantum Hall effectTheoretical Prediction of a Rotating Magnon Wave Packet in FerromagnetsRotational motion of magnons and the thermal Hall effectThermal Hall effect of magnons in magnets with dipolar interaction

[1]	Rousochatzakis I, Reuther J, Thomale R, Rachel S, and Perkins N B 2015 Phys. Rev. X 5 041035
[2]	Lee E K H, Schaffer R, Bhattacharjee S, and Kim Y B 2014 Phys. Rev. B 89 045117
[3]	Nasu J, Udagawa M, and Motome Y 2014 Phys. Rev. Lett. 113 197205
[4]	Takayama T, Kato A, Dinnebier R, Nuss J, Kono H, Veiga L S I, Fabbris G, Haskel D, and Takagi H 2015 Phys. Rev. Lett. 114 077202
[5]	Jahromi S S, Kargarian M, Masoudi S F, and Langari A 2016 Phys. Rev. B 94 125145
[6]	Jahromi S S, Orús R, Kargarian M, and Langari A 2018 Phys. Rev. B 97 115161
[7]	Kishimoto M, Morita K, Matsubayashi Y, Sota S, Yunoki S, and Tohyama T 2018 Phys. Rev. B 98 054411
[8]	Kimchi I and Vishwanath A 2014 Phys. Rev. B 89 014414
[9]	Dzyaloshinsky I 1958 J. Phys. Chem. Solids 4 241
[10]	Moriya T 1960 Phys. Rev. 120 91
[11]	Gao Y H, Hickey C, Xiang T, Trebst S, and Chen G 2019 Phys. Rev. Res. 1 013014
[12]	Zhang L C, Zhu F, Go D, Lux F R, dos S F J, Lounis S, Su Y, Blügel S, and Mokrousov Y 2021 Phys. Rev. B 103 134414
[13]	Joshi D G 2018 Phys. Rev. B 98 060405
[14]	Chern L E, Zhang E Z, and Kim Y B 2021 Phys. Rev. Lett. 126 147201
[15]	McClarty P A, Dong X Y, Gohlke M, Rau J G, Pollmann F, Moessner R, and Penc K 2018 Phys. Rev. B 98 060404
[16]	Li K 2022 arXiv:2209.04133 [cond-mat.str-el]
[17]	Katsura H, Nagaosa N, and Lee P A 2010 Phys. Rev. Lett. 104 066403
[18]	Onose Y, Ideue T, Katsura H, Shiomi Y, Nagaosa N, and Tokura Y 2010 Science 329 297
[19]	Hirschberger M, Chisnell R, Lee Y S, and Ong N P 2015 Phys. Rev. Lett. 115 106603
[20]	Chisnell R, Helton J S, Freedman D E, Singh D K, Bewley R I, Nocera D G, and Lee Y S 2015 Phys. Rev. Lett. 115 147201
[21]	Owerre S A 2016 J. Phys.: Condens. Matter 28 386001
[22]	Zhang L F, Ren J, Wang J S, and Li B W 2013 Phys. Rev. B 87 144101
[23]	Li K K, Li C Y, Hu J P, Li Y, and Fang C 2017 Phys. Rev. Lett. 119 247202
[24]	Li K K and Hu J P 2017 Chin. Phys. Lett. 34 077501
[25]	Cao X D, Chen K, and He D H 2015 J. Phys.: Condens. Matter 27 166003
[26]	Morita K, Kishimoto M, and Tohyama T 2018 Phys. Rev. B 98 134437
[27]	Yang Y, Perkins N B, Koç F, Lin C H, and Rousochatzakis I 2020 Phys. Rev. Res. 2 033217
[28]	Holstein T and Primakoff H 1940 Phys. Rev. 58 1098
[29]	Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[30]	Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[31]	Xiao D, Chang M C, and Niu Q 2010 Rev. Mod. Phys. 82 1959
[32]	Hatsugai Y 1993 Phys. Rev. Lett. 71 3697
[33]	Matsumoto R and Murakami S 2011 Phys. Rev. Lett. 106 197202
[34]	Matsumoto R and Murakami S 2011 Phys. Rev. B 84 184406
[35]	Matsumoto R, Shindou R, and Murakami S 2014 Phys. Rev. B 89 054420