Band Topology in Quantum Magnets: From nontrivial excitations to non-Hermitian topology and spintronics

We demonstrate that interactions can substantially alter the free-band description of magnons in ferromagnets on geometrically frustrated lattices. The anharmonic coupling facilitated by the Dzyaloshinsky-Moria interaction and a highly-degenerate structure of the two-particle continuum induce a non-perturbative damping of the high-energy magnon modes. We provide a detailed account of the effect of the interaction-induced magnon damping for the S=1/2 ferromagnet on the kagome lattice and propose further experiments.

Recently, there has been a growing interest in studying magnetic systems with nontrivial magnon band topology. However, examples discussed so far have been mostly limited to systems which can be thought of as a magnonic analog of integer quantum Hall systems. In this talk, I will talk about a class of magnonic systems that are the natural counterparts of time-reversal symmetric topological insulators in two and three dimensions [1,2]. The feature of these systems is that each pair of bands related by pseudo-time-reversal symmetry carries a Z2 topological invariant. I will demonstrate that the Z2 invariant so defined characterizes the presence/absence of helical edge/surface modes. If time permits, I will touch on magnonic analogs of topological crystalline insulators [3]. [1] H. Kondo, Y. Akagi, and H. Katsura, Phys. Rev. B 99, 041110(R) (2019). [2] H. Kondo, Y. Akagi, and H. Katsura, Phys. Rev. B 100, 144401 (2019). [3] H. Kondo and Y. Akagi, Phys. Rev. Lett. 127, 177201 (2021).

When magnons propagate across a non-collinear magnetic texture, they collect a Berry phase that can be interpreted as the flux of a synthetic orbital magnetic field. This Berry flux is related to the topological skyrmion density of the magnetic texture. For a topologically non-trivial skyrmion configuration, the magnon thus experiences a synthetic Lorentz force that bends the quasi-classical trajectories of the magnons. For a skyrmion lattice, the average synthetic field is finite such that the magnons are confined to cyclotron orbits of the corresponding magnon Landau levels. This is reflected in a magnon band structure with finite Chern numbers. Here, we report on a collaboration between experiment and theory that explored this topologically non-trivial magnon band structure by means of inelastic neutron scattering experiments on the cubic chiral magnet MnSi [1]. For this material, the spacing of the emergent Landau levels can be estimated to be 10 μeV on average. The resolution of such a small energy scale poses a challenge to state-of-the-art neutron spectroscopy, and, for this reason, three distinct scattering methods were applied: time-of-flight, polarized triple-axis, and neutron spin-echo spectroscopy. The combined data is quantitatively compared to theoretical predictions of the spin structure factor taking into account the resolution of the various instruments. We find excellent agreement of the whole data set with distinct contributions of transversal and longitudinal spin fluctuations. It is found that the emergent Landau levels exhibit a parabolic superstructure that is also apparent in the spectral weight distribution. We attribute this observation to a spatial variation of the local synthetic field whose standard deviation exceeds its average by a factor of five [2]. This gives rise to spatial regions of positive and negative synthetic fields such that the systems in fact resembles more a Chern insulator than a Hall insulator. This gives rise to run-away orbits of the quasi-classical magnon dynamics that are at the origin of the parabolic superstructure of the Landau levels. [1] T. Weber, D. M. Fobe, J. Waizner, P. Steffens, G. S. Tucker, M. Böhm, L. Beddrich, C. Franz, H. Gabold, R. Bewley, D. Voneshen, M. Skoulatos, R. Georgii, G. Ehlers, A. Bauer, C. Pfleiderer, P. Böni, M. Janoschek, M. Garst, Science 375, 1025 (2022). [2] Benjamin Wolba, PhD thesis, KIT.

The great potential of non-trivial topology for magnonics will be pointed out [1]. It will be illustrated in a ferromagnetic Shastry-Sutherland lattice [2]. For antiferromagnets, the necessary generalization of the scalar product of bosonic creation and annihilation operators to a symplectic product is pointed out. In a quasi one-dimensional model without long-range order suitable for BiCu$_2$PO$_6$ the occurrence of a non-trivial Zak-phase is shown [3]. In spite of this phase no edge states are found. We discuss to which extent this finding puts the bulk-boundary correspondence into perspective [4]. [1] M. Malki, G.S. Uhrig, Topological magnetic excitations, EPL Perspective, 132, 20003 (2020) [2] M. Malki, G.S. Uhrig, Topological magnon bands for magnonics, Phys. Rev. B 99, 174412 (2019) [3] M. Malki, L. Müller, G.S. Uhrig, Absence of localized edge modes in spite of a non-trivial Zak phase in BiCu$_2$PO$_6$, Phys. Rev. Res. 1, 033197 (2019) [4] M. Malki, G.S. Uhrig, Delocalization of edge states in topological phases, EPL 127, 27001 (2019)

The theory of Topological Quantum Chemistry (TQC) has been successful at diagnosing electronic band topology using only the symmetry group of the system. We begin extending TQC to low-dimensional, magnetic systems in two ways. First, we derive the elementary band (co-)representations of magnetic line groups derived from the first and fifth families of ordinary line groups. The (magnetic) line groups are frequently overlooked, despite being the symmetry groups corresponding to quasi-1D systems, which are of fundamental importance due to their use as illustrative models as well as being building blocks of higher-dimensional systems. Then we begin an exploration of quasi-2D systems described by the (magnetic) diperiodic groups. A well-known example of a system with such symmetries is the archetypal Haldane model. The validity of the TQC approach for bosonic systems is briefly explored.

There have been considerable advances in the fabrication of atomically thin magnetic layers potentially opening new experimental avenues in truly two-dimensional quantum magnetism. However, probing single-layer magnetism remains significantly challenging. We propose exploiting the reduced dimensionality and the presence of magnetic impurities to carry out a magnetic analogue of Quasi-Particle Interference (QPI) to obtain information about the magnon dispersion relations. We also consider the problem of probing Chern magnon bands in 2D magnets showing that tuneable impurities close to the boundary may allow one to probe the presence of the surface states in such systems. Authors: Asimpunya Mitra (IIT Kharagpur, India), Alberto Corticelli (MPI-PKS, Germany), Pedro Ribeiro (IST Lisbon, Portugal) and Paul A. McClarty (MPI-PKS, Germany). Reference: Magnon Interference Tunneling Spectroscopy as a Probe of 2D Magnetism (arXiv:2110.02662)

In this talk, I will discuss multiple possible phases in magnetic (electronic) band structure topology that go beyond symmetry indicators. I will in particular elucidate magnetic fragile topology, and how this depends on various magnetic orderings. This will reveal a magnetic correspondence between fragile phases and Chern phases. I will then discuss the extension to 3D and show how such phases can be hidden in band structures that look globally trivial - the so-called subdimensional topology. These results will be supplemented by edge and hinge spectra, and a discussion of bulk-boundary correspondences. I will also briefly comment on possible future directions and relations to magnon topology. This talk is based on two recent papers, (Physical Review B 103 (24), 245127 and Physical Review B 103 (19), 195145) which were co-authored with Dr. Adrien Bouhon and Dr. Robert-Jan Slager at the University of Cambridge.

We discuss various examples of topological magnons in quantum magnets on honeycomb lattice. These include topological magnons in generalized Kitaev models on two-dimensional honeycomb lattice and the corresponding thermal Hall transport, higher order magnon topology and hinge states in stacked honeycomb magnets, and Weyl/nodal magnons in Kitaev magnets on three-dimensional hyperhoneycomb lattice.

Topological magnets support magnetic excitations with a topologically nontrivial spectrum. As a result, they exhibit chiral edge states akin to those known from the quantum Hall effect. These edge states are envisioned to facilitate backscattering-free information channels for magnetic signals [1]. Since spin excitations do not carry charge, they do not suffer from Joule heating and allow for ultra-low energy computation. However, in contrast to electrons, there is no conservation law for spin excitations. This gives rise to particle number-nonconserving many-body interactions the influence of which on quasiparticle topology is an open issue of fundamental interest in the field of topological quantum materials. Herein, I discuss several aspects of quantum many-body effects caused by particle-number nonconservation. These include (i) the quantum damping due to spontaneous quasiparticle decay [2], (ii) interaction-stabilized topological gaps in the single-particle spectrum [3], and (iii) a topological hybridization of single-particle and few-particle states [4]. [1] Alexander Mook, Sebastián A. Díaz, Jelena Klinovaja, and Daniel Loss, "Chiral Hinge Magnons in Second-Order Topological Magnon Insulators," Phys. Rev. B 104, 024406 (2021) [2] Alexander Mook, Jelena Klinovaja, and Daniel Loss, "Quantum damping of skyrmion crystal eigenmodes due to spontaneous quasiparticle decay," Phys. Rev. Research 2, 033491 (2020) [3] Alexander Mook, Kirill Plekhanov, Jelena Klinovaja, and Daniel Loss, "Interaction-Stabilized Topological Magnon Insulator in Ferromagnets," Phys. Rev. X 11, 021061 (2021) [4] Alexander Mook, Rhea Hoyer, Jelena Klinovaja, Daniel Loss, "Topological Hybrids of Magnons and Magnon Bound Pairs," arXiv:2203.12374

Strain applied to a condensed-matter system can be used to engineer its excitation spectrum via artificial gauge fields, or it may tune the system through transitions between different phases. In the talk I will describe different settings where inhomogeneous strain is applied to local-moment magnets. For a classical spin liquid, the application of strain lifts the extensive degeneracies, and a sequence of novel phases may emerge, which I will illustrate for the Heisenberg model on the kagome lattice. In cases where the magnetic ground state is stable against small deformations, strain will mainly change the excitation spectrum, and I will discuss different types of emergent pseudo-Landau levels and their properties.

Band topology in electronic systems is known to have profound consequences on various observable properties in semi-metals and certain insulators. Insights from this  eld have reached into many areas of physics and here we describe certain uni- versal signatures of band topology in Dirac magnon materials. Here we report high- resolution inelastic neutron scattering measurements of the magnetic dynamics in XY-like CoTiO3 and Ising-like FeTiO3 with stacked honeycomb layers. In both sys- tems we observe a winding of the inelastic neutron scattering intensity in the vicinity of the nodal points. Furthermore, we show that  ne details of the magnon band struc- ture reveal bond dependent exchange couplings and an order-by-disorder mechanism in the XY-like CoTiO3, and we contrast the results with the Ising-like FeTiO3.

We theoretically investigate topological transports of magnon-phonon hybrid excitations in two-dimensional magnetic systems. The magnetoelastic interaction opens band gaps and allows the interband transitions between different excitation states. As a result, the topological transports of the magnon-polarons and their spin angular momenta can occur. When the time-reversal symmetry is broken, the magnon-polarons are topologically nontrivial, possessing finite Berry curvature. However, the magnon-polarons show finite spin Berry curvature with vanishing Berry curvature in the antiferromagnet preserving the time-reversal symmetry. We find that the intrinsic spin Nernst conductivity can be large in the clean antiferromagnets. Our results show that a simple magnet on a square lattice supports topologically nontrivial magnon-polarons their spin angular momenta, generalizing topological excitations in conventional magnetic systems.

We investigate the magnetic excitations of elemental gadolinium (Gd), a hexagonal close-packed ferromagnet with near-perfect exchange isotropy and $S=7/2$ moments. Using inelastic neutron scattering we show that Gd is a Dirac magnon material with nodal lines at K and nodal planes at half integer ℓ. We find an anisotropic intensity winding around the K-point Dirac magnon cone, which is interpreted to indicate Berry phase physics. Using linear spin wave theory calculations, we show the nodal lines, which are protected by inversion and effective time-reversal symmetry, have nontrivial Berry phases and topological surface modes. We also discuss the protection of the nodal plane by a combination of a screw-axis symmetry and the effective time-reversal symmetry, and introduce a $\mathbb{Z}_2$ topological invariant characterizing its presence and effect on the scattering intensity. Our results suggest that the entire class of rare earth hcp ferromagnets present a simple model system for topological magnetism. References: A. Scheie, Pontus Laurell, P. A. McClarty, G. E. Granroth, M. B. Stone, R. Moessner, and S. E. Nagler, Phys. Rev. Lett. 128, 097201 (2022). A. Scheie, Pontus Laurell, P. A. McClarty, G. E. Granroth, M. B. Stone, R. Moessner, and S. E. Nagler, Phys. Rev. B 105, 104402 (2022).

RuCl$_3$ is thought to exhibit a strong Kitaev-type exchange interaction, but whether this interaction produces a spin liquid state with chiral Majorana edge modes is still unclear. We will discuss our new temperature thermal Hall conductivity measurements where we observe a strongly temperature-dependent signal that is inconsistent with the Majorana-based picture reported by other groups. Using the quantitative framework developed for magnon-derived Hall effects, we show that this temperature-dependence can be fit to a bosonic model and that this fit produces values that are consistent with known parameters for RuCl3. Specifically, this fitting model corresponds to a magnon Chern insulator-like state in the system’s spin-polarized phase that was previously predicted.

An ability to control spin currents is important for probing many spin related phenomena in the field of spintronics and for designing logic and memory devices with low dissipation. Spin-orbit torque is an important example in which spin current flows across magnetic interface and helps to control magnetization dynamics. In this talk, I will first discuss transport, Hall-like responses of magnons in antiferromagnetic insulators, ranging from collinear antiferromagnets [1,2] to breathing pyrochlore noncollinear antiferromagnets [3,4]. The theory also applies to noncollinear antiferromagnets, such as kagome, where we predict both the spin Nernst response [3] and the generation of nonequilibrium spin polarization [4] by temperature gradients, the latter effect constitutes the magnonic analogue of the Edelstein effect of electrons. I will further discuss the spin superfluid transport associated with collective modes in magnetic insulators [5,6]. We observe that in two dimensional systems at finite temperatures spin superfluidity is affected by the presence of topological defects. We further propose to use the Hall response of topological defects, such as merons and antimerons, to spin currents in 2D magnetic insulator with in-plane anisotropy for identification of the Berezinskii-Kosterlitz-Thouless (BKT) transition in a transistor-like geometry. Our numerical results relying on a combination of Monte Carlo and spin dynamics simulations show transition from spin superfluidity to conventional spin transport, accompanied by the universal jump of the spin stiffness and exponential growth of the transverse vorticity current. We propose a superfluid spin transistor in which the spin and vorticity currents are modulated by tuning the in-plane magnet across BKT transition, e.g., by changing the exchange interaction, magnetic anisotropy, or temperature [7]. [1] V. Zyuzin, A.A. Kovalev, Phys. Rev. Lett. 117, 217203 (2016). [2] B. Li, A.A. Kovalev. Phys. Rev. Lett. 125, 257201 (2020) [3] B. Li, S. Sandhoefner, A.A. Kovalev, Phys. Rev. Research 2, 013079 (2020) [4] B. Li, A. Mook, A. Raeliarijaona, A.A. Kovalev, Phys. Rev. B 101, 024427 (2020) [5] G. G. Baez Flores, A.A. Kovalev, M. van Schilfgaarde, K. D. Belashchenko, Phys. Rev. B 101, 224405 (2020) [6] B. Li, A.A. Kovalev, Phys. Rev. B 103, 060406 (2021) [7] E. Schwartz, B. Li, A.A. Kovalev, arXiv:2112.14241v1

“Electric monopole” is an exotic quantum excitation in three-dimensional U(1) spin liquid, and its emergence is purely of quantum origin and has no classical analog. We predict topological thermal Hall effect (TTHE) of “electric monopoles” and present this prediction through non-Kramers doublets on a pyrochlore lattice. We observe that when the external magnetic field polarizes the Ising component of the local moment, internally this corresponds to the induction of emergent dual U(1) gauge flux for the “electric monopoles.” The motion of “electric monopoles” is then twisted by the induced dual U(1) gauge flux. This emergent Lorentz force on “electric monopoles” is the fundamental origin of TTHE. Therefore, TTHE would be a direct evidence of the “monopole” gauge coupling and the emergent U(1) gauge structure in pyrochlore U(1) spin liquid. Our result does not depend strongly on our choice of non-Kramers doublets for our presentation and can be well extended to Kramers doublets. Our prediction can be readily tested among the pyrochlore spin liquid candidate materials. We give a discussion about the expectation for different pyrochlore magnets.

The surface of a topological insulator hosts Dirac electronic states with the spin-momentum locking, which constrains spin orientation perpendicular to electron momentum. As a result, collective plasma excitations in the interacting Dirac liquid manifest themselves as coupled charge- and spin-waves. Here we demonstrate that the presence of the spin component enables effective coupling between plasma waves and spin waves at interfaces between the surface of a topological insulator and insulating magnet. Moreover, the helical nature of spin-momentum locking textures provides the phase winding in the coupling between the spin and plasma waves that makes the spectrum of hybridized spin-plasma modes to be topologically nontrivial. We also show that such topological modes lead to a large thermal Hall response.

In certain magnetic systems, magnon bands can exhibit topologically nontrivial phases, which are often referred to as magnonic topological insulators due to their similarities to electronic topological insulators. The conventional methods to manipulate the magnon-band topology have hitherto required the bulk manipulation of the systems via, e.g., the application of an external magnetic field. In this work, we lift the previous limitation of the magnon topology control on the bulk manipulation by showing that the magnon-band topology can be controlled from the boundary by injecting a spin chirality from the boundary into the bulk, which can be realized, e.g., by utilizing the proximate heavy metal and the resultant interfacial spin Hall effects. More specifically, we propose a scheme to manipulate magnonic topological phases of ferromagnets and antiferromagnets on a honeycomb lattice via chirality injection from the boundaries of the sample [1]. We find that the magnon Hamiltonian subjected to the spin chirality injection possesses a sublattice-symmetry-breaking mass term resulting from the coupling between the spin chirality and the Dzyaloshinskii-Moriya interaction on top of the previously studied bosonic counterpart of the Haldane model [2, 3]. This chirality-induced mass term competes with the known Haldane mass term, which allows us to nonlocally tune the bulk topology from nontrivial to trivial and vice versa by the boundary manipulation. For experimental probes of the chirality-induced topological phase transition, we propose the magnon thermal Hall effect arising from the band topology. The “shoulders” in the thermal Hall conductivity can herald the presence of the proposed phase boundary. Our work leads us to envision that the interfacial chirality injection may offer a nonintrusive means to manipulate the static and the dynamics bulk properties of general magnetic systems. [1] G. Go, S. Lee, and S. K. Kim, "Generation of Nonreciprocity of Gapless Spin Waves by Chirality Injection," arXiv:2110.01867 [2] S. K. Kim, H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, “Realization of the Haldane-Kane-Mele Model in a System of Localized Spins,” Phys. Rev. Lett. 117, 227201 (2016) [3] S. A. Owerre, “A first theoretical realization of honeycomb topological magnon insulator,” J. Phys. Condens. Matter 28, 386001 (2016)

Localized electrons can transport spin and heat, although charge transport is inhibited. This phenomenon is based on electronic correlations which may result in magnetic order and its collective excitations. These spin excitations emerge as chargeneutral bosonic quasiparticles with spin 1, termed magnons, which transport spin and heat. As a result, extensively studied and attractive phenomena in electron physics can be carried over to magnons and many analogues have been identified, e.g., spin Hall effect and nontrivial band topology. The spin Hall effect in nonmagnetic materials originates from spin-orbit coupling which manifests itself in an intrinsic spin Berry curvature. Magnons, on the other hand, are inherently premised on magnetism, which brings about an unconventional spin Hall effect, i.e., the magnetic spin Hall effect (MSHE). The MSHE enables new kinds of spin currents and unconventional transport geometries and it can be manipulated by magnetic fields. Substituting electrons for magnons and an electric field for a thermal gradient, the predicted “magnetic” spin currents are induced by a temperature gradient and carried by magnons; they include the “magnetic spin Nernst effect” (MSNE), the thermal analogue of the MSHE. In this talk I shall interpret the MSHE in terms of “spin-selective whirls in the Fermi sea” which deflect incoming electrons to the transversal direction. A classification in terms of magnetic and toroidal multipoles proves useful for predicting MSHE and MSNE. By applying the theory to a concrete model I shall discuss the potential of chiral antiferromagnets as a platform for the MSNE and magnetic spin currents. Most collinear antiferromagnets (AFMs) do not display a MSNE because they are incompatible with the indispensable multipoles. However, these AFMs can be driven via a spin-flop phase to a field-polarized phase by an external magnetic field. As a result, topological phases emerge that manifest themselves in the existence of topological magnons. These chiral edge states are robust and propagate only in one direction without backscattering. Detecting these topological magnons is notoriously difficult because there is no Fermi level and, thus, no quantized transport for magnons. Here, I demonstrate that transitions into the topological phases can be identified in transport experiments. Upon crossing magnetic and topological phase transitions the measured thermal Hall conductivity changes by several orders of magnitude. In addition, the variation of temperature can be utilized for discerning experimentally the two types of phase transitions, which renders the thermal Hall effect across phase transitions an indicator of topological magnons.

CrBr$_3$ is an excellent realization of the two-dimensional honeycomb ferromagnet, which offers a bosonic equivalent of graphene with Dirac magnons and topological character. I this talk I discuss our results of inelastic neutron scattering (INS) measurements performed using state-of-the-art instrumentation, with which we update 50-year-old experimental data and thus establish definitively both the scope of recent theoretical predictions based on interacting spin-wave theory (SWT) and the basis for experimental claims of a significant gap at the Dirac point. We demonstrate that CrBr$_3$ has next-neighbor $J_2$ and $J_3$ interactions approximately 5~/%/ of $J_1$ and a near-ideal Dirac dispersion at the K point that has no gap within the instrumental resolution. The magnon lifetime and the thermal band renormalization show the universal $T^2$ evolution expected from SWT, but the measured dispersion lacks the predicted van Hove features, pointing to the need for more sophisticated theoretical analysis.

We investigate the topology of magnon excitations in magnets with chiral space groups. The presence of (magnetic) screw rotations leads to symmetry enforced Weyl points and nodal planes in the magnon band structure. In addition, there are multifold crossings pinned at high-symmetry points.We systematically analyze the band topology of these crossings and calculate their topological charges. In particular, we find that the magnon nodal planes carry a quantized topological charge similar to magnon Weyl points. Analogous to the protected spin currents on the surface of topological insulators, the topologically nontrivial magnon crossings result in protected surface modes of heat quanta. We propose several candidate materials and calculate their magnon band structures, topological invariants,and topologically protected surface modes. Further, we investigate the excitations of a quantum paramagnet on a one-dimensional ladder, and show that by applying an external magnetic field a topological phase transition is enforced. Because the triplon excitations of the ladder are gapped, fewer scattering channels are available and the topological exciations are potentially more robust than their spinwave counterparts.

Non-collinear spin structures including domain walls, spin spirals and skyrmions represent a modulation of the magnetic configuration, giving rise to complex magnon band structures where topological phenomena may be observed [1]. The anisotropic environment created by the non-collinear structure leads to the elliptic polarization or squeezing of magnons [2]. Here, it is shown that the elliptic polarization causes an effective enhancement of the damping in non-collinear magnetic configurations, thereby reducing the lifetime of magnons [3]. The excitations may also become overdamped [4], giving rise to non-Hermitian exceptional points in the dispersion relation. The observation of these magnon modes in resonance experiments is often limited by selection rules. It is demonstrated here that reducing the symmetry of the structures lifts the restrictions imposed by the selection rules and leads to a hybridization between different magnon modes, enabling the experimental detection of previously dark modes [5] and the opening of gaps in the band structure where topological boundary states may be formed. These effects are illustrated through the example of various types of magnetic skyrmions. [1] M. Garst, J. Waizner, D. Grundler, J. Phys. D: Appl. Phys. 50, 293002 (2017). [2] H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, P. Yan, arXiv:2111.14241. [3] L. Rózsa, J. Hagemeister, E. Y. Vedmedenko, R. Wiesendanger, Phys. Rev. B 98, 100404(R) (2018). [4] L. Rózsa, J. Hagemeister, E. Y. Vedmedenko, R. Wiesendanger, Phys. Rev. B 98, 224426 (2018). [5] L. Rózsa, M. Weißenhofer, U. Nowak, J. Phys.: Condens. Matter 33, 054001 (2020).

The discovery by Kane and Mele of a model of spinful electrons characterized by a $Z_2$ topological invariant had a lasting effect on the study of electronic band structures. Given this, it is natural to ask whether similar topology can be found in the band-like excitations of magnetic insulators, and recently models supporting $Z_2$ topological invariants have been proposed for both magnon [Kondo et al. Phys. Rev. B 99, 041110(R) (2019)] and triplet [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100, 020407 (2019)] excitations. In both cases, magnetic excitations form time--reversal (TR) partners, which mimic the Kramers pairs of electrons in the Kane-Mele model but do not enjoy the same type of symmetry protection. In this paper, we revisit this problem in the context of the triplet excitations of a spin model on the bilayer kagome lattice. Here the triplet excitations provide a faithful analog of the Kane-Mele model as long as the Hamiltonian preserves the TR×U(1) symmetry. We find that exchange anisotropies, allowed by the point group and typical in realistic models, break the required TR×U(1) symmetry and instantly destroy the $Z_2$ band topology. We further consider the effects of TR breaking by an applied magnetic field. In this case, the lifting of spin-degeneracy leads to a triplet Chern insulator, which is stable against the breaking of TR×U(1) symmetry. Kagome bands realize both a quadratic and a linear band touching, and we provide a thorough characterization of the Berry curvature associated with both cases. We also calculate the triplet-mediated spin Nernst and thermal Hall signals which could be measured in experiments. These results suggest that the $Z_2$ topology of band-like excitations in magnets may be intrinsically fragile compared to their electronic counterparts.