Topological transitions of magnons in three-dimensional strained chiral antiferromagnets and thermal Hall effect of magnons in honeycomb ferromagnet CrI{}_{3}

Topological transitions of magnons in three-dimensional strained chiral antiferromagnets and thermal Hall effect of magnons in honeycomb ferromagnet CrI

S. A. Owerre Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada.
July 19, 2019
Abstract

In this paper, we study two different insulating three-dimensional (3D) quantum magnets. In the first part of this paper, we propose a strain-induced topological phase transition in 3D topological insulating antiferromagnets. Without loss of generality, we utilize the physically realistic spin model of 3D insulating kagome chiral antiferromagnets. By applying (100) uniaxial strain, we establish that the 3D antiferromagnetic WMs in this system are in fact an intermediate phase between a strain-induced 3D MCI with integer Chern numbers and a 3D trivial magnon insulator with zero Chern number. Remarkably, the strain-induced 3D MCI exhibits another topological phase transitions with different lowest magnon Chern numbers for varying strain parameter. In addition, we show that strain suppresses the 3D topological thermal Hall conductivity of magnons. Our results provide a powerful mechanism for investigations of topological phase transitions in 3D chiral topological antiferromagnets. Motivated by recent experimental observation of topological magnons in CrI with a non-negligible interlayer coupling, in the second part of this paper we study the thermal Hall effect of magnons in the 3D insulating honeycomb ferromagnet CrI. We utilize the Heisenberg spin model with a non-negligible interlayer coupling and in-plane exchange interactions up to third nearest-neighbours and an easy-axis anisotropy as observed in the experiment. Using the experimentally deduced parameters, we compute the 3D thermal Hall conductivity of magnons in CrI and show that it is not negligible and it is positively enhanced by the interlayer coupling and the magnetic field. Our result provides an essential guide for future experimental measurements of magnon thermal Hall conductivity in CrI.

I Introduction

Three-dimensional (3D) topological semimetals are exotic phases of matter with gapless electronic excitations, which are protected by topology and symmetry. Their theoretical predictions and experimental discoveries have attracted considerable attention in condensed-matter physics wan (); bur (); xu (); lv (); wan1 (); wan2 (); tang (); nea (). They currently remain an active field of study. Nevertheless, the condensed matter realization of topological semimetals is essentially independent of the statistical nature of the quasiparticle excitations. In fact the notion of Weyl points was first observed experimentally in bosonic quasiparticle excitations lu ().

Figure 1: Color online. Schematic of the topological magnon phase diagram studied in this paper. The horizontal axis represents the effect of uniaxial strain which we model as a change in the spin exchange coupling along the direction. The vertical axis is the magnetic field along the direction. There are four magnon phases in this system. The 3D NLM and 3D TPM appear at zero magnetic field or in the conventional non-collinear spin structure. The 3D WM phase appears in the unstrained limit at nonzero magnetic field in the noncoplanar spin structure. The two new magnon phases that appear due to strain in the noncoplanar spin structure are the 3D MCI and the 3D trivial insulator.

There has been an intensive search for bosonic analogs of 3D topological semimetals in insulating quantum magnets with broken time-reversal symmetry bos1 (); bos2 (); bos3 (); bos4 (); bos5 (); bos6 (); bos7 (); bos8 (); bos9 (); bos10 (); bos11 (); bos12 (); bos13 (). Recently, topological Dirac magnons protected by a coexistence of inversion and time-reversal symmetry have been experimentally observed in a 3D collinear antiferromagnet CuTeO bos12 (); bos13 (). This has opened a great avenue for observing topological Weyl magnon (WM) points in 3D insulating quantum magnets. In magnetic bosonic systems, however, it is essentially important that the WM nodes occur at the lowest excitation if they were to make any significant contributions to observable thermal Hall transports. This is due to the population effect of bosonic quasiparticles at low temperatures. In this respect, WM nodes at the lowest excitation can be considered as the analog of electronic Weyl points close to the Fermi energy. The WMs in the 3D kagome chiral antiferromagnet exhibit a topological thermal Hall effect bos7 (). Currently, they are the only known antiferromagnetic system in which the WM nodes occur at the lowest acoustic magnon branch and contribute significantly to the thermal Hall transports.

It is well-known that electronic ferromagnetic Weyl semimetal occurs as an intermediate phase between an ordinary insulator and a 3D quantum anomalous Hall insulator bur (); HI (). To our knowledge, this interesting topological phase boundary has not been established in 3D topological antiferromagnets. Thus far, the topological Dirac and Wely nodes that appear in 3D insulating antiferromagnets bos1 (); bos6 (); bos7 (); bos8 (); bos10 (); bos11 (); bos12 (); bos13 () cannot transit to another topological magnon phase. In this respect, strain provides an effective way to tune the band structure of crystal in quantum materials. For instance, uniaxial strain can induce chiral anomaly and topological phase transitions in 3D topological Dirac and Wely semimetals stra (); straa (); strab (). Strain can also induce a 3D topological Dirac semimetal in epitaxially-grown -Sn films on InSb(111) stra1 (). We envision that such strain effects could be possible in 3D topological antiferromagnets.

In addition, recent experiment has observed topological magnons in the bulk honeycomb ferromagnet CrI at zero magnetic field cr (), which follows from earlier theoretical proposals in Ref. owerre () followed by Ref. skim (). However, the observed topological magnons show a non-vanishing dispersion along the out-of-plane axis and there is a non-negligible interlayer coupling. The fitted parameters also contain exchange interactions up to third nearest neighbours. Therefore, a complete 3D analysis of this system is desirable. Moreover, the experimentally observed topological ferromagnetic magnons in CrI should also exhibit the magnon thermal Hall effect, which has been studied in other ferromagnetic insulators th1 (); th2 (); th3 (); th4 (); th5 (); th6 (); th7 ().

In this paper, we study two different 3D insulating quantum magnets. In the first part of this paper, we propose a strain-induced topological magnon phase transition in 3D topological insulating antiferromagnets. Due to the nature of the topological magnon band distributions in the 3D kagome chiral antiferromagnet, we have chosen this system for our study. However, our results can be extended to other 3D topological insulating antiferromagnets such as CuTeO bos12 (); bos13 (). Under (100) uniaxial strain, we show that topological magnon phase transition exist in the 3D topological insulating kagome chiral antiferromagnet. The schematic of the topological phase transitions is depicted in Fig. (1).

We have identified four different magnon phases in this system. The 3D nodal-line magnon (NLM) and the triple point magnon (TPM) appear at zero magnetic field or at zero in-plane Dzyaloshinskii-Moriya interaction (DM) interaction dm (); dm1 () in the conventional 3D in-plane 120 non-collinear spin structure. They can be tuned by strain. The 3D WM phase appears in the unstrained limit at nonzero magnetic field or non-zero in-plane DM interaction in the noncoplanar chiral spin structure. The two new magnon phases that appear due to strained noncoplanar chiral spin structure are the fully gapped 3D MCI and the fully gapped 3D trivial insulator. We show that the former has integer Chern numbers, whereas the latter has zero Chern number. The study of the topological thermal Hall effect of magnons shows that the thermal Hall conductivity is suppressed in the fully gapped insulator phases. This implies that strain suppresses the thermal Hall conductivity of magnons.

In the second part of this paper, we provide a complete computation of the magnon thermal Hall conductivity in the bulk honeycomb ferromagnet CrI using a 3D Heisenberg spin model and the experimentally determined parameters cr (). We show that the magnitude of the magnon thermal Hall conductivity in CrI is not negligible and it is positively enhanced by the interlayer coupling and the magnetic field. We believe that our result provides a promising guide for future thermal transport experiments in CrI.

Ii Strained 3D kagome chiral antiferromagnet

We study 3D kagome chiral antiferromagnets in the presence of uniaxial strain and an external magnetic field along the direction. The Heisenberg spin model is given by

(1)

where is the spin vector at site in layer . The first term is the intralayer nearest-neighbour Heisenberg coupling. We model the effect of uniaxial strain along the direction by the assumption that only the Heisenberg spin interaction along the direction changes 111An alternative approximation is to consider isotropic Heisenberg interactions with lattice deformation in which only the primitive lattice vectors change.. In this case along the diagonal bonds and along the horizontal bonds, where is the strain as shown in Fig. 2(a). The second term is the out-of-plane () DM interaction due to inversion symmetry breaking between two sites on each kagome layer. The DM interaction alternates between the triangular plaquettes of the kagome lattice and it stabilizes the conventional in-plane non-collinear spin structure. Its sign determines the vector chirality of the non-collinear spin order men1 (). The third term is the nearest-neighbour interlayer antiferromagnetic coupling between the kagome layers, which is inevitably present in real kagome materials ka (); ka1 (); ka2 (); ka3 (); ka4 (). Finally, the last term is an external magnetic field along the stacking direction , where is the Landé g-factor and is the Bohr magneton.

In the absence of strain, i.e. , the 3D noncoplanar chiral kagome chiral antiferromagnets are intrinsic WM semimetals. The noncoplanar chiral spin texture with macroscopically broken time-reversal symmetry can be induced by an in-plane intrinsic DM interaction or an external magnetic field bos7 (). The WM phase in this system cannot transit to any other magnon phase by changing the parameters of the system at . Hence, it is strictly robust. In this paper, we will investigate the fate of the WM phase when a uniaxial strain is applied along the direction. A similar effect can be induced by applying pressure. First, let us understand the spin structure at . In this case, the in-plane spins on each kagome layer are canted by the angle , where for . There are various magnetic phases for different limiting cases of . Nevertheless, our main concern here is the regime where the in-plane spins are non-collinear and stable. This happens in the regime .

Figure 2: Color online. (a) Top view of stacked kagome-lattice antiferromagnets along the (001) direction. The distribution of the DM interaction is denoted by the crossed and dotted circles. The blue arrows denote the conventional non-collinear spin order at zero magnetic field. A uniaxial strain is applied along the direction. (b) Top view of stacked honeycomb-lattice ferromagnets along the (001) direction. The DM interaction is perpendicular to the midpoint of the second-nearest neighbours denoted by the blue lines. The dotted red lines denote the bonds of the third-nearest neighbours. (c) The bulk Brillouin zone of the hexagonal lattices.
Figure 3: Color online. Strain-induced topological magnon phase transitions in 3D kagome chiral antiferromagnets. (a) 3D trivial magnon insulator for , (b) 3D acoustic WM for , (c) 3D MCI for . The other parameters are set as , , .
Figure 4: Color online. Heat map of the energy gap between the two acoustic magnon branches as a function of the momentum along the high symmetry line - and magnetic field. (a) 3D trivial magnon insulator for . (b) 3D WMs for . (c) 3D MCI for . The other parameters are set as , .

ii.1 Field-induced noncoplanar chiral spin texture

Next, we induce noncoplanar chiral spin texture in the non-collinear regime by applying a magnetic field along the stacking direction. We note that a noncoplanar chiral spin texture can also be induced if the intrinsic in-plane DM interaction is present ka (); ka1 (); ka2 (); ka3 (); ka4 (). Due to the presence of an out-of-plane DM interaction, a magnetically-ordered phase is present at low temperatures. In the ordered phase the magnetic excitations are magnons (quantized spin waves). They can be captured clearly in the linear spin wave theory approximation as shown in the Appendix. Indeed, a finite magnetic field induces a noncoplanar chiral spin texture with finite scalar spin chirality given by , where and the saturation field is given by

(2)
Figure 5: Color online. The energy band gap between the two lowest acoustic magnon bands along the high symmetry line as a function of strain. The parameters are , , and .

Evidently, at the scalar spin chirality vanishes. In this case the unstrained 3D kagome chiral antiferromagnet at exhibits a coexistence of 3D NLM and 3D TPM due to the symmetry protection of the conventional 3D in-plane spin structure at zero magnetic field. These symmetry-protected nodal-line magnon phases can be tuned for within the three magnon branches. Because time-reversal symmetry is not broken macroscopically at , there are no topological magnon phases in this system. However, at a noncoplanar chiral spin texture with macroscopically broken time-reversal symmetry is induced. As we mentioned above, the magnon bands of unstrained 3D kagome chiral antiferromagnet at possesses robust WM points in the noncoplanar regime, and they are the only topological magnon phase in this regime.

ii.2 Strain-induced topological phase transitions

Remarkably, the strained 3D kagome chiral antiferromagnet in the noncoplanar regime exhibits a topological magnon phase transition with interesting features. We will now investigate different aspects of these topological phase transitions. First, let us consider the 3D magnon band structures with varying . In Fig. (3) we have shown the evolution of the 3D magnon bands along the Brillouin zone (BZ) paths in Fig. 2(c). We can see that the uniaxial strain along the direction gaps out the 3D WM phase at along the high symmetry lines and of the BZ.

Figure 6: Color online. Chern number of the lowest acoustic magnon band as a function of strain for different values of the magnetic field (a) and as a function of the magnetic field for different values of strain (b). The other parameters are set as , .

Let us define the gap between the two acoustic magnon branches as

(3)

At the WM points vanishes, and the fully gapped magnon insulators are characterized by a non-zero along the high symmetry lines of the BZ. Next, let us check if the regime is truly a fully gapped magnon insulator with for varying magnetic field in the noncoplanar regime. For this purpose, we have fixed the antiferromagnetic interlayer coupling to and the DM interaction to . We then plot the heat map of as a function of the momentum along the high symmetry lines and the magnetic field in the noncoplanar regime. The heat map of is shown in Figs. 4(a)–(c) for different regimes of . At the critical point we can see that circular gapless points appear between the two acoustic magnon branches, which signify the presence of WM points along - lines. In the regimes and there are no discernible gapless points along the - line of the BZ. Therefore these two regimes define fully gapped magnon insulators, however with different properties as we will see later. Furthermore, in Fig. (5) we have shown the evolution of the magnon energy band gap along the high symmetry line as a function of strain. We can see that the three distinct regions are clearly identified. We have checked that similar trends are manifested along the - line.

Now, we will consider the Chern number topological phase transition of the system. This will justify the topological and non-topological regimes of . We will focus on the lowest acoustic magnon branch in which the strain-induced topological phase transition occurs. In this case, we can formally define the 3D trivial magnon insulator as the state where the Chern number of the lowest acoustic magnon branch vanishes, and a 3D MCI as the state with non-zero integer Chern numbers. The 3D antiferromagnetic system can be considered as slices of 2D antiferromagnetic MCIs th8 (); th9 () interpolating between the and planes. For an arbitrary point the Chern number of the magnon energy branches can be defined as

(4)

where is the in-plane momentum vector and is the momentum space Berry curvature for a given magnon band defined as

(5)

where defines the velocity operators with and is the 3D momentum vector. are the columns of a paraunitary operator, see the Appendix.

For an arbitrary point and the Chern number is well-defined in the noncoplanar regime. The Chern number of all the 2D slices at an arbitrary point is the same because the planes at two points can be adiabatically connected without closing the gap for . For the Chern number is only well-defined for , where is the location of the Weyl points along the momentum direction. Based on this consideration, we have shown the plots of the Chern number of the lowest acoustic magnon band as a function of strain for different values of in Fig. 6(a), and as a function of the magnetic field for different values of in Fig. 6(b). In both figures, we can see that the regime has zero Chern numbers. Therefore, the system is a 3D trivial magnon insulator for . In the regime , however, the Chern number of lowest acoustic magnon band is nonzero and changes as . This signifies topological magnon phase transitions in the 3D MCI phase.

ii.3 3D topological thermal Hall effect

Having identified the different magnon phases in the strained 3D kagome chiral antiferromagnet, we will now study an experimentally feasible measurement that can be performed on this system. The topological thermal Hall effect of magnons refers to the generation of a transverse thermal Hall voltage in the presence of a longitudinal temperature gradient due to the presence of noncoplanar chiral spin textures. In principle, it does not necessarily require the DM interaction provided a noncoplanar chiral spin configuration can be established, for example by adding further nearest-neighbour interactions. Therefore, the topological thermal Hall effect of magnons is different from the conventional magnon thermal Hall effect in insulating ferromagnets which strictly requires the DM interaction th1 (); th2 (); th3 (); th4 (); th5 (); th6 (); th7 ().

Figure 7: Color online. 3D topological thermal Hall conductivity for different regimes of as a function of (a) magnetic field for and (b) temperature for . The other parameters are set as , .

In the 3D model, the topological thermal Hall conductivity has three contributions and , where the components are given by

(6)

where is the Bose occupation function, the Boltzmann constant which we will set to unity, is the temperature and , with being the dilogarithm.

Due to the Bose occupation function, the dominant contribution to comes from the lowest magnon branch, where the topological phase transitions occur in the current system. As the noncoplanar chiral spin configuration is induced along the direction, the first two components and vanish. The nonzero component can be written as

(7)

where is 2D thermal Hall conductivity for each slice of the plane, which is given by

(8)

We have performed the 3D integration to calculate the topological thermal Hall conductivity. In Figs. 7(a) and (b) we have shown the trends of as a function of the magnetic field and temperature respectively for different regimes of . We find that the magnitude of is suppressed in the 3D insulator phases. In other words, strain decreases the thermal Hall conductivity. We note that in the 3D WM phase the magnitude of is dominated by the states near the WM nodes at the lowest magnon branch due to large Berry curvatures. In this case the thermal Hall conductivity is proportional to the distance separating the WM nodes in momentum space in analogy to the anomalous Hall conductivity in Weyl semimetals bur (). In the 3D insulator phases the magnitude of is dominated by the states near the topological gaps at the lowest magnon branch. Despite zero Chern number in the 3D trivial insulator phase, we can see that is nonzero for . This is in stark contrast to electronic systems where the Fermi energy can guarantee a completely filled band and zero Hall conductivity in the trivial insulator phase. In magnonic (bosonic) systems the Bose occupation function eliminates this restriction as the bands are thermally populated. Therefore, the thermal Hall effect in insulating quantum magnets is not a direct consequence of Chern number protected topological bands. It depends solely on the Berry curvature of the magnon bands irrespective of their topological classifications.

Iii 3D honeycomb ferromagnet

Motivated by recent experiment cr (), we study the bulk three-dimensional honeycomb ferromagnet CrI. The Heisenberg spin model is given by

(9)

The first term is the intralayer Heisenberg coupling up to third nearest-neighbours . The second term is the DM interaction due to inversion symmetry breaking on the second nearest-neighbour bonds , where for clockwise and counter-clockwise hopping magnons on each honeycomb layer sublattices as depicted in Fig. 2(b). The third term is the easy-axis anisotropy. The fourth term is the nearest-neighbour ferromagnetic interlayer coupling. Finally, the last term is an external Zeeman magnetic field along the stacking direction , where is the Landé g factor and is the Bohr magneton. We consider congruently-stacked honeycomb ferromagnetic layers. Due to a nonzero easy-axis anisotropy in CrI, recent experiment shows that the magnetic spin moments are already polarized along the axis at zero magnetic field, which enables the observation of topological magnon dispersions without an external applied magnetic field cr (). This is in stark contrast to the in-plane kagome ferromagnet Cu(1,3-bdc) th2 (), where the magnetic field is required to polarize the spins along the axis.

By fitting the observed topological magnon dispersions, the following parameters were deduced cr (): meV, meV, meV, meV, meV, and meV.

Figure 8: Color online. Topological spin wave modes in CrI with the experimentally deduced parameters at zero magnetic field. The acoustic (lower) and optical (upper) spin wave dispersions carry a Chern number of respectively. Inset shows the evolution of the Chern numbers with varying .
Figure 9: Color online. The 3D magnon thermal Hall conductivity of CrI as a function of temperature for different values of using the experimentally deduced parameters at zero magnetic field. Inset shows the dependence of the magnon thermal Hall conductivity on for different values of .
Figure 10: Color online. Heat map of the 3D magnon thermal Hall conductivity of CrI in the plane of the magnetic field and temperature using the experimentally deduced parameters.

iii.1 Bosonic model of CrI

At Curie temperature K, ferromagnetic ordering appears with Cr spins () oriented along the axis cr (). Therefore, the magnon dispersions can be captured by the linearized Holstein Primakoff (HP) transformation

(10)

where are the bosonic creation (annihilation) operators, and denote the spin raising and lowering operators. The resulting noninteracting magnon Hamiltonian in momentum space can be written as

(11)

where is the mean-field energy, is the basis vector. The momentum space Hamiltonian is given by

(12)

where

(13)
(14)
(15)
(16)

and with .

iii.2 Topological magnons and Chern numbers in CrI

The topological magnon dispersions are depicted in Fig. (8) with the experimentally deduced parameters at zero magnetic field. We can see that the acoustic (lower) and the optical (upper) magnon modes are well-separated. As observed in the experiment cr (), the optical magnon mode extends up to meV in the plane and the acoustic magnon mode extends up to meV along the line . The small gap at the point is due to nonzero easy-axis anisotropy meV. The gaps at the Dirac points at and can be attributed to a nonzero DM interaction meV. This leads to nonzero Chern numbers. Using the experimentally deduced parameters cr (), we find for the lower and upper spin wave modes of the 3D system. The inset of Fig. (8) shows that the Chern numbers change with varying .

iii.3 Thermal Hall effect of magnons in CrI

The thermal Hall effect of magnons in insulating ferromagnets has been extensively studied both theoretically and experimentally in different insulating ferromagnets th1 (); th2 (); th3 (); th4 (); th5 (); th6 (); th7 (). However, the thermal Hall effect in CrI has not been studied using the 3D model in Eq. (9). The thermal Hall effect provides a way to access the magnetic excitations in quantum magnets, and it is believed to be due to topological magnons in ordered insulating ferromagnets. Therefore, a nonzero thermal Hall conductivity will solidify the belief that the observed magnon modes in CrI are indeed topological. Similar to the 3D chiral kagome antiferromagnets, the total intrinsic anomalous thermal Hall conductivity in the present case also has the three contributions and , but only the the contributions of are nonzero as the spins are fully polarized along the axis. Therefore, Eqs. (7) and (8) are also valid in the present case.

In Fig. (9) we have shown the temperature dependence of the 3D magnon thermal Hall conductivity of CrI. We can see that the magnon thermal Hall conductivity is negative and drops with increasing temperature (note that ). The magnon thermal Hall conductivity also increases positively with increasing . The inset of Fig. (9) shows that the fully 3D model has a positively enhanced magnon thermal Hall conductivity than the 2D system at . Indeed, experiment shows that the magnon dispersion along the axis does not vanish and cr (). In Fig. (10) we have shown the heat map of magnon thermal Hall conductivity of CrI in the plane of the magnetic field and temperature. The plot shows that the magnon thermal Hall conductivity is also positively enhanced by increasing magnetic field.

Iv Conclusion

We have proposed a strain-induced topological phase transition in 3D topological insulating antiferromagnets and studied the 3D magnon thermal Hall effect in CrI.

In the first part of our results, we showed in the presence of (100) uniaxial strain antiferromagnetic WM in 3D topological insulating antiferromagnets is an intermediate phase between a 3D antiferromagnetic MCI with integer Chern numbers and a 3D antiferromagnetic trivial insulator with zero Chern number. We further showed that the thermal Hall conductivity of magnons is suppressed in the 3D insulator phases. Besides, we found that the 3D trivial magnon insulator with zero Chern number possess a non-zero thermal Hall conductivity due to the bosonic nature of magnons. We believe that our results can be investigated experimentally in various 3D insulating antiferromagnets by applying uniaxial strain or pressure. For the 3D kagome chiral antiferromagnets, there are various promising materials that have been synthesized lately ka (); ka1 (); ka2 (); ka3 (); ka4 (). Furthermore, it will be interesting to experimentally investigate the effects of strain on the recently observed 3D topological Dirac magnons in the insulating antiferromagnet CuTeO bos12 (); bos13 (). Due to the similarity between 3D insulating and metallic kagome chiral antiferromagnets, we envision that the current results could also manifest in the 3D antiferromagnetic topological Weyl semimetals MnSn/Ge ele (); ele1 (); ele2 (); ele3 (); ele4 (); ele5 ().

In the second part of our results, we computed the magnon thermal Hall conductivity of CrI using the experimentally determined values of the parameters cr (). We found that the magnon thermal Hall conductivity of CrI can be observed in experiment and it can be also be enhanced by the interlayer coupling and the magnetic field. The temperature dependence of the magnon thermal Hall conductivity is also negative, which suggests electron-like. Therefore, if the observed magnon dispersions in CrI are topological, we believe that our results will pave the way for future experimental measurements of magnon thermal Hall conductivity in CrI.

Acknowledments

Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

Appendix A Spin transformation

In the low temperature regime, the underlying magnetic excitations of the Hamiltonian (1) are magnons. We can then perform spin wave theory. First, let us express the spins in terms of local axes, such that the -axis coincides with the spin direction. This can be done by performing a local rotation about the -axis by the spin orientated angles , where for . The rotation matrix is given by

(17)

The external magnetic field induces canting of spins in the out of plane direction. Therefore, we perform another rotation about -axis by the angle . The rotation matrix is given by

(18)

Hence, the total rotation matrix is given by , where

(19)

Now, the spins transform as

(20)

Appendix B Spin wave theory

The terms that contribute to linear spin wave theory are given by

(21)
(22)
(23)

where . The magnetic field term is given by

(24)

The magnetic field induces a noncoplanar chiral spin texture. The emergent field-induced scalar spin chirality is given by

(25)

The classical ground state energy is given by

(26)

where is the number of sites per unit cell. The magnetic field has been rescaled in unit of . By minimizing the classical energy we get the canting angle , where is given by Eq. (2).

Next, we perform the linearized Holstein Primakoff (HP) transformation

(27)

where are the bosonic creation (annihilation) operators, and denote the spin raising and lowering operators. The resulting magnon Hamiltonian in momentum space can be written as

(28)

where is the basis vector.

(29)

where and are matrices.

Appendix C Generalized Bogoliubov transformation

The Hamiltonian (29) can be diagonalized by paraunitary operators . This is equivalent to diagonalizing the bosonic Bogoliubov Hamiltonian with , where is an identity matrix. The magnon Hamiltonian can be diagonalized using the bosonic Bogoliubov transformation: , where are the Bogoliubov quasiparticles, and is a paraunitary matrix defined as

(30)

where and are matrices that satisfy

(31)

The matrix satisfies the relations,

(32)

where

(33)

with energy band and band index .

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