# Topological phase diagrams of in-plane field polarized Kitaev magnets

Li Ern Chern  T.C.M. Group, Cavendish Laboratory, University of Cambridge,
Cambridge CB3 0HE, United Kingdom  Claudio Castelnovo  T.C.M. Group, Cavendish
Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom

###### Abstract

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 a three-fold 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.

###### pacs:

Introduction.—Kitaev spin liquid (KSL) is the ground state of an exactly
soluble spin model on a honeycomb lattice with bond dependent Ising
interactions [1]. It features Majorana fermions and $\mathbb{Z}_{2}$ gauge
fields, both of which originate from fractionalizations of $S=1/2$ degrees of
freedom. Without an external magnetic field, the spectrum of Majorana fermions
is gapless at the $\mathrm{K}$ and $\mathrm{K}^{\prime}$ points, forming conic
singularities similar to Dirac cones in the electronic dispersion of graphene.
By applying a perturbative field $\mathbf{h}=(h_{x},h_{y},h_{z})$, the
spectrum acquires a gap $\Delta\sim\lvert h_{x}h_{y}h_{z}\rvert/K^{2}$, and
the lower band a Chern number $\nu=\mathrm{sgn}(h_{x}h_{y}h_{z})$, see Figs.
1a and 1b. The resulting non-Abelian KSL is predicted to exhibit a half
quantized thermal Hall conductivity $\kappa_{xy}^{\mathrm{2D}}/T=(\nu/2)(\pi
k_{\mathrm{B}}^{2}/6\hbar)$ due to chiral Majorana edge modes
111$\kappa_{xy}^{\mathrm{2D}}\equiv\kappa_{xy}d$ where $d$ is the interlayer
distance between the honeycomb planes in a three dimensional structure..

(a)

(b)

(c)

(d)

Figure 1: (a) Majorana spectrum of the ferromagnetic Kitaev model subjected to
a perturbative magnetic field, with $K=-1$ and $\lvert
h_{x}h_{y}h_{z}/K^{3}\rvert\sim 0.01$. (b) For the non-Abelian Kitaev spin
liquid, the Chern number of the lower Majorana band depends on the field
direction through $\nu=\mathrm{sgn}(h_{x}h_{y}h_{z})$. Red (blue) areas
indicate $\nu=+1$ ($-1$), while black curves indicate the vanishing of the
band gap. (c) Magnon spectrum of the polarized state in a realistic nearest
neighbor spin model (1) of the Kitaev magnet $\alpha$-RuCl3 under an in-plane
magnetic field along the $a$ axis, with
$(J,K,\Gamma,\Gamma^{\prime})=(-1/8,-1,+1/2,-1/8)$ and $h=0.2$. The lower
(upper) magnon band carries Chern number $+1$ ($-1$). (d) For the polarized
state under an in-plane magnetic field, we find nontrivial patterns traced out
by the variation of the magnon Chern number over the space of couplings and
the field angle, see Figs. 3a-3f.

In recent years, there have been tremendous efforts in the search for KSL in
candidate materials, ranging from iridates [3, 4, 5] to ruthenium halides [6,
7, 8] to cobaltates [9, 10]. These so called Kitaev magnets [11, 12, 13, 14],
which have partially filled $d$ orbitals and a significant spin orbit
coupling, may realize a dominant Kitaev interaction $K$ via Jackeli-Khaliullin
[15] or related [16, 17, 18] mechanism. Here, we focus on arguably the most
popular amongst them, $\alpha$-RuCl3 [19]. Like most Kitaev magnets, the zero
field ground state of $\alpha$-RuCl3 is a zigzag (ZZ) magnetic order [20, 21],
due to the presence of other symmetry allowed interactions than $K$ [22].
However, an external magnetic field is found to suppress the ZZ order and
promote a disordered phase, which may be the non-Abelian KSL, as evidenced by
the half integer thermal Hall effect reported in Refs. [23, 24, 25, 26, 27].
Since the field-temperature window of the half quantization is rather small,
it is suggested that the field angle dependence of thermal Hall conductivity
[25] or heat capacity [28, 29] can lend further support for the case of non-
Abelian KSL. Meanwhile, other experiments [30, 31] report that the thermal
Hall conductivity in the field induced phase behaves rather like a smooth
function without any plateau, and it decreases rapidly as the temperature
approaches zero, which suggest emergent heat carriers of a bosonic nature.

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. In this case, the collective excitations are
magnons, which can give rise to experimentally measurable transport signals at
finite temperatures. 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 $\kappa_{xy}$
at low temperatures is not directly proportional to $\nu$ (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] will be valuable
for identifying topological magnons and interpreting thermal transport
measurements at high fields, see 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 $\alpha$-RuCl3 reaches the polarized state
more easily under in-plane fields than an out-of-plane field, likely due to an
anisotropic $g$ tensor [46, 47, 48, 49] and a positive $\Gamma$ interaction
[50], which discounts the out-of-plane field strength and disfavors an out-of-
plane magnetization, respectively [41, 51]. Also, the case of subjecting
Kitaev magnets with nearest neighbor interactions to strong out-of-plane
fields, which preserve the $C_{3}$ symmetry, has been studied in the
pioneering work Ref. [52] (see also Ref. [53]). It is found that, for the
linear spin wave theory of the polarized state, $\Gamma$ and $\Gamma^{\prime}$
can be absorbed into $J$, $K$, and $h$, which effectively reduces the
$JK\Gamma\Gamma^{\prime}$ model to a $JK$ model and greatly simplifies the
problem. The $C_{3}$ symmetry also plays an important role in the diagnosis of
magnon band topology based on topological quantum chemistry [54, 55], where
one examines if the elementary band representation splits into disconnected
bands. As demonstrated in Ref. [56], the magnon bands must be topological
whenever a gap exists in between, which is consistent with the results from
explicit calculations [52]. These make the out-of-plane field rather special.

In this Letter, we consider the nearest neighbor $JK\Gamma\Gamma^{\prime}$
model under sufficiently high in-plane magnetic fields, such that the
corresponding polarized states are stable. Unlike the case of an out-of-plane
field, the model has to be taken in full as none of the parameters can be made
redundant. To map out the phase diagrams of topological magnons, we first
solve analytically for the parameter regions where the gap between the two
bands closes, which are potential phase boundaries for topological
transitions, at generic in-plane field directions. For ease of visualizations,
we then set $J=0$, compute the Chern number $\nu$ where it is well defined,
and plot the results at several field angles. As long as the field is not
along the armchair direction, we always find parameter regions that are
topological ($\nu=\pm 1$) as well as those that are trivial ($\nu=0$). If
instead the field is along the armchair direction, then all gapped parameter
regions are topologically trivial due to a $C_{2}$ symmetry [40]. Rotating the
field in the honeycomb plane, we find that the total area of parameter regions
with topological magnons is the largest when the field is along the zigzag
direction. We also provide an understanding as to why certain parameter
regions are topologically trivial, via an effective Hamiltonian at high fields
[52]. Finally, we discuss the implications of our results to thermal Hall
conductivities and model parameters of Kitaev magnets, which will provide
helpful guidances to related experiments. Detailed calculations are presented
in Supplemental Material [57].

Linear spin wave theory.—The most generic nearest neighbor spin model of
Kitaev magnets is the $JK\Gamma\Gamma^{\prime}$ model [22]. In an external
magnetic field $\mathbf{h}$, it reads

$\displaystyle H=$ $\displaystyle\sum_{\lambda=x,y,z}\sum_{\langle
ij\rangle\in\lambda}\big{[}J\mathbf{S}_{i}\cdot\mathbf{S}_{j}+KS_{i}^{\lambda}S_{j}^{\lambda}+\Gamma(S_{i}^{\mu}S_{j}^{\nu}+S_{i}^{\nu}S_{j}^{\mu})$
(1)
$\displaystyle+\Gamma^{\prime}(S_{i}^{\mu}S_{j}^{\lambda}+S_{i}^{\lambda}S_{j}^{\mu}+S_{i}^{\nu}S_{j}^{\lambda}+S_{i}^{\lambda}S_{j}^{\nu})\big{]}-\sum_{i}\mathbf{h}\cdot\mathbf{S}_{i},$

where $(\lambda,\mu,\nu)$ is a cyclic permutation of $(x,y,z)$. For
convenience of analysis, we measure the field strength
$\lvert\mathbf{h}\rvert\equiv hS$ in portions of the spin magnitude
$S\equiv\lvert\mathbf{S}_{i}\rvert$. We also take $S$ to be dimensionless so
that $J,K,\Gamma,\Gamma^{\prime}$ and $h$ have units of energy. An in-plane
field can be parametrized as $(h_{a},h_{b},h_{c})=h(\cos\beta,\sin\beta,0)$,
where $\beta\in[0,2\pi)$ is the azimuthal angle in the honeycomb plane, with
$\beta=0$ ($\pi/2$) corresponding to the $a$ ($b$) direction 222The
crystallographic $abc$ axes, which distinguish the in-plane and out-of-plane
directions, and the cubic $xyz$ axes, according to which the spin components
in (1) are defined, are related by $a\parallel[11\bar{2}]$,
$b\parallel[\bar{1}10]$, and $c\parallel[111]$., see Fig. 2a. We apply the
linear spin wave theory [59, 60] to the in-plane field polarized state of (1),
and obtain an analytical expression of the magnon spectrum
$\omega_{\pm}(\mathbf{k})=S\sqrt{E(\mathbf{k})\pm\Delta(\mathbf{k})}/2$, where

$\displaystyle
E(\mathbf{k})=4(h-c_{1})^{2}+\big{\lvert}c_{2}f_{\mathbf{k}}+c_{4}g_{\mathbf{k}}\big{\rvert}^{2}-\big{\lvert}c_{5}f_{\mathbf{k}}-c_{4}g_{\mathbf{k}}\big{\rvert}^{2}-4\big{\lvert}c_{6}(f_{\mathbf{k}}-3)+c_{7}g_{\mathbf{k}}\big{\rvert}^{2}+c_{3}(c_{2}+c_{5})(f_{\mathbf{k}}+f_{\mathbf{k}}^{*}),$
(2a)
$\displaystyle\begin{split}&\Delta^{2}(\mathbf{k})=16(h-c_{1})^{2}\big{\lvert}c_{2}f_{\mathbf{k}}+c_{3}+c_{4}g_{\mathbf{k}}\big{\rvert}^{2}+(c_{2}+c_{5})\Big{\\{}(c_{2}+c_{5})\big{[}f_{\mathbf{k}}(c_{3}+c_{4}g_{\mathbf{k}}^{*})-f_{\mathbf{k}}^{*}(c_{3}+c_{4}g_{\mathbf{k}})\big{]}^{2}\\\
&\qquad\quad\;\>+8\big{[}f_{\mathbf{k}}(3c_{6}-c_{7}g_{\mathbf{k}}^{*})-f_{\mathbf{k}}^{*}(3c_{6}-c_{7}g_{\mathbf{k}})\big{]}\big{[}c_{6}f_{\mathbf{k}}(c_{3}+c_{4}g_{\mathbf{k}}^{*})-c_{6}f_{\mathbf{k}}^{*}(c_{3}+c_{4}g_{\mathbf{k}})+(g_{\mathbf{k}}-g_{\mathbf{k}}^{*})(c_{3}c_{7}+3c_{4}c_{6})\big{]}\\\
&\qquad\quad\;\>+4(c_{2}-c_{5})\big{[}f_{\mathbf{k}}(3c_{6}-c_{7}g_{\mathbf{k}}^{*})-f_{\mathbf{k}}^{*}(3c_{6}-c_{7}g_{\mathbf{k}})\big{]}^{2}\Big{\\}},\end{split}$
(2b)
$\displaystyle\begin{split}&c_{1}=3J+K-\Gamma-2\Gamma^{\prime},c_{2}=\frac{1}{6}\left[12J+4K+2\Gamma+4\Gamma^{\prime}+(K+2\Gamma-2\Gamma^{\prime})\cos(2\beta)\right],c_{3}=-\frac{\cos(2\beta)}{2}\left(K+2\Gamma-2\Gamma^{\prime}\right),\\\
&c_{4}=\frac{\sin(2\beta)}{2\sqrt{3}}\left(K+2\Gamma-2\Gamma^{\prime}\right),c_{5}=\Gamma+2\Gamma^{\prime}-\frac{\cos(2\beta)}{6}\left(K+2\Gamma-2\Gamma^{\prime}\right),c_{6}=\frac{\sin\beta}{3\sqrt{2}}\left(K-\Gamma+\Gamma^{\prime}\right),c_{7}=\frac{\cos\beta}{\sqrt{6}}\left(K-\Gamma+\Gamma^{\prime}\right),\end{split}$
(2c)

$f_{\mathbf{k}}=1+\exp(ik_{1})+\exp(ik_{2})$,
$g_{\mathbf{k}}=\exp(ik_{1})-\exp(ik_{2})$, and $k_{1},k_{2}\in[0,2\pi)$ are
components of the crystal momentum defined according to $\mathbf{a}_{1}$ and
$\mathbf{a}_{2}$ in Fig. 2a 333Ref. [19] provides a coarser expression for the
magnon spectrum under $\mathbf{h}\parallel a$ and $b$, which does not fit our
purpose of identifying phase boundaries of topological transitions.. Let
$\Delta(\mathbf{k})=\sqrt{\Delta^{2}(\mathbf{k})}\geq 0$, so that
$\omega_{-}(\mathbf{k})$ [$\omega_{+}(\mathbf{k})$] corresponds to the lower
(upper) band. As a clarification, we refer to the gap between the upper and
lower bands,
$\min_{\mathbf{k}}[\omega_{+}(\mathbf{k})-\omega_{-}(\mathbf{k})]\geq 0$, as
the band gap, which is not to be confused with the excitation gap,
$\min_{\mathbf{k}}\omega_{-}(\mathbf{k})>0$. Chern number is a topological
invariant that can never change as long as a finite band gap is maintained
[62], i.e., a topological phase transition can only occur when the band gap
vanishes, which happens if and only if $\Delta(\mathbf{k})=0$ for some
$\mathbf{k}=\mathbf{k}_{*}$.

(a)

(b)

Figure 2: (a) The three types of nearest neighbor bonds, $x$, $y$, and $z$, of Kitaev magnets are indicated by red, green, and blue colors, respectively. The in-plane crystallographic axes $a$ and $b$ are along the zigzag and armchair directions, respectively. The out-of-plane $c$ axis is not shown here. An external magnetic field $\mathbf{h}$ is applied in-plane at the azimuthal angle $\beta$. The choices of primitive lattice vectors $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are also indicated. (b) The point group of the $JK\Gamma\Gamma^{\prime}$ model is $\bar{3}m$ or $D_{3d}$. This diagram also represents how the magnon Chern number $\nu$ changes as $\mathbf{h}$ transforms under elements of $\bar{3}m$ and time reversal $\mathcal{T}$. If a filled circle is mapped to an empty circle or vice versa, then $\nu$ flips sign. If a circle is mapped to another circle of the same type, then $\nu$ remains invariant. Due to symmetry, it is sufficient to study field angles in the range $0\leq\beta\leq\pi/6$. Table 1: For field angles $\beta\neq\pi/6\,(\mathrm{mod}\,\pi/3)$, the band gap closes if the parameters of the $JK\Gamma\Gamma^{\prime}$ model satisfy any of the following equations. They define the phase boundaries of topological transitions. For field angles $\beta=\pi/6\,(\mathrm{mod}\,\pi/3)$, the band gap is zero whenever (I) or (4) is satisfied. I | $K+2\Gamma-2\Gamma^{\prime}=0$
---|---
II | $6J+2K+\Gamma+2\Gamma^{\prime}=0$
III | $6J+2K+\Gamma+2\Gamma^{\prime}+2(K+2\Gamma-2\Gamma^{\prime})\cos(2\beta)=0$
IV | $6J+2K+\Gamma+2\Gamma^{\prime}-2(K+2\Gamma-2\Gamma^{\prime})\cos(2\beta+\pi/3)=0$
V | $6J+2K+\Gamma+2\Gamma^{\prime}-2(K+2\Gamma-2\Gamma^{\prime})\cos(2\beta-\pi/3)=0$
VI |  $3J+K+2\Gamma+4\Gamma^{\prime}=0$ if (4) holds
VII |  $K-\Gamma+\Gamma^{\prime}=0$ if (4) holds

We assume a stable polarized state, where the excitation gap increases with
$h$, so that the system becomes more stable as $h$ grows, rather than
undergoing a magnon instability where the excitation gap goes to zero. As
demonstrated in Supplemental Material [57], this requires $h>c_{1}$, from
which we deduce the following. For a given set of parameters
$\\{J,K,\Gamma,\Gamma^{\prime}\\}$, (i) if the band gap is zero, then it
remains zero as $h$ varies, and (ii) if the band gap is finite, then it
remains finite as $h$ varies, unless $h\longrightarrow\infty$ while the
couplings stay finite. These imply that 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 (LABEL:Dksquaredefine), which
define the boundaries, then choose a sufficiently high field to compute the
Chern numbers [63, 64, 65, 66] for the parameters away from these zeros.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Topological phase diagrams, which indicate the Chern number $\nu$ of
the lower magnon band, of the in-plane field polarized states, over the space
of couplings parametrized by
$(J,K,\Gamma,\Gamma^{\prime})=(0,\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi)$,
at field angles $\beta$ equal to (a,b) $0$, (c) $\pi/24$, (d) $\pi/12$, (e)
$\pi/8$, and (f) $\pi/6$. Red, white, and blue areas indicate $\nu=+1$, $0$,
and $-1$, respectively, while black curves or areas indicate the vanishing of
the band gap. The latter defines the phase boundaries, which are labeled by
roman numerals as in Table 1. Grey dashed lines indicate latitudes at selected
zenith angles $\theta$. In each diagram, the center is either the $K>0$
($\theta=0$) limit, as in (a), or the $K<0$ ($\theta=\pi$) limit, as in (b-f),
while the left, right, top, and bottom ends on the equator $\theta=\pi/2$ are
the $\Gamma>0$ ($\phi=0$), $\Gamma<0$ ($\phi=\pi$), $\Gamma^{\prime}>0$
($\phi=\pi/2$), and $\Gamma^{\prime}<0$ ($\phi=3\pi/2$) limits, respectively.

Topological phase diagrams.—At finite in-plane fields, the band gap closes if
and only if the set of parameters $\\{J,K,\Gamma,\Gamma^{\prime}\\}$ meets any
of the criteria listed in Table 1. For parameters with a finite band gap, let
the Chern number of the lower (upper) band be $\nu$ ($-\nu$). From the
symmetries of the $JK\Gamma\Gamma^{\prime}$ model, one can relate the phase
diagrams at different field angles by the following rules [57]. Fixing the
couplings, (i) $\nu\longrightarrow\nu$ if $\mathbf{h}$ is rotated by $2\pi/3$
about the $c$ axis, (ii) $\nu\longrightarrow-\nu$ if $\mathbf{h}$ is rotated
by $\pi$ about the $b$ axis, and (iii) $\nu\longrightarrow-\nu$ if
$\mathbf{h}\longrightarrow-\mathbf{h}$ is reversed [41], while the phase
boundaries are invariant under each of these actions. In other words, the
Chern number transforms according to the $A_{2g}$ representation of the point
group $\bar{3}m$ [67, 68], and flips sign under time reversal $\mathcal{T}$,
as in the case of non-Abelian KSL [28]. Hence, $\beta\in[0,\pi/6]$ serves as
an independent unit, to which all other angles can be related by symmetries,
see Fig. 2b. On the other hand, flipping the signs of all couplings leaves the
Chern number invariant [57].

Figure 4: The total area $A$ of parameter regions with topological magnons on
the $K\Gamma\Gamma^{\prime}$ sphere, as a function of the field angle $\beta$,
over the range $0\leq\beta\leq\pi/3$. There is a discontinuity at
$\beta=\pi/6$ (indicated by the empty circle) as $A=0$ is enforced by a
$C_{2}$ symmetry, while large swathes of the parameter space become critical,
see Fig. 3f.

For visualizations, we set $J=0$, and calculate $\nu$ over the spherical
parameter space defined by $K^{2}+\Gamma^{2}+{\Gamma^{\prime}}^{2}=1$, at the
field angles $\beta=0,\pi/24,\pi/12,\pi/8,\pi/6$. The results are plotted
444We have used the stereographic projection $(\Gamma/(1\pm
K),\Gamma^{\prime}/(1\pm K))$, where the plus (minus) sign is for $K>0$
($K<0$). in Figs. 3a-3f, mainly on the hemisphere with $K<0$ relevant to
Kitaev magnets, while that with $K>0$ can be related by
$(\theta,\phi)\longrightarrow(\pi-\theta,\pi+\phi)$ as discussed in the
previous paragraph. We make two observations, with the understanding that all
angles mentioned below are defined modulo $\pi/3$. First, there exist both
parameter regions with topological magnons and those without for
$\beta\neq\pi/6$, unlike the case of an out-of-plane field where topological
magnons exist throughout the parameter space whenever the band gap is finite,
due to the $C_{3}$ symmetry about the $c$ axis [52, 56]. On the other hand,
topological magnons are forbidden at $\beta=\pi/6$ due to the $C_{2}$ symmetry
about the $b$ axis [40, 70]. Second, the total area $A$ of parameter regions
with topological magnons is maximal at $\beta=0$. As $\beta$ increases, $A$
first decreases and reaches a local minimum at $\beta=\pi/12$, then increases
again and approaches $A_{\mathrm{max}}$ as $\beta\longrightarrow\pi/6$. There
is a discontinuity at $\beta=\pi/6$ as $A$ is forced to $0$ by symmetry. We
plot $A$ as a function of $\beta$ for the range $\beta\in[0,\pi/3]$ in Fig. 4.
The second observation implies that one is most likely to find topological
magnons in a Kitaev magnet dominated by nearest neighbor anisotropic
interactions when the in-plane field is along the $a$ axis.

Now, we attempt to understand why magnons are topologically trivial in certain
parameter regions for $\beta\neq\pi/6$. When the field strength far exceeds
the interaction energy scale, we have $E(\mathbf{k})\sim h^{2}$ and
$\Delta(\mathbf{k})\sim h$ for the magnon dispersion
$\omega_{\pm}(\mathbf{k})$, see (2a) and (LABEL:Dksquaredefine). Following
Ref. [52], we analyze the linear spin wave theory at high fields by
systematically integrating out the magnon pairing terms with $1/h$ as a small
parameter. This is achieved via a Schrieffer-Wolff transformation [71], from
which we obtain an effective hopping model of the form
$\mathcal{H}^{\mathrm{eff}}(\mathbf{k})=d_{0}(\mathbf{k})\mathbf{1}_{2\times
2}+\mathbf{d}(\mathbf{k})\cdot\bm{\sigma}$. The energy eigenvalues are
$\omega_{\pm}^{\mathrm{eff}}(\mathbf{k})=d_{0}(\mathbf{k})\pm\lvert\mathbf{d}(\mathbf{k})\rvert$,
so the gap vanishes if and only if $\mathbf{d}(\mathbf{k})=\mathbf{0}$. When
$\mathbf{d}(\mathbf{k})\neq\mathbf{0}$, the Chern number of the lower band is
given by the winding number of the map
$\hat{\mathbf{d}}(\mathbf{k})\equiv\mathbf{d}(\mathbf{k})/\lvert\mathbf{d}(\mathbf{k})\rvert$
from the Brillouin zone (a torus) to a sphere [63],

$\nu=\frac{1}{4\pi}\int_{\mathrm{FBZ}}\mathrm{d}^{2}\mathbf{k}\left[\hat{\mathbf{d}}(\mathbf{k})\cdot\frac{\partial\hat{\mathbf{d}}(\mathbf{k})}{\partial
k_{x}}\times\frac{\partial\hat{\mathbf{d}}(\mathbf{k})}{\partial
k_{y}}\right].$ (3)

One finds that the third component of $\mathbf{d}(\mathbf{k})$ vanishes
throughout the Brillouin zone when $K-\Gamma+\Gamma^{\prime}=0$ [57], which
defines the phase boundary (VII) within the parameter region

$\big{\lvert}\lvert c_{2}+c_{4}\rvert-\lvert
c_{2}-c_{4}\rvert\big{\rvert}\leq\lvert c_{2}+c_{3}\rvert\leq\lvert
c_{2}+c_{4}\rvert+\lvert c_{2}-c_{4}\lvert.$ (4)

On the other hand, there exist parameters outside (4) that satisfy
$K-\Gamma+\Gamma^{\prime}=0$ and at the same time possess a finite band gap
555One can imagine the extension of (VII) to the white areas in Figs. 3a-3e..
At these parameters, the triple product on the right hand side of (3) is thus
identically zero, and consequently $\nu=0$. Any other parameter that can be
continuously connected to these parameters without a gap closing must be
topologically trivial as well.

(a)

(b)

Figure 5: (a) Candidate parametrizations $p$ [73], $n$ [74], and $z$ [75] of
$\alpha$-RuCl3, with $K$ set to $-1$ and other interactions scaled
accordingly, and topological phase diagrams in their neighborhoods under a
magnetic field $\mathbf{h}$ along the $a$ axis ($\beta=0$). (b) Thermal Hall
conductivities of $p$, $n$, and $z$ due to magnons in the polarized state
under $\mathbf{h}\parallel a$, at field strengths $h$ starting from $0.18$,
$0.10$, and $0.56$, respectively, and increasing to $0.32$, $0.24$, and $0.70$
in steps of $0.02$. Lighter colors indicate higher fields. $S=1/2$ is used.

Thermal Hall effect.—We discuss how the topological phase diagrams relate to
experimentally measurable quantities by connecting the Chern number to the
thermal Hall conductivity 666We assume that the heat current is applied along
the $a$ axis, while the transverse temperature gradient is measured along the
$b$ axis, as in the experiments. $\kappa_{xy}$ should be more appropriately
called $\kappa_{ab}$, but we use the former notation in line with the existing
literature., which is given by [77, 78, 79]

$\kappa_{xy}=-\frac{k_{\mathrm{B}}^{2}T}{\hbar
V}\sum_{n=1}^{\mathcal{N}}\sum_{\mathbf{k}\in\mathrm{FBZ}}c_{2}\left[g\left(\frac{\hbar\omega_{n\mathbf{k}}}{k_{\mathrm{B}}T}\right)\right]\Omega_{n\mathbf{k}}$
(5)

for magnons, where $n$ is the band index ranging from $1$ to $\mathcal{N}$
($=2$ in our case), $c_{2}(x)=\int_{0}^{x}\mathrm{d}t\,\ln^{2}[(1+t)/t]$,
$g(x)=1/(e^{x}-1)$ is the Bose-Einstein distribution, and
$\Omega_{n\mathbf{k}}$ is the momentum space Berry curvature [57]. While the
Chern number $\nu_{n}$ is given by the summation of $\Omega_{n\mathbf{k}}$
over $\mathbf{k}$, $\kappa_{xy}$ is given by a weighted summation of
$\Omega_{n\mathbf{k}}$ with non-positive weights, as $c_{2}(x)\geq 0$ and
there is a minus sign in front. One also notes that $c_{2}(x)\longrightarrow
0$ as $x\longrightarrow 0$, which implies $\kappa_{xy}\longrightarrow 0$ as
$T\longrightarrow 0$, and the contribution of high energy magnons to
$\kappa_{xy}$ is less important than low energy ones for $T>0$. Therefore,
though $\kappa_{xy}$ is not directly proportional to $\nu$, one can very often
use the latter to infer the sign of the former at low temperatures where only
the lower band is thermally populated. More precisely, $\nu>0$ ($\nu<0$) means
that there is an excess of positive (negative) Berry curvatures in the lower
band, and by (5) the sign of $\kappa_{xy}$ is expected to be opposite to $\nu$
777There may be a situation where $\nu>0$ but $\Omega_{n\mathbf{k}}<0$ is
concentrated at the lowest energy magnons, so $\kappa_{xy}$ have the same sign
as $\nu$ in a small temperature window near $T=0$. However, if we examine an
extended temperature range, we have predominantly
$\mathrm{sgn}(\kappa_{xy})=-\mathrm{sgn}(\nu)$.. On the other hand, $\nu=0$
means that the net Berry curvature is zero, so $\kappa_{xy}$ is generically
small though not necessarily zero, and its sign is arbitrary.

We illustrate these ideas with three proposed parametrizations of
$\alpha$-RuCl3 in the literature, $(J,K,\Gamma,\Gamma^{\prime})=(-1,-8,4,-1)$
[73], $(-1.5,-40,5.3,-0.9)$ [74], and $(0,-6.8,9.5,0)$ [75], where energies
are given in units of meV. For $\mathbf{h}\parallel a$, these parametrizations
are located in the $\nu=+1$, $-1$, and $0$ regimes, respectively, so we label
them by $p$, $n$, and $z$, see Fig. 5a. For each of them, we calculate
$\kappa_{xy}$ as a function of $T$ at several values of $h$, see Fig. 5b. We
find that $\kappa_{xy}$ is negative (positive) for $p$ ($n$) as expected,
while $\kappa_{xy}$ for $z$ is one order of magnitude smaller. If we assume
that the measured $\kappa_{xy}>0$ in the field induced phase under
$\mathbf{h}\parallel-a$ in $\alpha$-RuCl3 [25] is indeed determined by a
dominant magnon contribution, then $p$ 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 $\kappa_{xy}$ for the three parametrizations, as follows. (i) The bands are
topological, i.e., they carry finite Chern numbers. (ii) The excitation gap
should not be too large, so that the lower band is thermally populated at low
temperatures. (iii) The band gap should not be 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
$p$, $n$, and $z$ are $0.16$, $0.19$, and $0.10$, while the band gaps are
$0.07$, $0.25$, and $0.33$, in units of $\lvert K\rvert S$. $n$ and $p$
fulfill (i) and are comparable in (ii), but $n$ does better than $p$ in (iii),
so $n$ yields a larger $\kappa_{xy}$. On the other hand, $z$ does better than
$p$ and $n$ in (ii) and (iii), but it fails in (i), so its $\kappa_{xy}$ is
small. As $h$ increases, the excitation gap becomes larger and $\kappa_{xy}$
decreases.

Discussion.—In summary, we have mapped out topological phase diagrams of
Kitaev magnets polarized by in-plane magnetic fields, which reveal the Chern
numbers of the magnon bands over a large parameter space. We have also
discussed that topological magnons are generally expected to yield a sizable
thermal Hall conductivity with sign opposite to the Chern number, when the
band (excitation) gap is large (small), at low temperatures. Our results will
be helpful in the search for topological magnons, as well as understanding
their thermal transport signatures, in Kitaev magnets including
$\alpha$-RuCl3. While the window of a field induced KSL might be shut in many
of the candidate materials, the door to topological magnons is more likely
open and accessible via high fields. Although we have focused solely on
magnons in this Letter, we appreciate that alternative sources of heat
carriers in Kitaev magnets, such as spinons [81, 82, 83], triplons [84],
phonons [85], visons [86], and some combinations [87, 88, 89, 90, 91], have
been proposed in the literature. One particularly interesting future direction
is to investigate the interplay between different types of topological
excitations, whether they cooperate (cancel) with each other and lead to a
large (small) thermal Hall effect [92].

###### Acknowledgements.

For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission. We thank Kyusung Hwang and Hana Schiff for useful
discussions. This work was supported by Engineering and Physical Sciences
Research Council grants No. EP/T028580/1 and No. EP/V062654/1.

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Supplemental Material: Topological phase diagrams of in-plane field polarized
Kitaev magnets

Li Ern Chern1 and Claudio Castelnovo1

1T.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 $z$-axis align with the spin [60],
while the $x$ and $y$ axes can be chosen freely as long as they are orthogonal
and $\hat{\mathbf{z}}=\hat{\mathbf{x}}\times\hat{\mathbf{y}}$ [70]. For the
polarized state with field angle $\beta$, the axes of the rotated coordinates
are defined by
$\hat{\mathbf{z}}=\hat{\mathbf{a}}\cos\beta+\hat{\mathbf{b}}\sin\beta$,
$\hat{\mathbf{x}}=-\hat{\mathbf{c}}$,
$\hat{\mathbf{y}}=-\hat{\mathbf{a}}\sin\beta+\hat{\mathbf{b}}\cos\beta$, where
$\hat{\mathbf{a}}$, $\hat{\mathbf{b}}$, and $\hat{\mathbf{c}}$ are unit
vectors along the crystallographic axes $a$, $b$, and $c$, respectively, see
Fig. 2a. One then performs Holstein-Primakoff transformation [59] to represent
the spins in terms of bosons (i.e., magnons), perform a $1/S$ expansion of the
Hamiltonian ($\sim S^{2}$ in the classical limit), and discard terms are of
orders lower than $S$. These procedures are well established and described in
details elsewhere (see Refs. [78, 62, 40] for example), so we do not repeat
them here.

For the $JK\Gamma\Gamma^{\prime}$ model (1) under an in-plane magnetic field
$(h_{a},h_{b},h_{c})=h(\cos\beta,\sin\beta,0)$, the linear spin wave
Hamiltonian of the polarized state is given by
$H=(S/2)\sum_{\mathbf{k}}\Psi_{\mathbf{k}}^{\dagger}\mathcal{H}_{\mathbf{k}}\Psi_{\mathbf{k}}$,
where
$\Psi_{\mathbf{k}}=(b_{1\mathbf{k}},b_{2\mathbf{k}},b_{1-\mathbf{k}}^{\dagger},b_{2-\mathbf{k}}^{\dagger})$
and $\mathcal{H}_{\mathbf{k}}$ is a four dimensional Hermitian matrix,

$\displaystyle\mathcal{H}_{\mathbf{k}}=\begin{pmatrix}\mathcal{A}_{\mathbf{k}}&\mathcal{B}_{\mathbf{k}}\\\
\mathcal{B}_{-\mathbf{k}}^{*}&\mathcal{A}_{-\mathbf{k}}^{\mathrm{T}}\end{pmatrix},$
(S1a)
$\displaystyle\mathcal{A}_{\mathbf{k}}=\frac{1}{2}\begin{pmatrix}2(h-c_{1})&c_{2}f_{\mathbf{k}}^{*}+c_{3}+c_{4}g_{\mathbf{k}}^{*}\\\
c_{2}f_{\mathbf{k}}+c_{3}+c_{4}g_{\mathbf{k}}&2(h-c_{1})\end{pmatrix},$ (S1b)
$\displaystyle\mathcal{B}_{\mathbf{k}}=\frac{1}{2}\begin{pmatrix}0&c_{5}f_{\mathbf{k}}^{*}-c_{3}-c_{4}g_{\mathbf{k}}^{*}+2i\big{[}c_{6}(f_{\mathbf{k}}^{*}-3)+c_{7}g_{\mathbf{k}}^{*}\big{]}\\\
c_{5}f_{\mathbf{k}}-c_{3}-c_{4}g_{\mathbf{k}}+2i\big{[}c_{6}(f_{\mathbf{k}}-3)+c_{7}g_{\mathbf{k}}\big{]}&0\end{pmatrix},$
(S1c)

where $c_{i}$ are (real) linear combinations of the couplings, while
$f_{\mathbf{k}}$ and $g_{\mathbf{k}}$ are functions of the (crystal) momentum,
see (LABEL:cidefine) and related discussions in the main text. To obtain the
linear spin wave dispersion, $\mathcal{H}_{\mathbf{k}}$ has to be diagonalized
by a Bogoliubov transformation $T_{\mathbf{k}}$ satisfying $T_{\mathbf{k}}\eta
T_{\mathbf{k}}^{\dagger}=\eta$, where $\eta=\mathrm{diag}(1,1,-1,-1)$, in
order to preserve the commutation relation of bosons. The magnon bands are
given by

$\omega_{\pm}(\mathbf{k})=\frac{S}{2}\sqrt{E(\mathbf{k})\pm\Delta(\mathbf{k})},$
(S2)

where $E(\mathbf{k})$ and $\Delta(\mathbf{k})$ are defined in (2a) and
(LABEL:Dksquaredefine) in the main text.

We consider a stable polarized state, where the excitation gap
$\min_{\mathbf{k}}\omega_{-}(\mathbf{k})$ 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

$h-c_{1}>0$ (S3)

always, which is argued as follows. Eq. (S3) obviously holds for $c_{1}\leq 0$
for all $h$. For $c_{1}>0$, assuming that a finite excitation gap is possible
for some $h<c_{1}$, we can then dial up $h$ such that $h=c_{1}$. At the
$\mathrm{K}$ or $\mathrm{K}^{\prime}$ point, where $f_{\mathbf{k}}=0$, we will
have
$\omega_{\pm}(\mathbf{k})=\sqrt{-4\lvert-3c_{6}+c_{7}g_{\mathbf{k}}\rvert^{2}}$,
which is either zero or imaginary, neither being physically sensible.
Therefore, magnon stability is only consistent with $h-c_{1}>0$.

A quick inspection of (LABEL:Dksquaredefine) reveals that
$\Delta^{2}(\mathbf{k})$ consists of a field dependent part and a field
independent part. With the assumption (S3), we now claim that if
$\Delta(\mathbf{k})=0$, then it is only physically sensible that
$c_{2}f_{\mathbf{k}}+c_{3}+c_{4}g_{\mathbf{k}}=0$, unless the system sits at a
critical point where a transition to the polarized state occurs. Suppose that
the contrary is true, i.e., $\Delta(\mathbf{k})=0$ and
$c_{2}f_{\mathbf{k}}+c_{3}+c_{4}g_{\mathbf{k}}\neq 0$ at some $h=h_{*}$, with
$h_{*}-c_{1}>0$. The field dependent part, which is positive, must cancel the
field independent part exactly in $\Delta^{2}(\mathbf{k})$. Let
$h^{\prime}=h_{*}-\epsilon$ with $0<\epsilon<h_{*}-c_{1}$. We have
$h^{\prime}-c_{1}>0$, but $\Delta^{2}(\mathbf{k})<0$ at $h=h^{\prime}$, i.e.,
$\omega_{\pm}(\mathbf{k})$ 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

Proposition 1. Under the stability requirement $h-c_{1}>0$,
$c_{2}f_{\mathbf{k}}+c_{3}+c_{4}g_{\mathbf{k}}=0$ is a necessary condition for
$\Delta(\mathbf{k})=0$.

We refer to the gap between the upper and lower magnon bands,
$\min_{\mathbf{k}}[\omega_{+}(\mathbf{k})-\omega_{-}(\mathbf{k})]\geq 0$, 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 $\Delta(\mathbf{k})=0$ for
some $\mathbf{k}=\mathbf{k}_{*}$. For a fixed set of couplings
$\\{J,K,\Gamma,\Gamma^{\prime}\\}$, 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 $h\longrightarrow\infty$ 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
$JK\Gamma\Gamma^{\prime}$ parameter space where the band gap vanishes, which
are potential phase boundaries for topological transitions.

$\beta=0$. With $c_{4}=0$ and $c_{6}=0$, (LABEL:Dksquaredefine) reads

$\Delta^{2}(\mathbf{k})=16(h-c_{1})^{2}\big{\lvert}c_{2}f_{\mathbf{k}}+c_{3}\big{\rvert}^{2}+(c_{2}+c_{5})\big{[}(c_{2}+c_{5})c_{3}^{2}(f_{\mathbf{k}}^{*}-f_{\mathbf{k}})^{2}+8c_{3}c_{7}^{2}(g_{\mathbf{k}}f_{\mathbf{k}}^{*}-g_{\mathbf{k}}^{*}f_{\mathbf{k}})(g_{\mathbf{k}}-g_{\mathbf{k}}^{*})+4(c_{2}-c_{5})c_{7}^{2}(g_{\mathbf{k}}f_{\mathbf{k}}^{*}-g_{\mathbf{k}}^{*}f_{\mathbf{k}})^{2}\big{]}.$
(S4)

The band gap is zero if and only if there is at least one $\mathbf{k}$ such
that $\Delta(\mathbf{k})=0$. According to Proposition 1, we require

$c_{2}f_{\mathbf{k}}+c_{3}=0,$ (S5)

which makes the field dependent part of $\Delta^{2}(\mathbf{k})$ zero. The
field independent part should be zero as well. We solve these conditions on a
case by case basis.

Case 1. $c_{2}=0$. We must have $c_{3}=0$. When
$\mathbf{k}=\mathrm{K},\mathrm{K}^{\prime}$, $f_{\mathbf{k}}=0$, and
$\Delta^{2}(\mathbf{k})=0$ is satisfied.

Case 2. $c_{2}\neq 0$.

Case 2.1. $c_{3}=0$. When $\mathbf{k}=\mathrm{K},\mathrm{K}^{\prime}$,
$f_{\mathbf{k}}=0$, and $\Delta^{2}(\mathbf{k})=0$ is satisfied.

Case 2.2. $c_{3}\neq 0$. The real and imaginary parts of (S5) respectively
read

$\displaystyle c_{2}(1+\cos k_{1}+\cos k_{2})+c_{3}$ $\displaystyle=0,$ (S6a)
$\displaystyle c_{2}(\sin k_{1}+\sin k_{2})$ $\displaystyle=0.$ (S6b)

(S6b) implies $k_{2}=-k_{1}+2\pi$ or $k_{2}=k_{1}+\pi$.

Case 2.2.1. $k_{2}=-k_{1}+2\pi$. Eq. (S6a) becomes $c_{2}(1+2\cos
k_{1})+c_{3}=0$, which only admits a solution when $-1\leq-c_{3}/c_{2}\leq 3$.
In this case, $f_{\mathbf{k}}=-c_{3}/c_{2}$, and (S4) becomes

$\Delta^{2}(\mathbf{k})=\left(\frac{8c_{3}}{c_{2}}\right)^{2}c_{7}^{2}(c_{2}+c_{5})^{2}\sin^{2}k_{1}.$
(S7)

The first bracket on the right hand side is assumed to be nonzero, so
$\Delta^{2}(\mathbf{k})=0$ if and only if (i) $c_{7}=0$, (ii) $c_{2}+c_{5}=0$,
or (iii) $\sin k_{1}=0$, which implies $k_{1}=0,\pi$, which in turn implies
$3c_{2}+c_{3}=0$ or $-c_{2}+c_{3}=0$, respectively.

Case 2.2.2. $k_{2}=k_{1}+\pi$. Eq. (S6a) reads $c_{2}+c_{3}=0$. In this case,
$f_{\mathbf{k}}=1$ and $g_{\mathbf{k}}$ is real at $k_{1}=0,\pi$, which yield
$\Delta^{2}(\mathbf{k})=0$ by (S4).

Collecting all the results, the parameter regions where the band gap vanishes
satisfy one of the following equations: (I) $c_{3}=0$, (II) $3c_{2}+c_{3}=0$,
(III) $-c_{2}+c_{3}=0$, (IV,V) $c_{2}+c_{3}=0$, (VI) $c_{2}+c_{5}=0$ if
$-1\leq-c_{3}/c_{2}\leq 3$, and (VII) $c_{7}=0$ if $-1\leq-c_{3}/c_{2}\leq 3$.
The reason that we use two labels IV and V for the equation $c_{2}+c_{3}=0$
will be clear when we discuss the case $0<\beta\leq\pi/6$.

$0<\beta\leq\pi/6$. The band gap is zero if and only if there is at least one
$\mathbf{k}$ such that $\Delta(\mathbf{k})=0$ [see (LABEL:Dksquaredefine) in
the main text]. According to Proposition 1, we require

$c_{2}f_{\mathbf{k}}+c_{3}+c_{4}g_{\mathbf{k}}=0,$ (S8)

which makes the field dependent part of $\Delta^{2}(\mathbf{k})$ zero. The
field independent part should be zero as well. We solve these conditions on a
case by case basis.

Case 1. $K+2\Gamma-2\Gamma^{\prime}=0$. Then, $c_{3}=0$ and $c_{4}=0$. At
$\mathbf{k}=\mathrm{K},\mathrm{K}^{\prime}$, $f_{\mathbf{k}}=0$, and
$\Delta^{2}(\mathbf{k})=0$ from (LABEL:Dksquaredefine).

Case 2. $K+2\Gamma-2\Gamma^{\prime}\neq 0$. Then, $c_{3}\neq 0$ and $c_{4}\neq
0$. The real and imaginary parts of (S8) respectively read

$\displaystyle(c_{2}+c_{3})+(c_{2}+c_{4})\cos k_{1}+(c_{2}-c_{4})\cos k_{2}$
$\displaystyle=0,$ (S9a) $\displaystyle(c_{2}+c_{4})\sin
k_{1}+(c_{2}-c_{4})\sin k_{2}$ $\displaystyle=0.$ (S9b)

Case 2.1. $c_{2}=c_{4}$. Eq. (S9b) implies $k_{1}=0,\pi$. Substituting
$k_{1}=\pi$ in (S9a) leads to $c_{3}-c_{4}=0$, or
$K+2\Gamma-2\Gamma^{\prime}=0$, a contradiction. We thus discard $k_{1}=\pi$.
Substituting $k_{1}=0$ in (S9a) leads to $c_{3}+3c_{4}=0$, which holds only at
$\beta=\pi/12$. We can then choose $k_{2}=0,\pi$ so that $f_{\mathbf{k}}$ and
$g_{\mathbf{k}}$ are real, and $\Delta^{2}(\mathbf{k})=0$ is satisfied.

Case 2.2. $c_{2}=-c_{4}$. Eq. (S9b) implies $k_{2}=0,\pi$. Substituting
$k_{2}=0$ in (S9a) leads to $c_{3}-3c_{4}=0$, or
$K+2\Gamma-2\Gamma^{\prime}=0$, a contradiction. We thus discard $k_{2}=0$.
Substituting $k_{2}=\pi$ in (S9a) leads to $c_{3}+c_{4}=0$, which holds only
at $\beta=\pi/6$. We can then choose $k_{1}=0,\pi$ so that $f_{\mathbf{k}}$
and $g_{\mathbf{k}}$ are real, and $\Delta^{2}(\mathbf{k})=0$ is satisfied.

Case 2.3. $\lvert c_{2}\rvert\neq\lvert c_{4}\rvert$.

Case 2.3.1. $(k_{1},k_{2})\in\\{(0,0),(0,\pi),(\pi,0),(\pi,\pi)\\}$. Eq. (S9b)
is satisfied. Eq. (S9a) implies $3c_{2}+c_{3}=0$, $c_{2}+c_{3}+2c_{4}=0$,
$c_{2}+c_{3}-2c_{4}=0$, or $-c_{2}+c_{3}=0$, respectively. Since
$f_{\mathbf{k}}$ and $g_{\mathbf{k}}$ are real, $\Delta^{2}(\mathbf{k})=0$ is
satisfied.

Figure S1: Eq. (S11) can be interpreted as the sum of two vectors.

Case 2.3.2. $(k_{1},k_{2})\notin\\{(0,0),(0,\pi),(\pi,0),(\pi,\pi)\\}$. We
write (S9a) and (S9b) as

$\displaystyle\lvert c_{2}+c_{4}\rvert\cos\tilde{k}_{1}+\lvert
c_{2}-c_{4}\rvert\cos\tilde{k}_{2}$ $\displaystyle=-(c_{2}+c_{3}),$ (S10a)
$\displaystyle\lvert c_{2}+c_{4}\rvert\sin\tilde{k}_{1}+\lvert
c_{2}-c_{4}\rvert\sin\tilde{k}_{2}$ $\displaystyle=0,$ (S10b)

where $\tilde{k}_{1}=k_{1}$ if $c_{2}+c_{4}>0$ and $\tilde{k}_{1}=k_{1}+\pi$
if $c_{2}+c_{4}<0$; $\tilde{k}_{2}$ is defined in a similar way. Squaring both
sides of (S10a) and (S10b), and adding up the results lead to

$\lvert c_{2}+c_{4}\rvert^{2}+\lvert c_{2}-c_{4}\rvert^{2}+2\lvert
c_{2}+c_{4}\rvert\lvert
c_{2}-c_{4}\rvert\cos(\tilde{k}_{1}-\tilde{k}_{2})=\lvert
c_{2}+c_{3}\rvert^{2}.$ (S11)

(S11) can be interpreted as a summation of two vectors, one of length $\lvert
c_{2}+c_{4}\rvert$ and the other $\lvert c_{2}-c_{4}\rvert$, with an angle
$\tilde{k}_{1}-\tilde{k}_{2}$ in between, which results in a vector of length
$\lvert c_{2}+c_{3}\rvert$, see Fig. S1. With this interpretation, we deduce
that (S11) admits a solution when

$\big{\lvert}\lvert c_{2}+c_{4}\rvert-\lvert
c_{2}-c_{4}\rvert\big{\rvert}\leq\lvert c_{2}+c_{3}\rvert\leq\lvert
c_{2}+c_{4}\rvert+\lvert c_{2}-c_{4}\lvert.$ (S12)

The right (left) equality holds when the two vectors (anti-)align, i.e.,
$\tilde{k}_{1}-\tilde{k}_{2}=0$ ($\pi$), where the resulting vector reaches
its longest (shortest) possible length. The equalities in (S12), however,
require $(k_{1},k_{2})\in\\{(0,0),(0,\pi),(\pi,0),(\pi,\pi)\\}$, which can be
seen from (S10a). This is a contradiction, but we note that these momenta have
been covered in Case 2.3.1. We can thus focus on the inequalities in (S12),
and assume that they are satisfied. From (S11),

$\displaystyle\cos(\tilde{k}_{1}-\tilde{k}_{2})$ $\displaystyle=\frac{\lvert
c_{2}+c_{3}\rvert^{2}-\lvert c_{2}+c_{4}\rvert^{2}-\lvert
c_{2}-c_{4}\rvert^{2}}{2\lvert c_{2}+c_{4}\rvert\lvert c_{2}-c_{4}\rvert},$
(S13a) $\displaystyle\sin(\tilde{k}_{1}-\tilde{k}_{2})$
$\displaystyle=\pm\sqrt{1-\cos^{2}(\tilde{k}_{1}-\tilde{k}_{2})}.$ (S13b)

Substituting (S10b) in (S10a) yields

$\displaystyle\sin\tilde{k}_{1}$ $\displaystyle=-\frac{\lvert
c_{2}-c_{4}\rvert}{c_{2}+c_{3}}\sin(\tilde{k}_{1}-\tilde{k}_{2}),$ (S14a)
$\displaystyle\sin\tilde{k}_{2}$ $\displaystyle=\frac{\lvert
c_{2}+c_{4}\rvert}{c_{2}+c_{3}}\sin(\tilde{k}_{1}-\tilde{k}_{2}).$ (S14b)

Notice that $c_{2}+c_{3}\neq 0$ due to (4). From (S14a) and (S14b),

$\displaystyle-\sin(\tilde{k}_{1}-\tilde{k}_{2})\cos\tilde{k}_{1}+\cos(\tilde{k}_{1}-\tilde{k}_{2})\sin\tilde{k}_{1}=\frac{\lvert
c_{2}+c_{4}\rvert}{c_{2}+c_{3}}\sin(\tilde{k}_{1}-\tilde{k}_{2})\iff\cos\tilde{k}_{1}=-\frac{\lvert
c_{2}+c_{4}\rvert}{c_{2}+c_{3}}-\frac{\lvert
c_{2}-c_{4}\rvert}{c_{2}+c_{3}}\cos(\tilde{k}_{1}-\tilde{k}_{2}),$ (S15a)
$\displaystyle\sin(\tilde{k}_{1}-\tilde{k}_{2})\cos\tilde{k}_{2}+\cos(\tilde{k}_{1}-\tilde{k}_{2})\sin\tilde{k}_{2}=-\frac{\lvert
c_{2}-c_{4}\rvert}{c_{2}+c_{3}}\sin(\tilde{k}_{1}-\tilde{k}_{2})\iff\cos\tilde{k}_{2}=-\frac{\lvert
c_{2}-c_{4}\rvert}{c_{2}+c_{3}}-\frac{\lvert
c_{2}+c_{4}\rvert}{c_{2}+c_{3}}\cos(\tilde{k}_{1}-\tilde{k}_{2}).$ (S15b)

In order for (S14a), (S14b), (S15a), and (S15b) to admit a solution
$(\tilde{k}_{1},\tilde{k}_{2})$, we need to verify that the absolute values of
the right hand sides are less than or equal to unity. We calculate

$\displaystyle\sin^{2}\tilde{k}_{1}+\cos^{2}\tilde{k}_{1}=\frac{\lvert
c_{2}-c_{4}\rvert^{2}+\lvert c_{2}+c_{4}\rvert^{2}+2\lvert
c_{2}+c_{4}\rvert\lvert
c_{2}-c_{4}\rvert\cos(\tilde{k}_{1}-\tilde{k}_{2})}{\lvert
c_{2}+c_{3}\rvert^{2}}=1,$ (S16a)
$\displaystyle\sin^{2}\tilde{k}_{2}+\cos^{2}\tilde{k}_{2}=\frac{\lvert
c_{2}+c_{4}\rvert^{2}+\lvert c_{2}-c_{4}\rvert^{2}+2\lvert
c_{2}-c_{4}\rvert\lvert
c_{2}+c_{4}\rvert\cos(\tilde{k}_{1}-\tilde{k}_{2})}{\lvert
c_{2}+c_{3}\rvert^{2}}=1,$ (S16b)

where the last equalities follow from (S11). If the squares of two real
numbers add up to unity, then each must be less than or equal to unity. We
have demonstrated that there indeed exists $(\tilde{k}_{1},\tilde{k}_{2})$,
defined implicitly via (S14a), (S14b), (S15a), and (S15b) in conjunction with
(S13a) and (S13b), which solves (S10a) and (S10b).

Substituting (S8) in (LABEL:Dksquaredefine),

$\Delta^{2}(\mathbf{k})=-\frac{4(c_{2}+c_{5})^{2}(c_{3}c_{7}+3c_{4}c_{6})^{2}(f_{\mathbf{k}}-f_{\mathbf{k}}^{*})^{2}}{c_{4}^{2}}=-\frac{\cos^{2}(3\beta)(c_{2}+c_{5})^{2}(K-\Gamma+\Gamma^{\prime})^{2}(K+2\Gamma-2\Gamma^{\prime})^{2}(f_{\mathbf{k}}-f_{\mathbf{k}}^{*})^{2}}{6c_{4}^{2}}.$
(S17)

If $\beta=\pi/6$, then $\cos(3\beta)=0$, and $\Delta^{2}(\mathbf{k})=0$ is
satisfied. For $0<\beta<\pi/6$, $\cos(3\beta)\neq 0$.
$f_{\mathbf{k}}-f_{\mathbf{k}}^{*}$ vanishes if and only if $f_{\mathbf{k}}$
is real, but this implies, via (S9b), $c_{4}=0$ or
$(k_{1},k_{2})\in\\{(0,0),(0,\pi),(\pi,0),(\pi,\pi)\\}$, both of which violate
the initial assumptions. Thus $f_{\mathbf{k}}-f_{\mathbf{k}}^{*}\neq 0$.
Furthermore, $K+2\Gamma-2\Gamma^{\prime}\neq 0$ by assumption. Therefore,
$\Delta^{2}(\mathbf{k})=0$ if and only if (i) $c_{2}+c_{5}=0$ or (ii)
$K-\Gamma+\Gamma^{\prime}=0$.

Collecting all the results, for $0<\beta<\pi/6$, the parameter regions where
the band gap vanishes satisfy one of the following equations: (I)
$K+2\Gamma-2\Gamma^{\prime}=0$, (II) $3c_{2}+c_{3}=0$, (III) $-c_{2}+c_{3}=0$,
(IV) $c_{2}+c_{3}+2c_{4}=0$, (V) $c_{2}+c_{3}-2c_{4}=0$, (VI) $c_{2}+c_{5}=0$
if (S12) holds, and (VII) $K-\Gamma+\Gamma^{\prime}=0$ if (S12) holds. Setting
$\beta=0$, these criteria are equivalent to those solved earlier for
$\beta=0$, which was the reason that we used two labels IV and V for
$c_{2}+c_{3}=0$, as $c_{4}=0$ there. On the other hand, for $\beta=\pi/6$, the
gap is zero whenever (I) or (S12) holds. (I-VII) are expressed in terms of the
couplings and the field angle in Table 1.

##  S2 Computation of Chern number

The Berry curvature of the $n^{\mathrm{th}}$ magnon band at the momentum
$\mathbf{k}$ is defined in terms of the Bogoliubov transformation
$T_{\mathbf{k}}$ as [78]

$\Omega_{n\mathbf{k}}=i\epsilon_{\mu\nu}\left(\eta\frac{\partial
T_{\mathbf{k}}^{\dagger}}{\partial k_{\mu}}\eta\frac{\partial
T_{\mathbf{k}}}{\partial k_{\nu}}\right)_{nn},$ (S18)

where $\epsilon$ is the totally antisymmetric tensor and
$\mu,\nu\in\\{x,y\\}$. (Caution: The expression within the brackets on the
right hand side is a matrix, and the subscript $nn$ means the entry at the
$n^{\mathrm{th}}$ row and the $n^{\mathrm{th}}$ column; $n$ is not a dummy
index that is being summed over.) Integrating (S18) over the magnetic
Brillouin zone gives the Chern number of the $n^{\mathrm{th}}$ band,

$\nu_{n}=\frac{1}{2\pi}\int_{\mathrm{MBZ}}\mathrm{d}k_{x}\mathrm{d}k_{y}\,\Omega_{n\mathbf{k}}.$
(S19)

This section explains the method that we use to compute the Chern number in a
discretized Brillouin zone, which was introduced in Ref. [65] and based on a
$U(1)$ lattice gauge theory (see also Refs. [53, 66]). The Berry curvature
(S18) multiplied by the integral measure,
$\Omega_{n\mathbf{k}}\,\mathrm{d}k_{x}\mathrm{d}k_{y}$, is invariant under a
coordinate transformation [64], e.g.

$\displaystyle\Omega_{n}(k_{x},k_{y})\,\mathrm{d}k_{x}\mathrm{d}k_{y}=F_{n}(k_{1},k_{2})\,\mathrm{d}k_{1}\mathrm{d}k_{2},$
(S20a) $\displaystyle F_{n}(k_{1},k_{2})\equiv i\left(\eta\frac{\partial
T_{\mathbf{k}}^{\dagger}}{\partial k_{1}}\eta\frac{\partial
T_{\mathbf{k}}}{\partial k_{2}}-\eta\frac{\partial
T_{\mathbf{k}}^{\dagger}}{\partial k_{2}}\eta\frac{\partial
T_{\mathbf{k}}}{\partial k_{1}}\right)_{nn}.$ (S20b)

(More formally, the differential 2-form
$\Omega_{n\mathbf{k}}\,\mathrm{d}k_{x}\wedge\mathrm{d}k_{y}$ is coordinate
independent.) Let $\mathcal{N}$ be the number of sites per magnetic unit cell,
in particular $\mathcal{N}=2$ for the polarized state. If there is a finite
magnon pairing term, then the $2\mathcal{N}$ dimensional Hamiltonian matrix
$\mathcal{H}_{\mathbf{k}}$ has a particle-hole redundancy by construction. The
columns of the $2\mathcal{N}$ dimensional Bogoliubov transformation matrix
$T_{\mathbf{k}}$ are arranged such that the first (last) $\mathcal{N}$ columns
belong to the particle (hole) sector.

Let $\lvert
n(\mathbf{k})\rangle\equiv(\mathbf{u}_{n}(\mathbf{k}),\mathbf{v}_{n}(\mathbf{k}))$
be the $n^{\mathrm{th}}$ vector of $T_{\mathbf{k}}$. We have introduced the
$\mathcal{N}$ dimensional vector $\mathbf{u}_{n}(\mathbf{k})$
[$\mathbf{v}_{n}(\mathbf{k})$] as the first [second] half of $\lvert
n(\mathbf{k})\rangle$. In the rest of this section, we focus on the particle
sector, i.e., $1\leq n\leq\mathcal{N}$. Next, we define the Berry connection
of the $n^{\mathrm{th}}$ band at the momentum $\mathbf{k}$ as

$A_{n,\lambda}(\mathbf{k})=i\langle
n(\mathbf{k})\rvert\eta\partial_{k_{\lambda}}\lvert
n(\mathbf{k})\rangle=i\left[\mathbf{u}_{n}^{\dagger}(\mathbf{k})\partial_{k_{\lambda}}\mathbf{u}_{n}(\mathbf{k})-\mathbf{v}_{n}^{\dagger}(\mathbf{k})\partial_{k_{\lambda}}\mathbf{v}_{n}(\mathbf{k})\right],$
(S21)

where $\lambda=1,2$. Since $\langle n(\mathbf{k})\rvert\eta\lvert
n(\mathbf{k})\rangle=1$, $\langle
n(\mathbf{k})\rvert\eta\partial_{k_{\lambda}}\lvert n(\mathbf{k})\rangle$ is
purely imaginary and hence $A_{n,\lambda}$ is purely real. Using the
definition (S20b), it can be straightforwardly verified that

$F_{n}(\mathbf{k})=\left[\partial_{k_{1}}A_{n,2}(\mathbf{k})-\partial_{k_{2}}A_{n,1}(\mathbf{k})\right].$
(S22)

On the discretized Brillouin zone, suppose that the spacings of momenta along
the $k_{1}$ and $k_{2}$ directions are $\delta k_{1}$ and $\delta k_{2}$,
respectively. If we make $\delta k_{\lambda}$ small enough, we can approximate
(S21) and (S22) as

$\displaystyle A_{n,\lambda}\,\delta k_{\lambda}$ $\displaystyle\approx
i\left[\langle n(\mathbf{k})\lvert\eta\rvert n(\mathbf{k}+\delta
k_{\lambda}\hat{\mathbf{\lambda}})\rangle-1\right],$ (S23a) $\displaystyle
F_{n}(\mathbf{k})\,\delta k_{1}\delta k_{2}$
$\displaystyle\approx\left[A_{n,2}(\mathbf{k}+\delta
k_{1}\hat{1})-A_{n,2}(\mathbf{k})\right]\delta
k_{2}-\left[A_{n,1}(\mathbf{k}+\delta
k_{2}\hat{2})-A_{n,1}(\mathbf{k})\right]\delta k_{1}.$ (S23b)

We define the $U(1)$ link variable

$U_{\lambda}(\mathbf{k})=\frac{\langle n(\mathbf{k})\rvert\eta\lvert
n(\mathbf{k}+\delta k_{\lambda}\hat{\lambda})\rangle}{\left\lvert\langle
n(\mathbf{k})\rvert\eta\lvert n(\mathbf{k}+\delta
k_{\lambda}\hat{\lambda})\rangle\right\rvert}=\frac{\langle
n(\mathbf{k})\rvert\eta\lvert n(\mathbf{k}+\delta
k_{\lambda}\hat{\lambda})\rangle}{1+O(\delta
k_{\lambda}^{2})}\approx\exp\left[-iA_{n,\lambda}(\mathbf{k})\delta
k_{\lambda}\right],$ (S24)

where, to obtain the second equality, we have expanded $\langle
n(\mathbf{k})\rvert\eta\lvert n(\mathbf{k}+\delta
k_{\lambda}\hat{\lambda})\rangle\approx 1+\langle
n(\mathbf{k})\rvert\eta\partial_{k_{\lambda}}\lvert n(\mathbf{k})\rangle\delta
k_{\lambda}$ and used the fact that $\langle
n(\mathbf{k})\rvert\eta\partial_{k_{\lambda}}\lvert n(\mathbf{k})\rangle$ is
imaginary. Eq. (S23b) can be expressed in terms of (S24) as

$F_{n}(\mathbf{k})\delta k_{1}\delta k_{2}\approx
i\ln\left[U_{1}(\mathbf{k})U_{2}(\mathbf{k}+\delta
k_{1}\hat{1})U_{1}^{-1}(\mathbf{k}+\delta
k_{2}\hat{2})U_{2}^{-1}(\mathbf{k})\right].$ (S25)

Finally, the Chern number of the $n^{\mathrm{th}}$ magnon band (S19) is
calculated as

$\nu_{n}\approx\frac{1}{2\pi}\sum_{\mathbf{k}\in\mathrm{MBZ}}F_{n}(\mathbf{k})\,\delta
k_{1}\delta k_{2}.$ (S26)

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
$\lvert n(\mathbf{k})\rangle\longrightarrow\exp[-i\chi(\mathbf{k})]\lvert
n(\mathbf{k})\rangle$ as desired, while the latter requires explicit gauge
fixings when taking the differences of $T_{\mathbf{k}}$ to approximate the
derivatives.

We mention in passing that the thermal Hall conductivity (5) can also be
calculated within this framework,

$\kappa_{xy}^{\mathrm{2D}}\approx-\frac{k_{\mathrm{B}}^{2}T}{(2\pi)^{2}\hbar}\sum_{n=1}^{\mathcal{N}}\sum_{\mathbf{k}\in\mathrm{MBZ}}c_{2}\left[g\left(\frac{\hbar\omega_{n\mathbf{k}}}{k_{\mathrm{B}}T}\right)\right]F_{n}(\mathbf{k})\,\delta
k_{1}\delta k_{2},$ (S27)

with $F_{n}(\mathbf{k})$ 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
$JK\Gamma\Gamma^{\prime}$ model, which, in the absence of an external magnetic
field, includes a time reversal $\mathcal{T}$ symmetry, a $C_{3}$ symmetry
about the $c$ axis, and a $C_{2}$ symmetry about the $b$ axis. Let
$\mathbf{J}=(J,K,\Gamma,\Gamma^{\prime})$. The Hamiltonian matrix (S1a) is a
function of the parameter $\mathbf{J}$, the field $\mathbf{h}$, and the
momentum $\mathbf{k}$, so we write it as
$\mathcal{H}_{\mathbf{k}}(\mathbf{J},h,\beta)$.

Consider a $C_{2}$ rotation of the field, i.e.,
$\beta=\pi/2-\tilde{\beta}\longrightarrow\pi/2+\tilde{\beta}$, at a fixed
parameter $\mathbf{J}$. Under $C_{2}$, we also have the mapping
$k_{1}\longleftrightarrow k_{2}$, or equivalently $k_{x}\longrightarrow-k_{x}$
and $k_{y}\longrightarrow k_{y}$. By this observation or by explicit
calculation [40], one can show that
$\mathcal{H}_{(k_{x},k_{y})}(\mathbf{J},h,\pi/2+\tilde{\beta})=\mathcal{H}_{(-k_{x},k_{y})}(\mathbf{J},h,\pi/2-\tilde{\beta})$.
As a consequence, if the band gap is zero (finite) at
$(\mathbf{J},h,\pi/2+\tilde{\beta})$, then it is also zero (finite) at
$(\mathbf{J},h,\pi/2-\tilde{\beta})$. Assume that the band gap is finite. The
Bogoliubov transformations are related by
$T_{(k_{x},k_{y})}(\mathbf{J},h,\pi/2+\tilde{\beta})=T_{(-k_{x},k_{y})}(\mathbf{J},h,\pi/2-\tilde{\beta})$.
By (S18) and (S19), we have
$\Omega_{n(k_{x},k_{y})}(\mathbf{J},h,\pi/2+\tilde{\beta})=-\Omega_{n(-k_{x},k_{y})}(\mathbf{J},h,\pi/2-\tilde{\beta})$
and
$\nu_{n}(\mathbf{J},\pi/2+\tilde{\beta})=-\nu_{n}(\mathbf{J},\pi/2-\tilde{\beta})$
for the Berry curvatures and the Chern numbers. In other words, the Chern
number of the $n^{\mathrm{th}}$ magnon band in the polarized state switches
sign when the field direction is changed to its $C_{2}$ counterpart. Theorems
1 and 2 in Ref. [40] follow from this.

Consider a $C_{3}$ rotation of the field, i.e.,
$\beta\longrightarrow\beta+2\pi/3$, at a fixed parameter $\mathbf{J}$. This
amounts to holding the field fixed in space while cyclically permuting the
$x$, $y$, and $z$ spin components. Therefore, $C_{3}$ preserves the band gap
(which can be either zero or finite) and the Chern number.

Consider the action of time reversal $\mathcal{T}$, i.e.,
$\mathbf{h}\longrightarrow-\mathbf{h}$ or, specifically for in-plane fields,
$\beta\longrightarrow\beta+\pi$, at a fixed parameter $\mathbf{J}$. We have
$\mathcal{H}_{\mathbf{k}}(\mathbf{J},h,\beta+\pi)=\mathcal{H}_{-\mathbf{k}}^{*}(\mathbf{J},h,\beta)$
by a theorem proved in Ref. [41]. Therefore, $\mathcal{T}$ 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.,
$\mathbf{J}\longrightarrow-\mathbf{J}$, at a fixed field direction $\beta$.
Following Ref. [52], we introduce

$U=\begin{pmatrix}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&-1&0\end{pmatrix}=\mathbf{1}\otimes(i\sigma_{2}),$ (S28)

where $\mathbf{1}$ is the two dimensional identity matrix and $\sigma_{1,2,3}$
are the Pauli matrices. We then carry out the transformation
$\mathcal{H}_{\mathbf{k}}(\mathbf{J},h,\beta)\longrightarrow
U\mathcal{H}_{\mathbf{k}}(\mathbf{J},h,\beta)U^{\dagger}\equiv\tilde{\mathcal{H}}_{\mathbf{k}}(\mathbf{J},h,\beta)$,
$\Psi_{\mathbf{k}}\longrightarrow
U\Psi_{\mathbf{k}}\equiv\tilde{\Psi}_{\mathbf{k}}$, which leaves the
Hamiltonian
$H=(S/2)\sum_{\mathbf{k}}\Psi_{\mathbf{k}}^{\dagger}\mathcal{H}_{\mathbf{k}}(\mathbf{J},h,\beta)\Psi_{\mathbf{k}}$
invariant. The matrix $U$ is unitary, so it preserves the hermicity of
$\mathcal{H}_{\mathbf{k}}(\mathbf{J},h,\beta)$. In addition, $U$ preserves the
bosonic commutation relation, i.e., $\tilde{\Psi}_{\mathbf{k}}$ obeys the same
commutation rule as $\Psi_{\mathbf{k}}$, which can be seen from

$U\eta
U^{\dagger}=(\mathbf{1}\otimes\sigma_{2})(\sigma_{3}\otimes\mathbf{1})(\mathbf{1}\otimes\sigma_{2})=\eta.$
(S29)

Importantly, it can be shown that
$\mathcal{H}_{\mathbf{k}}(-\mathbf{J},h,\beta)=\tilde{\mathcal{H}}_{-\mathbf{k}}(\mathbf{J},h+2c_{1},\beta)$,
where $c_{1}=3J+K-\Gamma-2\Gamma^{\prime}$ as defined in (LABEL:cidefine). We
can always choose a sufficiently large $h$ such that both
$(-\mathbf{J},h,\beta)$ and $(\mathbf{J},h+2c_{1},\beta)$ yield a stable
polarized state. Let $T_{\mathbf{k}}(\mathbf{J},h,\beta)$ and
$\tilde{T}_{\mathbf{k}}(\mathbf{J},h,\beta)$ be the Bogoliubov transformations
that diagonalize $\mathcal{H}_{\mathbf{k}}(\mathbf{J},h,\beta)$ and
$\tilde{\mathcal{H}}_{\mathbf{k}}(\mathbf{J},h,\beta)$, respectively, which
are related by
$T_{\mathbf{k}}(-\mathbf{J},h,\beta)=\tilde{T}_{-\mathbf{k}}(\mathbf{J},h+2c_{1},\beta)$.
If the band gap is finite at the parameter $\mathbf{J}$, by (S18) and (S19),
we have
$\Omega_{n\mathbf{k}}(-\mathbf{J},h,\beta)=\tilde{\Omega}_{n-\mathbf{k}}(\mathbf{J},h+2c_{1},\beta)$
and $\nu_{n}(-\mathbf{J},h,\beta)=\nu_{n}(\mathbf{J},h+2c_{1},\beta)$ for the
Berry curvatures and the Chern numbers. By Corollary 2, the Chern number of
the $n^{\mathrm{th}}$ magnon band in the polarized state at $-\mathbf{J}$ is
same as that at $\mathbf{J}$. On the other hand, if the band gap vanishes at
$\mathbf{J}$, then it also vanishes at $-\mathbf{J}$ by Corollary 1.

##  S4 Schrieffer-Wolff Transformation

The Schrieffer-Wolff transformation discussed in the main text is given by
[52]

$\displaystyle H\longrightarrow e^{W}He^{-W}=H+[W,H]+\ldots,$ (S30a)
$\displaystyle
W=\sum_{\mathbf{k}}\Psi_{\mathbf{k}}^{\dagger}\mathcal{W}_{\mathbf{k}}\Psi_{\mathbf{k}},\quad\mathcal{W}_{\mathbf{k}}=\frac{1}{2h}\begin{pmatrix}0&\mathcal{B}_{\mathbf{k}}\\\
-\mathcal{B}_{\mathbf{k}}^{\dagger}&0\end{pmatrix},$ (S30b)

which eliminates magnon pairings up to $O(1/h)$, and absorbs their effects in
a pure hopping model. From

$[W,H]=\frac{S}{2}\sum_{\mathbf{k}}\Psi_{\mathbf{k}}^{\dagger}\left(\mathcal{W}_{\mathbf{k}}\,\eta\,\mathcal{H}_{\mathbf{k}}-\mathcal{H}_{\mathbf{k}}\,\eta\,\mathcal{W}_{\mathbf{k}}\right)\Psi_{\mathbf{k}},$
(S31)

we obtain

$\displaystyle\mathcal{A}_{\mathbf{k}}\longrightarrow\mathcal{A}_{\mathbf{k}}-\frac{1}{h}\mathcal{B}_{\mathbf{k}}\mathcal{B}_{\mathbf{k}}^{\dagger}=\begin{pmatrix}h-c_{1}-\lvert
b_{-\mathbf{k}}\rvert^{2}/4h&a_{-\mathbf{k}}/2\\\
a_{\mathbf{k}}/2&h-c_{1}-\lvert
b_{\mathbf{k}}\rvert^{2}/4h\end{pmatrix}\equiv\mathcal{A}_{\mathbf{k}}^{\mathrm{eff}},$
(S32a)
$\displaystyle\mathcal{B}_{\mathbf{k}}\longrightarrow\mathcal{B}_{\mathbf{k}}-\frac{1}{2h}\left(\mathcal{B}_{\mathbf{k}}\mathcal{A}_{\mathbf{k}}+\mathcal{A}_{\mathbf{k}}\mathcal{B}_{\mathbf{k}}\right)=-\frac{1}{8h}\begin{pmatrix}a_{\mathbf{k}}b_{-\mathbf{k}}+a_{-\mathbf{k}}b_{\mathbf{k}}&-4c_{1}b_{-\mathbf{k}}\\\
-4c_{1}b_{\mathbf{k}}&a_{\mathbf{k}}b_{-\mathbf{k}}+a_{-\mathbf{k}}b_{\mathbf{k}}\end{pmatrix}\equiv\mathcal{B}^{\mathrm{eff}}_{\mathbf{k}},$
(S32b) $\displaystyle a_{\mathbf{k}}\equiv
c_{2}f_{\mathbf{k}}+c_{3}+c_{4}g_{\mathbf{k}},b_{\mathbf{k}}\equiv
c_{5}f_{\mathbf{k}}-c_{3}-c_{4}g_{\mathbf{k}}+2i\left[c_{6}(f_{\mathbf{k}}-3)+c_{7}g_{\mathbf{k}}\right],$
(S32c)

and the effective Hamiltonian $\mathcal{H}_{\mathbf{k}}^{\mathrm{eff}}$ is
given by (S1a) with $\mathcal{A}_{\mathbf{k}}$ and $\mathcal{B}_{\mathbf{k}}$
replaced by $\mathcal{A}_{\mathbf{k}}^{\mathrm{eff}}$ and
$\mathcal{B}_{\mathbf{k}}^{\mathrm{eff}}$, respectively. We can further
neglect $\mathcal{B}_{\mathbf{k}}$, as argued in the following. The linear
spin wave dispersion $\omega_{\pm}^{\mathrm{eff}}(\mathbf{k})$ satisfies the
characteristic polynomial of $\eta\,\mathcal{H}_{\mathbf{k}}^{\mathrm{eff}}$,
which has the form

$\lambda^{4}+(c_{A}+c_{B})\lambda^{2}+c_{\mathrm{mix}}=0.$ (S33)

The explicit expressions of $c_{A}$, $c_{B}$, and $c_{\mathrm{mix}}$ are
omitted here for simplicity; we merely note that $c_{A}\sim h^{2}$ depends
only on the matrix components of $\mathcal{A}_{\mathbf{k}}^{\mathrm{eff}}$,
$c_{B}\sim 1/h^{2}$ only on $\mathcal{B}_{\mathbf{k}}^{\mathrm{eff}}$, and
$c_{\mathrm{mix}}$ on both. Eq. (S33) is solved by

$\lambda^{2}=\frac{-(c_{A}+c_{B})\pm\sqrt{(c_{A}+c_{B})^{2}-4c_{\mathrm{mix}}}}{2}.$
(S34)

One finds by explicit calculation that the contribution of
$\mathcal{B}_{\mathbf{k}}^{\mathrm{eff}}$ to the square root in (S34) is at
best $O(1/\sqrt{h})$: it can be of higher order, but not lower. From
$\omega_{\pm}^{\mathrm{eff}}\sim\sqrt{\lambda^{2}}$, one can perform a large
$h$ expansion and deduce that $\mathcal{B}_{\mathbf{k}}^{\mathrm{eff}}$ only
contributes at $O(1/h^{3/2})$ and beyond to
$\omega_{\mathbf{k}}^{\mathrm{eff}}$. Discarding
$\mathcal{B}_{\mathbf{k}}^{\mathrm{eff}}$ is thus justified, and we are left
with a pure hopping Hamiltonian $\mathcal{A}_{\mathbf{k}}^{\mathrm{eff}}$. The
magnon excitation energies calculated from
$\mathcal{A}_{\mathbf{k}}^{\mathrm{eff}}$ are equal to those calculated from
$\mathcal{H}_{\mathbf{k}}^{\mathrm{eff}}$ up to $O(1/h)$.

Since the effective Hamiltonian $\mathcal{A}_{\mathbf{k}}^{\mathrm{eff}}$ is a
$2\times 2$ hermitian matrix, it can be expressed as

$\displaystyle\mathcal{A}_{\mathbf{k}}^{\mathrm{eff}}$
$\displaystyle=d_{0}(\mathbf{k})\mathbf{1}+\mathbf{d}(\mathbf{k})\cdot\bm{\sigma},$
(S35a) $\displaystyle d_{1}(\mathbf{k})$
$\displaystyle=\frac{1}{2}\big{[}(c_{2}+c_{3})+(c_{2}+c_{4})\cos
k_{1}+(c_{2}-c_{4})\cos k_{2}\big{]},$ (S35b) $\displaystyle
d_{2}(\mathbf{k})$ $\displaystyle=\frac{1}{2}\big{[}(c_{2}+c_{4})\sin
k_{1}+(c_{2}-c_{4})\sin k_{2}\big{]},$ (S35c)
$\displaystyle\begin{split}d_{3}(\mathbf{k})&=\frac{1}{h}\bigg{\\{}\big{[}(c_{5}-c_{3})+(c_{5}-c_{4})\cos
k_{1}+(c_{5}+c_{4})\cos k_{2}\big{]}\big{[}(c_{6}+c_{7})\sin
k_{1}+(c_{6}-c_{7})\sin k_{2}\big{]}\\\ &\quad+\big{[}(c_{5}+c_{4})\sin
k_{1}+(c_{5}-c_{4})\sin k_{2}\big{]}\big{[}2c_{6}-(c_{6}+c_{7})\cos
k_{1}-(c_{6}-c_{7})\cos k_{2}\big{]}\bigg{\\}},\end{split}$ (S35d)

where the explicit expression of $d_{0}(\mathbf{k})$ is omitted as it does not
play a role in the band topology. When $\mathbf{d}(\mathbf{k})\neq\mathbf{0}$,
the Chern number of the lower magnon band is given by (3) in the main text. If
one of the components of $\mathbf{d}(\mathbf{k})$ 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 $\nu=0$ in our phase diagrams Figs. 3a-3e can now
be understood as being related to the vanishing of $d_{3}(\mathbf{k})$. We
first note from (LABEL:cidefine) that both $c_{6}$ and $c_{7}$ contain the
factor $K-\Gamma+\Gamma^{\prime}$. Therefore, if $K-\Gamma+\Gamma^{\prime}=0$,
then $c_{6}=0$ and $c_{7}=0$, which in turn imply $d_{3}(\mathbf{k})=0$ for
all $\mathbf{k}$ by (S35d). The rest of the argument is contained in the main
text.


(7.8) $\displaystyle\hat{\alpha}(Z)\lesssim\alpha_{r}(Z).$

Take $b=(A)_{B(Z)}$ in (7.5). Then

$|\nabla\hat{A}(X)|\leq\int_{\hat{B}(X)}|A(Y)-(A)_{B(Z)}||\nabla\phi_{\delta(X)}(X-Y)|dY.$

Let us estimate the gradient term inside the integral.

$\displaystyle\delta(X)^{n+1}|\nabla_{X}\phi_{\delta(X)}|$
$\displaystyle\lesssim|\nabla\delta(X)|\phi\left(\frac{X-Y}{\delta(X)}\right)+\bigg{|}\nabla\phi\left(\frac{X-Y}{\delta(X)}\right)\bigg{|}\bigg{(}|\nabla\delta(X)|\frac{|X-Y|}{\delta(X)}+1\bigg{)}$
$\displaystyle\leq\|\phi\|_{L^{\infty}}+2\|\nabla\phi\|_{L^{\infty}}\lesssim
1,$

since $|X-Y|\leq\delta(X)$ and $|\nabla\delta|\leq 1$. It follows that

$|\nabla_{X}\phi_{\delta(X)}|\lesssim\delta(X)^{-(n+1)}.$

This implies that for any $X\in\hat{B}(Z)$ we have that

$\displaystyle|\nabla\hat{A}(X)|$
$\displaystyle\lesssim\frac{1}{\delta(X)^{n+1}}\int_{\hat{B}(X)}|A(Y)-(A)_{B(Z)}|dY$
$\displaystyle\leq\frac{1}{\delta(X)^{n+1}}\int_{B(Z)}|A(Y)-(A)_{B(Z)}|dY$
$\displaystyle\approx\frac{1}{\delta(Z)}\fint_{B(Z)}|A(Y)-(A)_{B(Z)}|dY$
$\displaystyle\leq\frac{1}{\delta(Z)}\left(\fint_{B(Z)}|A(Y)-(A)_{B(Z)}|^{r}dY\right)^{1/r}=\frac{1}{\delta(Z)}\alpha_{r}(Z).$

From this

$\hat{\alpha}(Z)=\delta(Z)\sup_{X\in\hat{B}(Z)}|\nabla\hat{A}|\lesssim\alpha_{r}(Z),$

as desired.

It remains to apply our two perturbation results for $A_{0}=\hat{A}$ and
$A_{1}=A$. Clearly $\hat{A}$ $\lambda_{0}$-elliptic. Moreover we can see that
$\|\hat{A}\|_{BMO(\Omega)}\lesssim\Lambda_{0}$.

To see this we distinguish two cases. First, consider a ball $B\subset\Omega$
such that $B\not\subset\hat{B}(X)$ is true for all $X\in\Omega$. Then we can
find a cover with balls $(\hat{B}(X_{i}))_{i}$ such that the balls
$\hat{B}(X_{i})$ have finite overlap, and
$|\bigcup_{i}\hat{B}(X_{i})|\lesssim|B|$. The constants in the last inequality
are independent of $B$. By Lemma 2.1 of [Jon80] we know that
$|(A)_{\hat{B}(X)}-(A)_{B(Z)}|\lesssim\Lambda_{0}$ for all $X\in\hat{B}(Z)$.
Hence

$\displaystyle\fint_{B}$
$\displaystyle|\hat{A}-(A)_{B}|dX\leq\fint_{B}|\hat{A}-A|dX+\fint_{B}|A-(A)_{B}|dX$
$\displaystyle\leq\fint_{B}\left|\int_{\hat{B}(X)}(A(Y)-A(X))\frac{\phi\left(\frac{X-Y}{\delta(X)}\right)}{\delta(X)^{n}}dY\right|dX+\Lambda_{0}$
$\displaystyle\leq\fint_{B}\fint_{\hat{B}(X)}|A(Y)-(A)_{\hat{B}(X)}|dY+|A(X)-(A)_{\hat{B}(X)}|dX+\Lambda_{0}$
$\displaystyle\leq\fint_{B}|A(X)-(A)_{\hat{B}(X)}|dX+2\Lambda_{0}$
$\displaystyle\lesssim\frac{1}{|B|}\sum_{i}\int_{\hat{B}(X_{i})}|A(X)-(A)_{B(X_{i})}|+|(A)_{B(X_{i})}-(A)_{\hat{B}(X)}|dX+2\Lambda_{0}$
$\displaystyle\lesssim\frac{1}{|B|}\sum_{i}|B(X_{i})|\fint_{B(X_{i})}|A(X)-(A)_{B(X_{i})}|+\Lambda_{0}dX+2\Lambda_{0}$
$\displaystyle\lesssim\lambda_{0}\left(2+\frac{1}{|B|}\sum_{i}|B(X_{i})|\right)\lesssim\Lambda_{0}.$

The second case is if $B\subset\hat{B}(X_{1})$ for some $X_{1}\in\Omega$. Then
we have

$\displaystyle\fint_{B}|\hat{A}-(A)_{B(X_{1})}|dX$
$\displaystyle\leq\fint_{B}\left|\int_{\hat{B}(X)}(A(Y)-(A)_{B(X_{1})})\frac{\phi\left(\frac{X-Y}{\delta(X)}\right)}{\delta(X)^{n}}dY\right|dX$
$\displaystyle\leq\fint_{B}\fint_{\hat{B}(X)}|A(Y)-(A)_{B(X_{1})}|dYdX$
$\displaystyle\lesssim\fint_{B}\fint_{B(X_{1})}|A(Y)-(A)_{B(X_{1})}|dYdX\lesssim\Lambda_{0}.$

Thus we can conclude that

$\inf_{M\in\mathbb{R}^{n\times n}\textrm{
constant}}\fint_{B}|\hat{A}(X)-M|dX\lesssim\Lambda_{0},$

which implies $\|\hat{A}\|_{BMO(\Omega)}\lesssim\Lambda_{0}$. It follows that
we indeed may apply our perturbation results.

Let

$\beta_{r}(Z)\coloneqq\left(\fint_{B(Z)}|\hat{A}(Y)-A(Y)|^{r}dY\right)^{1/r}.$

Our next objective is to show that

(7.9)
$\displaystyle\|\beta_{r}(Z)^{2}\delta^{-1}(Z)dZ\|_{\mathcal{C}}\lesssim\|\alpha_{r}(Z)^{2}\delta(Z)^{-1}dZ\|_{\mathcal{C}}.$

Assume for the moment this is indeed true. Then _Lemma 7.7_ and _Theorem 1.5_
imply _Theorem 1.9_. Similarty, _Lemma 7.7_ and _Theorem 1.6_ imply _Theorem
1.7_. Thus if we establish (7.9) we are done.

We start by observing that the following estimate holds:

$\displaystyle\left(\fint_{B(Z)}|\hat{A}-A|^{r}\right)^{1/r}$
$\displaystyle\leq\left(\fint_{B(Z)}|\hat{A}-(A)_{B(Z)}|^{r}\right)^{1/r}+\left(\fint_{B(Z)}|A-(A)_{B(Z)}|^{r}\right)^{1/r}$
$\displaystyle=\left(\fint_{B(Z)}|\hat{A}-(A)_{B(Z)}|^{r}\right)^{1/r}+\alpha_{r}(Z).$

The last term already has the required form. For the first term we see that

$\displaystyle\left(\fint_{B(Z)}|\hat{A}-(A)_{B(Z)}|^{r}\right)^{1/r}$
$\displaystyle\leq\bigg{(}\fint_{B(Z)}\left|\int_{\hat{B}(X)}|A(Y)-(A)_{B(Z)}|\phi_{\delta(X)}(X-Y)dY\right|^{r}dX\bigg{)}^{1/r}$
$\displaystyle\leq\|\phi\|_{L^{\infty}}\bigg{(}\fint_{B(Z)}\left|\delta(X)^{-n}\int_{\hat{B}(X)}|A(Y)-(A)_{B(Z)}|dY\right|^{r}dX\bigg{)}^{1/r}$
$\displaystyle\lesssim\bigg{(}\fint_{B(Z)}\left|\fint_{\hat{B}(X)}|A(Y)-(A)_{B(Z)}|dY\right|^{r}dX\bigg{)}^{1/r}$
$\displaystyle\lesssim\bigg{(}\fint_{B(Z)}|A(Y)-(A)_{B(Z)}|^{r}dX\bigg{)}^{1/r}=\alpha_{r}(Z).$

∎

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