Topological thermal Hall effect and magnonic edge states in kagome ferromagnets with bond anisotropy

The magnon band topology due to the Dzyaloshinskii–Moriya interaction (DMI) and its relevant topological thermal Hall effect has been extensively studied in kagome lattice magnets. In this theoretical investigation, we report a new mechanism for phase transitions between topological phases of magnons in kagome ferromagnets by tuning the anisotropic nearest-neighbor ferromagnetic interaction and DMI. Using the linear spin-wave theory, we calculate the Chern number and thermal Hall conductivity of magnons in low temperature regime. We show the magnon band structures and magnonic edge states in each topological phase. From the topological phase diagram, we find a sign reversal of the thermal Hall conductivity upon tuning the modulation factors. We explicitly demonstrate the correspondence of thermal Hall conductivity with the propagation direction of the magnonic edge states. Finally, we discuss candidate materials as experimental realizations of our theoretical model.


Introduction
Magnonics, which aims to investigate the properties and control the transport of magnons in magnetic insulators, has attracted great interests in modern condensed matter physics [1][2][3]. Due to their nanometer wavelength and low energy consumption, magnons are considered as promising information carriers with great potential for data processing devices in the near future [4][5][6].
Although a magnon is a charge-neutral quasiparticle immune to the Lorentz force usually experienced by charged particles, it experiences an effective magnetic field in momentum space induced by DMI [59,60]. The magnon thermal Hall effect (THE) is characterized by a transverse thermal conductivity in response to a longitudinal temperature gradient, which has been realized experimentally in the pyrochlore ferromagnets [61,62] and the kagome ferromagnets [46,47]. The magnon THE was firstly explained as a result of noncompensated magnon edge currents by Matsumoto and Murakami [59]. Subsequently, studies have found that the net edge currents are closely related to the nontrivial magnonic edge states, and the sign of the thermal Hall conductivity is associated with the propagation direction of edge states [22,63]. The unique nature of magnon THE has inspired lots of discoveries in phase transitions among magnonic topological phases. Heretofore the topological phase transitions for magnonic kagome lattices have been reported under a varying magnetic field [49,50], magnon-phonon coupling [64][65][66] and next-nearest-neighbor (NNN) exchange interactions [67]. Recently, topological magnons in XXZ kagome lattice antiferromagnets have been widely investigated [32,68,69]. When generalized into a kagome lattice, three different nearest-neighbor (NN) hopping parameters could form an alternating pattern, giving rise to a rich variety of magnonic topological phases in two dimensions. From this point of view, it is tempting to ask whether a magnonic analog of the topological insulator exists and, more importantly, what non-trivial transverse transport properties of magnons are shown in such a system.
In this paper, we study a kagome ferromagnet with anisotropic NN ferromagnetic interaction and DMI as illustrated in figure 1. By varying the modulation factors η 1 , η 2 and η 3 , we find five different magnonic topological phases and plot the topological phase diagram in η 2 -η 3 parameter plane. We moreover show the topological phase transition characterized by the Chern numbers of magnon bands, which can be detected by the change of magnon thermal Hall conductivity. We also demonstrate the correspondence between the thermal Hall conductivity and the propagation direction of the nontrivial edge states. Furthermore, we find sign changes of the thermal Hall conductivity coupled with the topological phase transition. The studies here will be useful for the future design of magnon-based devices, since the thermal Hall conductivity can be easily tuned via modulation factors.
The paper is organized as follows. In section 2 we introduce our theoretical model and method. Detailed numerical results are presented in section 3, including the topological phase diagram, topological magnon band structures, magnonic edge modes, the thermal Hall conductivity. Finally, we end the paper with a brief discussion on the candidate materials and a summary in section 4.

Spin model and topological magnons
Let us consider a collinear ferromagnet on a kagome lattice in the xy plane as schematically shown in figure 1. Its Hamiltonian reads where S i is the vector of spin operators at site i. The first two terms describe the NN and NNN ferromagnetic interactions (J ij , J 2 > 0), respectively. The third term is the out-of-plane NN DMI, where D ij is the DMI strength and ij = ±1 depends on the chirality of the triangles in the kagome lattice. Specifically, the NN interactions J ij and D ij are modulated around its equilibrium value J 1 and D with different coefficients η i depending on the bond direction as shown in figure 1: J ij D ij = η 1 J 1 (D) along α 1 , J ij D ij = η 2 J 1 (D) along α 2 and J ij D ij = η 3 J 1 (D) along α 3 directions, respectively. The anisotropic NN interactions can be realized by lattice distortion induced by a controllable voltage or mechanical strain.
Notice that such lattice deformations should change not only the NN exchange interactions but also the NNN ferromagnetic interactions. As discussed in the appendix A, the anisotropy of NNN ferromagnetic interactions does not produce new nontrivial band topology, introducing only restricted modifications to the band dispersion and distribution of Berry curvature. So for simplicity, we will treat the NNN ferromagnetic interactions as isotropic throughout this paper. In addition, the simultaneous changes of NN ferromagnetic interaction and DMI will preserve the collinear ferromagnetic ground state, because the DMI is well below the threshold value for a new spin configuration formation.
In the following, we apply a linear spin-wave theory to obtain the tight-binding magnon Hamiltonian in the momentum space. We express the spin operators in equation (1) in terms of magnon creation operator b † i and annihilation operatorb i by using the Holstein-Primakoff transformations [70]: We neglect the higher-order terms in the magnon operators from the magnon-magnon interactions, as we consider the low temperature regime here. After a Fourier transformation, equation (1) becomes respectively. The linking vectors α i and β i are illustrated in figure 1.
In contrast to electronic systems, the absence of Fermi surface in bosonic systems leads to the ill-defined Chern number of magnons. However, the Chern number of magnons for the nth magnonic bulk band can still be given in a similar way, and the Berry curvature of magnons Ω z nk is defined as where ψ nk and ε nk are the eigenvectors and eigenvalues of H (k) for the nth band, respectively. Another important quantity in topological systems is the winding number ν m in band gap m, which is given by the sum of Chern numbers up to the mth band [71], The winding number reflects the surface properties of the system, that is to say, we can find |ν m | pairs of topologically nontrivial edge states in the mth band gap and their propagation direction is determined by the sign of ν m . The topological nature of magnonic edge states can be probed by measuring of the magnon THE in magnonic systems. The magnon thermal Hall conductivity is calculated using the following formula [59,60] where The weighting function is given by , with Li 2 (x) being the polylogarithm function.

Results
For the sake of simplicity, the Hamiltonian equation (1)  In figures 3(a)-(e), we plot the distribution of Berry curvature Ω z nk associated with the lower magnon band for five selected sets of modulation parameters {η 2 , η 3 }, that drive the system into five different topological phases. It is remarkable that the Berry curvature exhibits a C 2 rotational symmetry in figures 3(a)-(d) due to the anisotropic NN interactions, instead of a C 6 rotational symmetry when η 1 = η 2 = η 3 shown in figure 3(e). The summation of Ω z nk over the first Brillouin zone gives the corresponding Chern number of the lower band C 1 . Then, we turn our attention to the exploration of the band topology in kagome ferromagnets. Figure 4 shows the magnon bulk bands along the high-symmetry directions (Γ-K-M-Γ) of the Brillouin zone (figure 3(f)) in each topological phase. Three well separated magnon bands, i.e. two well defined band gaps, can be found in all phases. The upper band is close to the middle band at M point with increasing η 3 when η 2 = 2 as shown in figures 4(b)-(e). We also note that the Berry curvature exhibits higher densities when the band gap is shifted towards closing, for example, the peaks of Berry curvature along Γ-K line and near K point in figures 3(c) and (e).
To better visualize the magnonic edge states, we solve the eigenvalue problem of a nanoribbon geometry with open boundary conditions, and then plot the band structures in figure 5. The winding numbers in  phase II are ν 1 = 1 and ν 2 = 0, so two crossed chiral edge states are clearly found in the lower band gap and two gapped trivial edge states are found in the upper band gap, as shown in figure 5(a). In phase III, as the two winding numbers are the same, ν 1 = ν 2 = 1, two equivalent pairs of nontrivial edge states emerge in each band gap in figure 5(b). As η 3 is further increasing, the nonzero winding numbers in phase IV are ν 1 = −1 and ν 2 = 1, and we can find two pairs of nontrivial edge states in the two band gaps in figure 5(c).

Topological thermal Hall effect
In this section, we discuss the topological nature of magnon THE. In magnonic systems, the topological phases and phase transitions can be detected by the measurement of magnon dispersion using inelastic neutron scattering [47]. In figure 6, the magnon thermal Hall conductivity κ xy in low temperature regime is plotted as a function of temperature with different modulation factors {η 2 , η 3 } in each topological phases, which all show a monotonically increasing behavior. We also find κ xy < 0 in phases I, II and III, conversely, κ xy > 0 in phases IV and V.
The thermal Hall conductivity κ xy as a function of η 3 when η 2 = 2 for four different temperatures are depicted in figure 7. The system undergoes the phase transition from phase II to phase V, as η 3 increases from 0 to 3. In addition, we find a sign reversal of κ xy when the topological phase transits between phases III and IV, which can be explained by connecting the propagation directions of the nontrivial edge states. From equations (4) and (6), we see that C n and κ xy only have different weighting functions in the integrand. Hence, the dominant contribution to the thermal Hall conductivity at low temperatures comes from the lowest band, since the lowest band is more occupied due to the c 2 function within the bosonic statistics. As a consequence, the sign of κ xy is governed by the sign of ν 1 in low temperature regime as ν 1 = C 1 . As we mentioned above, the propagation directions of edge states in the lower gap are opposite between the topological phases III and IV. Finally, the sign reversal of κ xy is accompanied by the sign reversal of ν 1 , even though the two pairs of nontrivial edge states also have opposite propagation directions in phase IV.

Discussion and summary
In this paper, a tunable topological magnon excitation in kagome lattice ferromagnets has been predicted. We shall further briefly comment on the candidate materials for experimental realizations of the above theoretical model. Recent experiments show that the materials Cu(1-3,bdc) [46,47], as well as the mineral haydeeite-α-MgCu 3 (OD) 6 Cl 2 [48], exhibit a stable ferromagnetic phase down to the monolayer limit. In these materials, magnetic ions are arranged on a kagome lattice with their magnetization pointing perpendicular to the plane. Meanwhile, the variations of the NN interactions can be easily achieved from the lattice distortions induced by applying external perturbations, such as mechanical strain or pressure, because the interactions depend sensitively on the inter-atomic distances [73][74][75][76]. The different types of strains studied in the present work can be achieved and tuned by applying a uniaxial mechanical strain field, either along different crystallographic directions. For instance, a uniaxial strain along α 1 direction shall result in η 2 = η 3 = η 1 . Using the parameters in reference [48], we can find a thermal Hall conductivity of −3.2 × 10 −3 W Km −1 with isotropic NN interaction (η 2 = η 3 = η 1 ) and 3.2 × 10 −3 W Km −1 with anisotropic NN interaction (η 2 = η 3 = 2η 1 ) at the temperature T = 50 K. Tuning the anisotropy parameters over a wide range to cover the diagram reported on figure 2 remains a technical challenge that can be achieved by depositing the kagome ferromagnet on a piezoelectric substrate for instance [77,78]. Detailed first-principles calculations are needed to identify suitable interfaces [79].
Thus far, we have studied the magnon Hall effect in a picture of magnons within the bosonic description. As mentioned in section 2, we truncate the bosonic Hamiltonian to quadratic order by  neglecting the magnon-magnon interaction in the low temperature regime. However, by using the Holstein-Primakoff transformations, it is unclear whether the quartic terms for the interaction between magnons could be ignored. In this perspective, another theoretical formalism of the THE was developed in the language of spin operators instead of boson operators [80]. In this formalism, the THE in a correlated paramagnetic phase can also be captured with reductions to the Schwinger-boson representation of spin operators by using self-consistent mean-field theory. Remarkably, the thermal Hall coefficients obtained from the linear-response theory for noninteracting models [59] and the mean-field Hamiltonian for the interacting spin model are identical. On the other hand, as low-energy charge-neutral excitations, phonons can also carry angular momenta [81][82][83] and induce a transverse phonon angular momentum flows at finite temperatures, the so-called phonon THE [14]. However, the phonon angular momenta in a phonon Hall current can only create negligible accumulation at the crystal edges in comparison with the magnon spin moment accumulation in a magnon Hall current [84,85]. In addition, since the magnon-phonon interaction may significantly modify the magnon-mediated THE due to symmetry breaking by an in-plane DMI and the magnetoelastic coupling [64,86,87], its role on the nontrivial topology of magnon excitation would be an interesting topic left to future studies [87]. Since magnon-magnon interactions and magnon-phonon interactions need to be taken into consideration at high temperatures, the linear spin-wave theory becomes inaccurate in this limit. However, self-consistent Holstein-Primakoff and Schwinger-boson methods within Hartree-Fock mean-field approximation will be applicable to computation of the thermal Hall response functions [80,88], which goes beyond the scope of this paper.
In summary, we have shown that tuning the relative strength of anisotropic NN ferromagnetic interaction and DMI in kagome ferromagnets gives rise to the topological phase transition between different topological phases characterized by the Chern numbers of magnon bands. We describe the band topology of magnons with the magnonic edge states in the band gaps. Furthermore, we have calculated the thermal Hall conductivity κ xy and expound its relevance to the topological invariant ν m . We find that κ xy can easily be controlled by tuning the modulation factors. In addition, a sign reversal of κ xy was found along with topological phase transitions. In this regard, the present studies open thrilling perspectives for experimentalists and can be applied in the field of thermal micro-sensors in the future.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors. with μ i = J i cos k · β i . Here, all other parameters have the same meanings and values as mentioned in the main text. Figure 8 shows the topological phase diagram as a function of η 2 and η 3 when taking anisotropic NNN ferromagnetic interactions into consideration. In comparison with the topological phase diagram in figure 2, both phase diagrams are divided into five identical phases listed in the main text. In other words, there's no other nontrivial topological phase creating. In addition to that, the anisotropic NNN ferromagnetic interactions redistribute the region of each phase, as the specific magnon band dispersions and distributions of Berry curvature are modified by the difference between the NNN ferromagnetic interactions.