Topological phase diagrams of in-plane field polarized Kitaev magnets

While the existence of a magnetic field induced quantum spin liquid in Kitaev magnets remains under debate, its topological properties often extend to proximal phases where they can lead to unusual behaviors of both fundamental and applied interests. Subjecting a generic nearest neighbor spin model of Kitaev magnets to a sufficiently strong in-plane magnetic field, we study the resulting polarized phase and the associated magnon excitations. In contrast to the case of an out-of-plane magnetic field where the magnon band topology is enforced by symmetry, we find that it is possible for topologically trivial and nontrivial parameter regimes to coexist under in-plane magnetic fields. We map out the topological phase diagrams of the magnon bands, revealing a rich pattern of variation of the Chern number over the parameter space and the field angle. We further compute the magnon thermal Hall conductivity as a weighted summation of Berry curvatures, and discuss experimental implications of our results to planar thermal Hall effects in Kitaev magnets.

While the existence of KSL at intermediate fields remains under debate [32][33][34][35][36][37][38][39], Kitaev magnets eventually polarize at sufficiently high fields, where the collective excitations are magnons, which can give rise to experimentally measurable transport signals.Furthermore, if the magnon bands are topological, the resulting thermal Hall conductivity can reach the same order of magnitude as the half quantized value [40][41][42][43].Although for magnons    at low temperatures is not directly proportional to the Chern number  of the lower band, the latter is very often a good indicator of the opposite sign of the former.Therefore, phase diagrams that reveal the magnon Chern number across generic model parameters of Kitaev magnets [44] are valuable to identify topological magnons and to interpret thermal transport measurements at high fields (Figs.1c and  1d).The main objective of this Letter is precisely to present such topological phase diagrams for in-plane magnetic fields, which are relevant to experiments of planar thermal Hall effect [25,26,30,31,45].
We note that Kitaev magnets such as -RuCl 3 are polarized more easily by in-plane fields than out-of-plane fields, likely due to an anisotropic  tensor [46][47][48][49] and a positive Γ interac- tion [50], which discounts the out-of-plane field strength and disfavors an out-of-plane magnetization, respectively [41,51].The case of polarizing Kitaev magnets with strong out-ofplane fields has been studied theoretically in Ref. [52] (see also Ref. [53]).It is found that, within the linear spin wave approximation, the ΓΓ ′ model can be effectively reduced to a  model.The  3 symmetry also plays an important role in the diagnosis of magnon band topology in Kitaev magnets, based on topological quantum chemistry or symmetry indicator theory [54][55][56][57][58].As demonstrated in Ref. [59], the magnon bands must be topological whenever a gap exists in between.In this Letter, we consider the nearest neighbor ΓΓ ′ model polarized by in-plane magnetic fields, which break the  3 symmetry, and map out the phase diagrams of topological magnons.Unlike the aforementioned case, none of the model parameters can be made redundant.We find that, as long as the field is not along the armchair direction, there exist parameter regions that are topological ( = ±1) as well as trivial ( = 0) ones, the latter of which can be understood via an effective Hamiltonian [52].We discuss the implications of our results to thermal Hall conductivities of Kitaev magnets at high fields, from which we propose a scheme to determine the relevant candidate parametrizations.
Model.-The most generic nearest neighbor spin Hamiltonian for Kitaev magnets is the ΓΓ ′ model [21].In an external magnetic field h, it reads where (, , ) is a cyclic permutation of (, , ).For convenience of analysis, we write the field strength |h| ≡ ℎ in terms of the spin magnitude  ≡ |S  | [60].An in-plane field can be parametrized as (ℎ  , ℎ  , ℎ  ) = ℎ(cos , sin , 0), where  ∈ [0, 2) is the azimuthal angle in the honeycomb plane, see Fig. 2a [61].We apply linear spin wave theory [62,63] to the in-plane field polarized state of (1), and obtain an analytical expression for the magnon spectrum where ] corresponds to the lower (upper) band.For clarity, we refer to the gap between the two bands, min k [ + (k) −  − (k)] ≥ 0, as the band gap, which is not to be confused with the excitation gap, min k  − (k) > 0. Chern number is a topological invariant that can never change as long as a finite band gap is maintained [64], i.e., a topological phase transition can only occur when Δ(k) = 0 for some k.
We assume a polarized state in which the excitation gap grows with ℎ, so that the system becomes more stable as ℎ increases, rather than undergoing a magnon instability.This requires ℎ >  1 [65], from which we deduce the following.For a given set of parameters {, , Γ, Γ ′ }, if the band gap is finite (zero), then it remains finite (zero) as ℎ varies, unless ℎ −→ ∞.Therefore, the topological phase diagrams are independent of the field strength, and, for a given field angle, we can map them out by first solving for the zeros of (2b) and then choosing a sufficiently high field to compute the Chern numbers [66][67][68][69] at parameters away from these zeros.
Topological phase diagrams.-Forfinite in-plane fields, the band gap closes iff the set of parameters {, , Γ, Γ ′ } meets any of the criteria listed in Table I.Whenever the band gap is finite, let the Chern number of the lower (upper) band be  (−), which transforms according to the  2 representation of the point group 3 [70,71], and flips sign under time reversal [41], as in the case of the non-Abelian KSL [28].More specifically, fixing the couplings, (i)  −→  if h is rotated by 2/3 about the out-of-plane axis, (ii)  −→ − if h is rotated by  about the  axis, and (iii)  −→ − if h −→ −h, while the phase boundaries are invariant under these actions [65].Hence,  ∈ [0, /6] serves as an independent unit, to which all other angles can be related by symmetries, see TABLE I.For field angles 0 ≤  < /6, the band gap closes iff the parameters of the ΓΓ ′ model satisfy any of the following equations.For  = /6, the band gap is zero whenever (I) or (4) is satisfied.For visualizations, we set  = 0 and calculate  over the spherical parameter space defined by  2 + Γ 2 + Γ ′2 = 1, at the field angles  = 0, /24, /12, /8, /6, see Figs. 3a-3f [72].We make two observations, with the understanding that all angles mentioned below are defined modulo /3.First, for  ≠ /6, there exist both parameter regions with topological magnons and those without.For  = /6, topological magnons are altogether forbidden due to a  2 symmetry [40,73].Second, the total area  of the parameter regions with  = ±1 is maximal at  = 0, which implies that, for a Kitaev magnet dominated by nearest neighbor anisotropic interactions, topological magnons are most likely found when the in-plane field is along the  axis [74].
To understand why magnons are topologically trivial in certain parameter regions, we analyze the linear spin wave theory at high fields by systematically integrating out the pairing terms [52].This is achieved via a Schrieffer-Wolff transformation [75], from which we obtain an effective hopping model of the form H eff One finds that the third component of d(k) vanishes throughout the Brillouin zone when  − Γ + Γ ′ = 0 [65], which defines the phase boundary (VII) within the parameter region On the other hand, there exist parameters outside (4) that satisfy  −Γ+Γ ′ = 0 and possess a finite band gap simultaneously [76].At these parameters, the triple product in (3) is identically zero, and consequently  = 0. Any other parameter that can be continuously connected to these parameters without a gap closing must be topologically trivial as well.Thermal Hall effect.-Wediscuss how the topological phase diagrams relate to experimentally measurable quantities by connecting the Chern number to the thermal Hall conductivity [80], which is given by [81][82][83]  for magnons, where  is the band index ranging from 1 to , and Ω k is the momentum space Berry curvature [65].While the Chern number   is given by the summation of Ω k over k,    is given by a weighted summation of Ω k with non-positive weights.Also, high-energy magnons contribute less to    than low-energy ones.Therefore, though    is not directly proportional to , one can very often use the latter to infer the sign of the former at low temperatures.More precisely,  > 0 ( < 0) means that there is an excess of positive (negative) Berry curvatures in the lower band, and by ( 5) the sign of    is expected to be opposite to  [84].On the other hand,  = 0 means that the net Berry curvature is zero, so    is generically small though not necessarily zero, and its sign is arbitrary.
We illustrate these ideas with three proposed parametrizations of -RuCl 3 in the literature, (, , Γ, Γ ′ ) = (−1, −8, 4, −1) [77], (−1.5, −40, 5.3, −0.9) [78], and (0, −6.8, 9.5, 0) [79], where energies are given in units of meV.For h ∥ , these parametrizations are located in the  = +1, −1, and 0 regimes, respectively, so we label them by , , and , see Fig. 4a.For each of them, we calculate    as a function of  at several values of ℎ, see Fig. 4b.We find that    is negative (positive) for  () as expected, while    for  is several times smaller.If we assume that the measured    > 0 in the field-induced phase under h ∥ − in -RuCl 3 [25] is indeed determined by a dominant magnon contribution, then  appears to be a more promising candidate parametrization.We also list three criteria that are conducive for a large magnon thermal Hall effect, which help us to understand the difference in    between the three parametrizations, as follows.(i) The bands are topological.(ii) The excitation gap is not too large, so that the lower band is thermally populated at low temperatures.(iii) The band gap is not too small, so that the population of the upper band remains negligible over an extended temperature range.For instance, at the respective lowest fields, the excitation gaps of , , and  are 0.16, 0.19, and 0.24, while the band gaps are 0.07, 0.25, and 0.78, in units of | |. and  fulfill (i) and are comparable in (ii), but  does better than  in (iii), so  yields a larger    .On the other hand,  is comparable to  and  in (ii) and does better in (iii), but  fails (i), so its    is small.As ℎ increases, the excitation gap becomes larger and    decreases.
Discussion.-In summary, we have mapped out topological phase diagrams of Kitaev magnets polarized by in-plane magnetic fields, which reveal the magnon Chern number over a large parameter space.Since topological magnons are generally expected to yield a sizable thermal Hall conductivity with sign opposite to the Chern number at low temperatures, our results will be helpful in determining the relevant parametrizations of Kitaev magnets including -RuCl 3 .We briefly address the effects of the third nearest neighbor Heisenberg exchange [85,86] and the magnon interactions [87][88][89][90][91][92][93][94] in the Supplemental Material [65].While the window of a field-induced KSL might be shut in many of the candidate materials, the door to topological magnons is most likely open and accessible via high fields.We appreciate that alternative sources of heat carriers in Kitaev magnets, such as spinons [95][96][97], triplons [98], phonons [99], and visons [100], as well as some effects arising from spin-lattice coupling [101][102][103][104][105][106], have been proposed.One particularly interesting future direction is to investigate the interplay between different types of topological excitations, whether they cooperate with one another and lead to a large thermal Hall conductivity [107] or other unusual properties.

Supplemental Material: Topological phase diagrams of in-plane field polarized Kitaev magnets
Li Ern Chern 1 and Claudio Castelnovo 1 1 T.C.M. Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom

S1. LINEAR SPIN WAVE THEORY
In the linear spin wave analysis, one first rotates the local coordinate frame defined at each magnetic site such that the  axis align with the spin [63], while the  and  axes can be chosen freely as long as they are orthogonal and ẑ = x × ŷ [73].For the polarized state with field angle , the axes of the rotated coordinates are defined by ẑ = â cos  + b sin , x = −ĉ, ŷ = −â sin  + b cos , where â, b, and ĉ are unit vectors along the crystallographic axes , , and , respectively, see Fig. 2a.One then performs Holstein-Primakoff transformation [62] to represent the spins in terms of bosons (i.e., magnons), perform a 1/ expansion of the Hamiltonian (∼  2 in the classical limit), and discard terms are of orders lower than .These procedures are well established and described in details elsewhere (see Refs. [40,64,82] for example), so we do not repeat them here.
For the ΓΓ ′ model ( 1) under an in-plane magnetic field (ℎ  , ℎ  , ℎ  ) = ℎ(cos , sin , 0), the linear spin wave Hamiltonian of the polarized state is given by ) and H k is a four dimensional Hermitian matrix, where   are (real) linear combinations of the couplings, while  k and  k are functions of the (crystal) momentum, see (2c) and related discussions in the main text.To obtain the linear spin wave dispersion, H k has to be diagonalized by a Bogoliubov transformation  k satisfying  k  † k = , where  = diag(1, 1, −1, −1), in order to preserve the commutation relation of bosons.The magnon bands are given by where  (k) and Δ(k) are defined in (2a) and (2b) in the main text.
We consider a stable polarized state, where the excitation gap min k  − (k) is greater than zero.As the field strength ℎ increases, the excitation gap should increase as well, so that the polarized state becomes more stable, rather than undergoing a magnon instability in which the excitation gap vanishes.Based on this physical expectation, we can assume always, which is argued as follows.Eq. (S3) obviously holds for  1 ≤ 0 for all ℎ.For  1 > 0, assuming that a finite excitation gap is possible for some ℎ <  1 , we can then dial up ℎ such that ℎ =  1 .At the K or K ′ point, where  k = 0, we will have which is either zero or imaginary, neither being physically sensible.Therefore, magnon stability is only consistent with ℎ −  1 > 0.
A quick inspection of (2b) reveals that Δ 2 (k) consists of a field dependent part and a field independent part.With the assumption (S3), we now claim that if Δ(k) = 0, then it is only physically sensible that  2  k +  3 +  4  k = 0, unless the system sits at a critical point where a transition to the polarized state occurs.Suppose that the contrary is true, i.e., Δ(k) = 0 and  2  k +  3 +  4  k ≠ 0 at some ℎ = ℎ * , with ℎ * −  1 > 0. The field dependent part, which is positive, must cancel the field independent part exactly in Δ 2 (k).Let ℎ ′ = ℎ * −  with 0 <  < ℎ * −  1 .We have ℎ ′ −  1 > 0, but Δ 2 (k) < 0 at ℎ = ℎ ′ , i.e.,  ± (k) develops an imaginary component, which is unphysical.Therefore, ℎ cannot be less than ℎ * .In case ℎ * marks the phase boundary, we can further increase the field to some ℎ > ℎ * to obtain a stable polarized state.We have thus established We refer to the gap between the upper and lower magnon bands, min k [ + (k) −  − (k)] ≥ 0, as the band gap, which is not to be confused with the excitation gap.From (S2), we see that the band gap vanishes if and only if Δ(k) = 0 for some k = k * .For a fixed set of couplings {, , Γ, Γ ′ }, the analysis in the previous two paragraphs implies, within a stable polarized state, Corollary 1.If the band gap is zero, then it remains zero as ℎ varies; Corollary 2. If the band gap is finite, then it remains finite as ℎ varies, unless ℎ −→ ∞ while the couplings stay finite.
To see why these observations are useful, we first note that the Chern number is a topological invariant that can never change as long as a finite band gap is maintained.A topological phase transition can only occur when the band gap vanishes.Therefore, within a stable in-plane field polarized state and for a finite ℎ, Corollaries 1 and 2 respectively imply Lemma 1.If a topological transition exists, the phase boundary, which must be a parameter region where the band gap goes to zero, is independent of the field strength; Lemma 2. The Chern number of each magnon band, which is well defined when the band gap is finite, is independent of the field strength.
We are now ready to solve analytically for regions in the ΓΓ ′ parameter space where the band gap vanishes, which are potential phase boundaries for topological transitions. = 0.With  4 = 0 and  6 = 0, (2b) reads The band gap is zero if and only if there is at least one k such that Δ(k) = 0.According to Proposition 1, we require which makes the field dependent part of Δ 2 (k) zero.The field independent part should be zero as well.We solve these conditions on a case by case basis.

S2. COMPUTATION OF CHERN NUMBER
The Berry curvature of the  th magnon band at the momentum k is defined in terms of the Bogoliubov transformation  k as [82] where  is the totally antisymmetric tensor and ,  ∈ {, }.(Caution: The expression within the brackets on the right hand side is a matrix, and the subscript  means the entry at the  th row and the  th column;  is not a dummy index that is being summed over.)Integrating (S18) over the magnetic Brillouin zone gives the Chern number of the  th band, This section explains the method that we use to compute the Chern number in a discretized Brillouin zone, which was introduced in Ref. [68] and based on a  (1) lattice gauge theory (see also Refs.[53,69]).The Berry curvature (S18) multiplied by the integral measure, Ω k d  d  , is invariant under a coordinate transformation [67], e.g.
(More formally, the differential 2-form Ω k d  ∧ d  is coordinate independent.)Let N be the number of sites per magnetic unit cell, in particular N = 2 for the polarized state.If there is a finite magnon pairing term, then the 2N dimensional Hamiltonian matrix H k has a particle-hole redundancy by construction.The columns of the 2N dimensional Bogoliubov transformation matrix  k are arranged such that the first (last) N columns belong to the particle (hole) sector.
Let |(k)⟩ ≡ (u  (k), v  (k)) be the  th vector of  k .We have introduced the N dimensional vector u  (k) [v  (k)] as the first [second] half of |(k)⟩.In the rest of this section, we focus on the particle sector, i.e., 1 ≤  ≤ N .Next, we define the Berry connection of the  th band at the momentum k as where  = 1, 2. Since ⟨(k)||(k)⟩ = 1, ⟨(k)|   |(k)⟩ is purely imaginary and hence  , is purely real.Using the definition (S20b), it can be straightforwardly verified that On the discretized Brillouin zone, suppose that the spacings of momenta along the  1 and  2 directions are  1 and  2 , respectively.If we make   small enough, we can approximate (S21) and (S22) as We define the  (1) link variable where, to obtain the second equality, we have expanded ⟩  and used the fact that ⟨(k)|   |(k)⟩ is imaginary.Eq. (S23b) can be expressed in terms of (S24) as Finally, the Chern number of the  th magnon band (S19) is calculated as The main advantage of using (S25) over (S20b) for computing Chern numbers is that the former is manifestly gauge invariant, i.e., it is unaffected by |(k)⟩ −→ exp[− (k)] |(k)⟩ as desired, while the latter requires explicit gauge fixings when taking the differences of  k to approximate the derivatives.
We mention in passing that the thermal Hall conductivity (5) can also be calculated within this framework, with   (k) given in (S25).

S3. SYMMETRIES
In this section, we discuss how topological phase diagrams for different in-plane field directions are related by symmetries of the ΓΓ ′ model, which, in the absence of an external magnetic field, includes a time reversal T symmetry, a  3 symmetry about the  axis, and a  2 symmetry about the  axis.Let J = (, , Γ, Γ ′ ).The Hamiltonian matrix (S1a) is a function of the parameter J, the field h, and the momentum k, so we write it as H k (J, ℎ, ).
Consider the action of time reversal T , i.e., h −→ −h or, specifically for in-plane fields,  −→  + , at a fixed parameter J.
We have H k (J, ℎ,  + ) = H * −k (J, ℎ, ) by a theorem proved in Ref. [41].Therefore, T preserves the band gap (which can be either zero or finite) but flips the sign of the Chern number.
Finally, consider flipping the signs of all couplings, i.e., J −→ −J, at a fixed field direction .Following Ref. [52], we introduce where 1 is the two dimensional identity matrix and  1,2,3 are the Pauli matrices.We then carry out the transformation The matrix  is unitary, so it preserves the hermicity of H k (J, ℎ, ).In addition,  preserves the bosonic commutation relation, i.e., Ψk obeys the same commutation rule as Ψ k , which can be seen from Importantly, it can be shown that We can always choose a sufficiently large ℎ such that both (−J, ℎ, ) and (J, ℎ + 2 1 , ) yield a stable polarized state.Let  k (J, ℎ, ) and Tk (J, ℎ, ) be the Bogoliubov transformations that diagonalize H k (J, ℎ, ) and Hk (J, ℎ, ), respectively, which are related by  k (−J, ℎ, ) = T−k (J, ℎ + 2 1 , ).If the band gap is finite at the parameter J, by (S18) and (S19), we have Ω k (−J, ℎ, ) = Ω−k (J, ℎ + 2 1 , ) and   (−J, ℎ, ) =   (J, ℎ + 2 1 , ) for the Berry curvatures and the Chern numbers.By Corollary 2, the Chern number of the  th magnon band in the polarized state at −J is same as that at J. On the other hand, if the band gap vanishes at J, then it also vanishes at −J by Corollary 1.

S4. SCHRIEFFER-WOLFF TRANSFORMATION
When the field strength far exceeds the interaction energy scale, we have  (k) ∼ ℎ 2 and Δ(k) ∼ ℎ, see (2a) and (2b).The Schrieffer-Wolff transformation discussed in the main text is given by [52] which eliminates magnon pairings up to  (1/ℎ), and absorbs their effects in a pure hopping model.From we obtain The explicit expressions of  A ,  B , and  mix are omitted here for simplicity; we merely note that  A ∼ ℎ 2 depends only on the matrix elements of A eff k ,  B ∼ 1/ℎ 2 only on those of B eff k , and  mix on both.Eq. ( S33) is solved by One finds by explicit calculation that the contribution of B eff k to the square root in (S34) is at best  ( √ ℎ): it can be of higher order, but not lower.From  eff ± ∼ √  2 , one can perform a large ℎ expansion and deduce that B eff k only contributes at  (1/ℎ 3/2 ) and beyond to  eff k .Discarding B eff k is thus justified, and we are left with a pure hopping Hamiltonian A eff k .The magnon excitation energies calculated from A eff k are equal to those calculated from H eff k up to  (1/ℎ).
Since the effective Hamiltonian A eff k is a 2 × 2 hermitian matrix, it can be expressed as where the explicit expression of  0 (k) is omitted as it does not play a role in the band topology.When d(k) ≠ 0, the Chern number of the lower magnon band is given by (3) in the main text.If one of the components of d(k) vanishes throughout the Brillouin zone, then the triple product on the right hand side of (3) is identically zero, and consequently the Chern number is zero as well.This provides a sufficient condition for topologically trivial magnons.
The parameter regions with  = 0 in our phase diagrams Figs.3a-3e can now be understood as being related to the vanishing of  3 (k).We first note from (2c) that both  6 and  7 contain the factor  − Γ + Γ ′ .Therefore, if  − Γ + Γ ′ = 0, then  6 = 0 and  7 = 0, which in turn imply  3 (k) = 0 for all k by (S35d).The rest of the argument is contained in the main text.
Let  be the total area of the parameter regions on the spherical surface  2 + Γ 2 + Γ ′2 = 1 with topological magnons.Fig. S2 shows the dependence of  on the field angle .One finds that  =  max is maximal at  = 0.As  increases,  first decreases and reaches a local minimum at  = /12, then increases again and approaches  max as  −→ /6.There is a discontinuity at  = /6 as  is forced to 0 by symmetry.

S5. THIRD NEAREST NEIGHBOR HEISENBERG INTERACTION
Some of the proposed parametrizations for Kitaev magnets in the literature contain a non-negligible third nearest neighbor Heisenberg interaction,  3 ⟨ ⟨ ⟨  ⟩ ⟩ ⟩ S  • S  , see Refs.[85,86] for example.In this section, we add a finite  3 term to the ΓΓ ′ model ( 1) under an in-plane magnetic field, perform a linear spin wave analysis of the polarized state, and explore its implications on the magnon band topology.
We first define  1 = 3 +  − Γ − 2Γ ′ + 3 3 ,  8 = 2 3 , and  k = exp[( 2 −  1 )] + exp[( 1 −  2 )] + exp[( 1 +  2 )], while retaining the same definitions of  2 , . . .,  7 and  k ,  k as before, see (2c) and the succeeding sentence.Then, the Hamiltonian matrix H k assumes the form (S1a) with and B k same as in (S1c).The magnon dispersion is given by where Making use of the fact that both  k and  k are zero at k = K, K ′ , we can follow the same reasoning as in Sec.S1 to show that magnon stability is only consistent with ℎ −  1 > 0. We then arrive at Proposition 1, which is modified in the presence of the third nearest neighbor Heisenberg interaction as follows: Under the stability requirement ℎ −  1 > 0,  2  k +  8  k +  3 +  4  k = 0 is a necessary condition for Δ(k) = 0. Within a stable polarized state, for a fixed set of couplings {, , Γ, Γ ′ ,  3 }, the modified Proposition 1 implies Corollaries 1 and 2, which in turn imply Lemmas 1 and 2, as in Sec.S1.In other words, upon including  3 , we still have the general property that the topological phase diagrams are independent of the field strength.However, solving for the critical regions where the band gap vanishes is no longer analytically tractable for generic model parameters and field angles.
Below, we provide the details of determining the critical regions in the aforementioned parameter space.Let Δ(k) = 0. Proposition 1 implies that the field dependent and independent parts of (S37b) are both zero, leading to the conditions Substituting  3 = −1/2 and  5 = 5/6 in (S39), writing down the real and imaginary parts of (S38a) separately, we obtain the following set of equations, 4(sin We solve (S40a)-(S40c) on a case by case basis.
Collecting all the results, for the specfic parameter space and field angle under consideration, the critical regions satisfy one of the following equations: (I) and (VI) 4 2 − 5 8 = 0 if 40 2 − 114 8 ≠ 0 and (S44) holds.We can then compute the Chern numbers at parameters away from (I-VI) with the method described in Sec.S2.

S6. BEYOND LINEAR SPIN WAVE THEORY
Our main results in this work are derived from linear spin wave theory (LSWT), which yields well defined band structures comprising single-magnon states.This allows us to study the magnon band topology for generic model parameters and field angles, and relate it to the thermal Hall conductivity given by the formula (5), which itself is also derived under the assumption of a non-interacting, quadratic Hamiltonian.When the system is near the phase transition between the high-field polarized state and some other low-field state, quantum fluctuations may be strong such that LSWT breaks down and one has to deal with magnon interactions.While addressing the interacting regime in full is beyond the scope of this work, we state that LSWT is nonetheless a valid approximation at sufficiently high fields where the single-magnon dispersion do not overlap with the multi-magnon continuum [64], in particular the two-magnon continuum.We demonstrate this expectation by treating higher order quantum fluctuations perturbatively via nonlinear spin wave theory (NLSWT), where we perform the Holstein-Primakoff expansion of the Hamiltonian beyond the harmonic level,  =  (0) +  (2) +  (3) +  (4) + . . .

(S45)
Upon factoring out  2 ,  (0) ∼ 1 is the classical energy,  (2) ∼ 1/ is the linear spin wave Hamiltonian, and  (3) ∼ 1/ 3/2 and  (4) ∼ 1/ 2 are the cubic and quartic anharmonicities.A useful quantity to study how the magnon spectrum is affected by interactions is the spectral function (k, ), which is proportional to the imaginary part of the Green's function  (k, ).We calculate self-energy corrections to  (k, ) up to O (1/ 2 ), and plot (k, ) for selected parametrizations under a magnetic field along the  axis, with the coupling constants scaled such that  = −1 as in Fig. 4a of the main text.For the parametrization  at ℎ = 1, part of the upper band is severed from the rest and broadened by the two-magnon continuum, see Fig. S4d.Increasing the field to ℎ = 2, we recover well defined single-magnon energies, though they are slightly renormalized, see Fig. S4f.The corresponding results from LSWT are also presented for comparison, see Figs.S4c and S4e.More remarkably, the connectivity of the upper and lower bulk bands in a slab geometry as calculated from NLSWT (lower panel of Fig. S5) is consistent with the presence or absence of chiral edge modes, which signify nontrivial or trivial band topology, as calculated from LSWT (upper panel of Fig. S5), at least in the high field regime.For the parametrizations  and  where LSWT yields topological magnons, there exist energy states that run continuously from the lower to upper bulk bands in NLSWT, see Figs.S5b, S5d, and S5f.For the parametrization  where LSWT yields trivial magnons, the upper and lower bulk bands are disconnected from each other in NLSWT, see Fig. S5h.Note that despite the partial overlap between the upper bulk bands and the two magnon continuum at high energies in Fig. S5b, the connection survives.
Below, we sketch the NLSWT calculations, noting that detailed and extensive accounts can be found in the existing literature, see Refs.[87-89, 93, 94] for example.Our goal is to evaluate the diagonal part of the (retarded) Green's function at zero temperature.Including self-energy corrections up to O (1/ 2 ), we have [89,93] where  = 1, 2 is the band index (not to be confused with the candidate parametrization),  k is the single-magnon dispersion from LSWT [which was previously denoted as  ± (k)], and Σ  (k, ) is the diagonal part of the self-energy.The corresponding spectral function is given by and we define (k, ) =    (k, ).We remark that while our discussion here assumes a genuine two-dimensional system with periodic boundary conditions in both directions, it can be straightforwardly extended to the slab geometry with open boundary condition in one direction and periodic boundary condition in the other.

FIG. 1 .
FIG. 1.(a) Majorana spectrum of Kitaev honeycomb model in a perturbative magnetic field.(b) For the non-Abelian KSL, the Chern number of the lower Majorana band depends on the field direction through  = sgn(ℎ  ℎ  ℎ  ).Red (blue) areas indicate  = +1 (−1), while black curves indicate the vanishing of the band gap.(c) Magnon spectrum of the polarized state in a realistic spin model (1) of Kitaev magnets under a magnetic field.(d) For the in-plane field polarized state, we find a nontrivial variation of the magnon Chern number over the parameter space and the field angle, see Figs. 3a-3f.

FIG. 2 .
FIG. 2. (a) The three bond types , , and  in Kitaev magnets, the in-plane crystallographic axes  and , and the primitive lattice vectors a 1 and a 2 .An external magnetic field h is applied in-plane at the azimuthal angle .(b) The 3 point group of the ΓΓ ′ model.If h transforms under a symmetry element that maps a filled circle to an empty circle or vice versa, then the Chern number  flips sign.If one circle is mapped to another of the same type, then  remains invariant.

Fig. 2b .
Fig. 2b.On the other hand, flipping the signs of all couplings leaves  invariant [65].For visualizations, we set  = 0 and calculate  over the spherical parameter space defined by  2 + Γ 2 + Γ ′2 = 1, at the field angles  = 0, /24, /12, /8, /6, see Figs. 3a-3f[72].We make two observations, with the understanding that all angles mentioned below are defined modulo /3.First, for  ≠ /6, there exist both parameter regions with topological magnons and those without.For  = /6, topological magnons are altogether forbidden due to a  2 symmetry[40,73].Second, the total area  of the parameter regions with  = ±1 is maximal at  = 0, which implies that, for a Kitaev magnet dominated by nearest neighbor anisotropic interactions, topological magnons are most likely found when the in-plane field is along the  axis[74].To understand why magnons are topologically trivial in certain parameter regions, we analyze the linear spin wave theory at high fields by systematically integrating out the pairing terms[52].This is achieved via a Schrieffer-Wolff transformation[75], from which we obtain an effective hopping model of the form H eff (k) =  0 (k)1 2×2 + d(k) • σ.The band gap vanishes iff d(k) = 0.When d(k) ≠ 0, the Chern number of the lower band is given by the winding number of the map d(k) ≡ d(k)/|d(k)| from the Brillouin zone to a sphere [66], Fig. 2b.On the other hand, flipping the signs of all couplings leaves  invariant [65].For visualizations, we set  = 0 and calculate  over the spherical parameter space defined by  2 + Γ 2 + Γ ′2 = 1, at the field angles  = 0, /24, /12, /8, /6, see Figs. 3a-3f[72].We make two observations, with the understanding that all angles mentioned below are defined modulo /3.First, for  ≠ /6, there exist both parameter regions with topological magnons and those without.For  = /6, topological magnons are altogether forbidden due to a  2 symmetry[40,73].Second, the total area  of the parameter regions with  = ±1 is maximal at  = 0, which implies that, for a Kitaev magnet dominated by nearest neighbor anisotropic interactions, topological magnons are most likely found when the in-plane field is along the  axis[74].To understand why magnons are topologically trivial in certain parameter regions, we analyze the linear spin wave theory at high fields by systematically integrating out the pairing terms[52].This is achieved via a Schrieffer-Wolff transformation[75], from which we obtain an effective hopping model of the form H eff (k) =  0 (k)1 2×2 + d(k) • σ.The band gap vanishes iff d(k) = 0.When d(k) ≠ 0, the Chern number of the lower band is given by the winding number of the map d(k) ≡ d(k)/|d(k)| from the Brillouin zone to a sphere [66],

FIG. S2 .
FIG.S2.The total area  of the parameter regions with topological magnons on the ΓΓ ′ sphere, as a function of the field angle , over the range 0 ≤  ≤ /3.There is a discontinuity at  = /6 (indicated by the empty circle) where  = 0 is enforced by a  2 symmetry, while large swathes of the parameter space become critical, see Fig.3fin the main text.