Fragility of $\mathcal{Z}_{2}$ topological invariant characterizing triplet excitations in a bilayer kagome magnet

The symmetry classification of the intra-dimer interactions according to the D_6h symmetry group yields three invariant terms. Two correspond to the Heisenberg exchange anisotropy distinguishing the in-plane and out-of-plane components

$$\mathcal{H}^{\leavevmode\hbox to3.08pt{\vbox to8.77pt{\pgfpicture\makeatletter\hbox{\hskip 229.15997pt\lower-1.5379pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }] {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{tikz@color}{rgb}{0,0,0}\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-227.62207pt}{0.0pt}\pgfsys@moveto{-226.48416pt}{0.0pt}\pgfsys@curveto{-226.48416pt}{0.62845pt}{-226.99362pt}{1.13791pt}{-227.62207pt}{1.13791pt}\pgfsys@curveto{-228.25052pt}{1.13791pt}{-228.75998pt}{0.62845pt}{-228.75998pt}{0.0pt}\pgfsys@curveto{-228.75998pt}{-0.62845pt}{-228.25052pt}{-1.13791pt}{-227.62207pt}{-1.13791pt}\pgfsys@curveto{-226.99362pt}{-1.13791pt}{-226.48416pt}{-0.62845pt}{-226.48416pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-227.62207pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{tikz@color}{rgb}{0,0,0}\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-227.62207pt}{5.69046pt}\pgfsys@moveto{-226.48416pt}{5.69046pt}\pgfsys@curveto{-226.48416pt}{6.31891pt}{-226.99362pt}{6.82837pt}{-227.62207pt}{6.82837pt}\pgfsys@curveto{-228.25052pt}{6.82837pt}{-228.75998pt}{6.31891pt}{-228.75998pt}{5.69046pt}\pgfsys@curveto{-228.75998pt}{5.06201pt}{-228.25052pt}{4.55255pt}{-227.62207pt}{4.55255pt}\pgfsys@curveto{-226.99362pt}{4.55255pt}{-226.48416pt}{5.06201pt}{-226.48416pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{-227.62207pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{tikz@color}{rgb}{0,0,0}\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{-227.62207pt}{1.13791pt}\pgfsys@moveto{-227.62207pt}{1.13791pt}\pgfsys@lineto{-227.62207pt}{6.82881pt}\pgfsys@lineto{-227.62207pt}{6.82881pt}\pgfsys@lineto{-227.62207pt}{1.13791pt}\pgfsys@closepath\pgfsys@moveto{-227.62207pt}{6.82881pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}_{\rm XXZ}=J_{\|}\sum_{j}(S^{x}_{j_{1}}S^{x}_{j_{2}}+S^{y}_{j_{1}}S^{y}_{j_{2}})+J_{\bot}\sum_{j}S^{z}_{j_{1}}S^{z}_{j_{2}}\;.$$

(2)

The third intra-dimer term is a symmetric exchange anisotropy, which we refer to as the bond-nematic interaction:

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(3)

The index $j$ runs over the dimers, and $1$ and $2$ denote the layer indices of dimer-$j$. The vectors $\mathbf{n}_{j}$ appearing in the nematic term have the form $\mathbf{n}_{A}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$, $\mathbf{n}_{B}=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$, and $\mathbf{n}_{C}=\left(-1,0\right)$, and $\mathbf{Q}^{\|}_{j_{1},j_{2}}$ denotes the vector $(Q^{x^{2}\!-\!y^{2}}_{j_{1},j_{2}},Q^{xy}_{j_{1},j_{2}})$ made of the nematic interactions

	$\displaystyle Q^{x^{2}\!-\!y^{2}}_{i,j}=S^{x}_{i}S^{x}_{j}-S^{y}_{i}S^{y}_{j}\;,$		(4a)
	$\displaystyle Q^{xy}_{i,j}=S^{x}_{i}S^{y}_{j}+S^{y}_{i}S^{x}_{j}\;.$		(4b)

We illustrate the nematic operators, $Q^{x^{2}\!-\!y^{2}}$ and $Q^{xy}$, in the fashion of the d-orbitals, $d^{x^{2}\!-\!y^{2}}$ and $d^{xy}$, as they transform in the same way. Fig. 2 introduces our schematic illustration of the in-plane nematic operators, $Q^{x^{2}\!-\!y^{2}}$ and $Q^{xy}$, and their linear combinations as appear in the intra-dimer interactions, together with the directions of the $\mathbf{n}$ vectors.

Due to the bond-inversion, the antisymmetric exchange anisotropy, i.e. the Dzyaloshinskii–Moriya interaction is not allowed on the dimers.

Classifying the first neighbor inter-dimer interactions, we find six independent operators that transform as the fully symmetric irreducible representation. Beside the anisotropy in the Heisenberg exchange, distinguishing the in-plane and out-of-plane components

$\displaystyle\mathcal{H}^{\rm 1st}_{\rm XXZ}$

$\displaystyle=$

$\displaystyle J^{\prime}_{\|}\sum_{\begin{subarray}{c}\langle i,j\rangle\\ l=1,2\end{subarray}}(S^{x}_{i_{l}}S^{x}_{j_{l}}+S^{y}_{i_{l}}S^{y}_{j_{l}})+J^{\prime}_{\bot}\sum_{\begin{subarray}{c}\langle i,j\rangle\\ l=1,2\end{subarray}}S^{z}_{i_{l}}S^{z}_{j_{l}},$

(5)

there are additional four operators, namely the in-plane and out-of-plane components of the nematic and DM interactions. The DM interaction has the form

$\displaystyle\mathcal{H}^{\rm 1st}_{\text{DM}}$

$\displaystyle=$

$\displaystyle\sum_{\langle i,j\rangle}\mathbf{D}^{\prime}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})\;,$

(6)

and the in-plane ($D^{\prime}_{\|}$) and out-of-plane ($D^{\prime}_{\bot}$) components of the vector $\mathbf{D}^{\prime}$ are shown in Fig. 3.

$D^{\prime}_{\bot}$ is uniform in the top and bottom layers, but the $D^{\prime}_{\|}$ components change sign under exchanging the layers. As discussed in Section III, only the uniform out-of-plane DM component $D^{\prime}_{\bot}$ appears in Bogoliubov–de Gennes Hamiltonian describing the triplet dynamics.

In addition, there are in-plane and out-of-plane bond-nematic terms. The in-plane component has the form of

$$\mathcal{H}^{\rm 1st}_{\|}=K^{\prime}_{\|}\sum_{\langle i,j\rangle}\sum_{l=1,2}\mathbf{n}_{ij}\cdot\mathbf{Q}^{\|}_{il,jl}\;,$$

(7)

where $\mathbf{Q}^{\|}_{il,jl}$ is defined the same way as in Eq 4, the index $l$ takes the value $1$ for the top, and $2$ for the bottom layer, while $i$ and $j$ denote first neighbor sites within the layers. The vectors $\mathbf{n}_{ij}$ have the form

	$\displaystyle\mathbf{n}_{A^{\prime}B}$	$\displaystyle=$	$\displaystyle\mathbf{n}_{AB^{\prime}}=\left(-1,0\right)\;,$		(8)
	$\displaystyle\mathbf{n}_{B^{\prime}C}$	$\displaystyle=$	$\displaystyle\mathbf{n}_{BC^{\prime}}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\;,$		(9)
	$\displaystyle\mathbf{n}_{C^{\prime}A}$	$\displaystyle=$	$\displaystyle\mathbf{n}_{CA^{\prime}}=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\;.$		(10)

The in-plane component of the inter-dimer nematic interactions on the first neighbors are shown in Fig. 4, where we use the same notation introduced in Fig. 2 (a) and (b).

As with the out–of–plane DM components, the in–plane nematic terms are uniform in the top and bottom layers, and so will give a finite contribution to a Hamiltonian for triplets.

The first-neighbor out-of-plane bond-nematic interaction is

$$\mathcal{H}^{\rm 1st}_{\bot}=K^{\prime}_{\bot}\sum_{\langle i,j\rangle}\mathbf{n}^{\bot}_{ij}\cdot\mathbf{Q}^{\bot}_{il,jl}\;,$$

(11)

where the vectors $\mathbf{n}^{\bot}_{ij}$ are shown in Fig. 5(b), and the components of $\mathbf{Q}^{\bot}_{i,j}=(Q^{zx}_{i,j},Q^{yz}_{i,j})$ correspond to the bond-nematic operators

	$\displaystyle Q^{xz}_{i,j}=S^{z}_{i}S^{x}_{j}+S^{x}_{i}S^{z}_{j}\;,$		(12a)
	$\displaystyle Q^{yz}_{i,j}=S^{y}_{i}S^{z}_{j}+S^{z}_{i}S^{y}_{j}\;.$		(12b)

We utilize the representation of the $d^{yz}$ and $d^{zx}$ orbitals, to illustrate the out-of-plane bond-nematic terms as shown in Fig. 5.

Once again, the out-of-plane inter-dimer symmetric exchange anisotropy term is the opposite in the top and bottom layers. For this reason, as with the in-plane DM vectors, it will cancel in the triplet hopping Hamiltonian discussed in Section III.

We also consider the effect of second–neighbour interactions within the planes of the Kagome lattice. As with the first–neighbor interactions, there are six different terms: XXZ exchange $J^{\prime\prime}_{\|}$ and $J^{\prime\prime}_{\bot}$; in–plane and out–of–plane DM interactions, $D^{\prime\prime}_{\|}$ and $D^{\prime\prime}_{\bot}$; and the in–plane and out–of–plane bond-nematic terms $K^{\prime\prime}_{\|}$ and $K^{\prime\prime}_{\bot}$. XXZ interactions are defined through

	$\displaystyle\mathcal{H}^{\rm 2nd}_{\rm XXZ}$	$\displaystyle=$	$\displaystyle J^{\prime\prime}_{\|}\sum_{\langle\langle i,j\rangle\rangle}\sum_{l=1,2}S^{x}_{i_{l}}S^{x}_{j_{l}}+S^{y}_{i_{l}}S^{y}_{j_{l}}$		(13)
			$\displaystyle+J^{\prime\prime}_{\bot}\sum_{\langle\langle i,j\rangle\rangle}\sum_{l=1,2}S^{z}_{i_{l}}S^{z}_{j_{l}}\;,$

where the sum $\langle\langle i,j\rangle\rangle$ runs over second–neighbor bonds within the Kagome layers. Similarly, DM interactions are defined through

$\displaystyle\mathcal{H}^{\rm 2nd}_{\text{DM}}$

$\displaystyle=$

$\displaystyle\sum_{\langle\langle i,j\rangle\rangle}\mathbf{D}^{\prime\prime}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})\;,$

(14)

where the components of the associated DM vectors in the top and bottom layers are shown in Fig. 3 (d) and (e), respectively.

The additional bond–nematic–type interactions have the form

$$\mathcal{H}^{\rm 2nd}_{\|}=K^{\prime\prime}_{\|}\sum_{\langle\langle i,j\rangle\rangle}\sum_{l=1,2}\mathbf{n}_{ij}\cdot\mathbf{Q}^{\|}_{il,jl}\;,$$

(15)

where the vector operator, $\mathbf{Q}^{\|}_{il,jl}$ is defined in Eq. (4); $l$ takes the values $1$ and $2$ for top and bottom layers, respectively and the vectors $\mathbf{n}_{ij}$ have the form

	$\displaystyle\mathbf{n}_{AB}$	$\displaystyle=$	$\displaystyle\mathbf{n}_{A^{\prime}B^{\prime}}=\left(1,0\right)\;,$		(16)
	$\displaystyle\mathbf{n}_{BC}$	$\displaystyle=$	$\displaystyle\mathbf{n}_{B^{\prime}C^{\prime}}=\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\;,$		(17)
	$\displaystyle\mathbf{n}_{CA}$	$\displaystyle=$	$\displaystyle\mathbf{n}_{C^{\prime}A^{\prime}}=\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)\;.$		(18)

The out-of-plane bond-nematic term is

$$\mathcal{H}^{\rm 2nd}_{\bot}=K^{\prime\prime}_{\bot}\sum_{\langle\langle i,j\rangle\rangle}\sum_{l=1,2}\mathbf{n}^{\bot}_{ij}\cdot\mathbf{Q}^{\bot}_{il,jl}\;,$$

(19)

where the components of $\mathbf{Q}^{\bot}_{i,j}=(Q^{zx}_{il,jl},Q^{yz}_{il,jl})$ are introduced in Eqs. 12 and the associated vectors $\mathbf{n}^{\bot}_{ij}$ are shown in Fig. 7(b).

We start from a model defined by

	$\displaystyle\mathcal{H}$	$\displaystyle=$	$\displaystyle\mathcal{H}^{\text{1,2}}_{\text{D${}_{6h}$}}+\mathcal{H}_{\text{Zeeman}}$		(20)
		$\displaystyle=$	$\displaystyle\mathcal{H}_{\text{XXZ}}+\mathcal{H}_{\text{DM}}+\mathcal{H}_{\text{Nematic}}+\mathcal{H}_{\text{Zeeman}}\;,$		(21)

where $\mathcal{H}^{\text{1,2}}_{\text{D${}_{6h}$}}$ [Eq. (1)] is the most general Hamiltonian for a spin–1/2 magnet on a bilayer Kagome lattice with first– and second–neighbour interactions [cf. Section II], and

$\displaystyle\mathcal{H}_{\text{Zeeman}}=-g_{z}h^{z}\sum_{i}S^{z}_{i}\;,$

(22)

encodes the effect of a magnetic field perpendicular to the plane of the Kagome lattice.

The largest term in this model is taken to be (approximately) Heisenberg exchange $J_{\parallel}\approx J_{\perp}$ on intra–dimer bonds [Section II.1]. Where these interactions are antiferromagnetic, and sufficiently large compared with other terms, the ground state of $\mathcal{H}_{\text{XXZ}}$ is a quantum paramagnet formed by a product of singlets on all dimer bonds

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(23)

where

$\displaystyle\left|s\right>_{j}=\frac{1}{\sqrt{2}}(\left|\uparrow_{1}\downarrow_{2}\right>-\left|\downarrow_{1}\uparrow_{2}\right>)_{j}\;.$

(24)

The low–lying excitations of this quantum paramagnet will be spin-1 triplet excitations,

	$\displaystyle\left|t_{1}\right>_{j}=\left|\uparrow_{1}\uparrow_{2}\right>_{j}$		(25a)
	$\displaystyle\left|t_{0}\right>_{j}=\frac{1}{\sqrt{2}}(\left|\uparrow_{1}\downarrow_{2}\right>+\left|\downarrow_{1}\uparrow_{2}\right)_{j}$		(25b)
	$\displaystyle\left|t_{-1}\right>_{j}=\left|\downarrow_{1}\downarrow_{2}\right>_{j}\;,$		(25c)

where the indices $1$ and $2$ denote the top and bottom sites of the dimer, and $j$ labels the dimer. In the presence of terms connecting different dimers, these triplet excitations will form dispersing bands of excitations, usually referred to as “triplons”.

The spin operators can be expressed using the singlet-triplet basis

	$\displaystyle S^{+}_{j_{1}}$	$\displaystyle\!\!=\!\!$	$\displaystyle\frac{1}{\sqrt{2}}\left(t^{\dagger}_{1,j}t^{\phantom{\dagger}}_{0,j}\!-\!t^{\dagger}_{1,j}s^{\phantom{\dagger}}_{j}+t^{\dagger}_{0,j}t^{\phantom{\dagger}}_{-1,j}\!+\!s^{\dagger}_{j}t^{\phantom{\dagger}}_{-1,j}\right)$		(26a)
	$\displaystyle S^{z}_{j_{1}}$	$\displaystyle\!\!=\!\!$	$\displaystyle\frac{1}{2}\left(t^{\dagger}_{1,j}t^{\phantom{\dagger}}_{1,j}+t^{\dagger}_{0,j}s^{\phantom{\dagger}}_{j}+s^{\dagger}_{j}t^{\phantom{\dagger}}_{0,j}-t^{\dagger}_{-1,j}t^{\phantom{\dagger}}_{-1,j}\right)$		(26b)
	$\displaystyle S^{-}_{j_{1}}$	$\displaystyle\!\!=\!\!$	$\displaystyle\frac{1}{\sqrt{2}}\left(t^{\dagger}_{0,j}t^{\phantom{\dagger}}_{1,j}\!-\!s^{\dagger}_{j}t^{\phantom{\dagger}}_{1,j}+t^{\dagger}_{-1,j}t^{\phantom{\dagger}}_{0,j}\!+\!t^{\dagger}_{-1,j}s^{\phantom{\dagger}}_{j}\right)$		(26c)
	$\displaystyle S^{+}_{j_{2}}$	$\displaystyle\!\!=\!\!$	$\displaystyle\frac{1}{\sqrt{2}}\left(t^{\dagger}_{1,j}t^{\phantom{\dagger}}_{0,j}\!+\!t^{\dagger}_{1,j}s^{\phantom{\dagger}}_{j}+t^{\dagger}_{0,j}t^{\phantom{\dagger}}_{-1,j}\!-\!s^{\dagger}_{j}t^{\phantom{\dagger}}_{-1,j}\right)$		(26d)
	$\displaystyle S^{z}_{j_{2}}$	$\displaystyle\!\!=\!\!$	$\displaystyle\frac{1}{2}\left(t^{\dagger}_{1,j}t^{\phantom{\dagger}}_{1,j}-t^{\dagger}_{0,j}s^{\phantom{\dagger}}_{j}-s^{\dagger}_{j}t^{\phantom{\dagger}}_{0,j}-t^{\dagger}_{-1,j}t^{\phantom{\dagger}}_{-1,j}\right)$		(26e)
	$\displaystyle S^{-}_{j_{2}}$	$\displaystyle\!\!=\!\!$	$\displaystyle\frac{1}{\sqrt{2}}\left(t^{\dagger}_{0,j}t^{\phantom{\dagger}}_{1,j}\!+\!s^{\dagger}_{j}t^{\phantom{\dagger}}_{1,j}+t^{\dagger}_{-1,j}t^{\phantom{\dagger}}_{0,j}\!-\!t^{\dagger}_{-1,j}s^{\phantom{\dagger}}_{j}\right)$		(26f)

where the indices $1$ and $2$ denote the top and bottom site (layer) of the vertical dimer, $t^{\dagger}_{m,j}\left|0\right>=\left|t_{m}\right>_{j}$, $s^{\dagger}_{j}\left|0\right>=\left|s\right>_{j}$ with $m=-1,0,1$ and the $x$ and $y$ components of the spin operators are $S^{x}_{j}=\frac{1}{2}(S^{+}_{j}+S^{-}_{j})$ and $S^{y}_{j}=-\frac{i}{2}(S^{+}_{j}-S^{-}_{j})$.

In general, all components of the different exchange interactions within the Kagome planes will contribute to triplon dispersion. For small to moderate spin–orbit coupling, the largest contributions will come from (approximately) Heisenberg interactions on first–neighbour bonds $J^{\prime}_{\parallel}\approx J^{\prime}_{\perp}$. However the Berry phase underpinning a topological bandstructure originates in the DM interactions, $\mathcal{H}_{\text{DM}}$. And the fate of these topological bands will in turn depend on the nematic interactions, $\mathcal{H}_{\text{Nematic}}$. A few comments are therefore due on how these enter the problem.

Our model allows for DM interactions on first–neighbour bonds, $\mathbf{D}^{\prime}$ [Eq. (6)], and second–neighbour bonds, $\mathbf{D}^{\prime\prime}$ [Eq. (14)], illustrated in Fig. 3. The inter-layer bond–inversion symmetry precludes any DM interaction acting on the local dimers, and consequently the mixing between the odd singlet and even triplet states. Furthermore, as long as the $\sigma_{h}$ mirror symmetry is preserved, only the uniform out-of-plane DM components, $\mathbf{D}^{\prime}=(0,0,D^{\prime})$ and $\mathbf{D}^{\prime\prime}=(0,0,D^{\prime\prime})$, survive in the triplet Hamiltonian to linear order.

Similar considerations apply to nematic interactions $\mathcal{H}_{\text{Nematic}}$. In this case, the sign of the out-of-plane nematic terms is opposite in the layers, leaving only the in-plane components in the triplet Hamiltonian. $\mathcal{H}_{\text{Nematic}}$ thus simplifies to only the intra–dimer and inter–dimer in–plane nematic interactions, introduced in Eq. (3), Eq. (7), and (15).

Without loss of generality, we consider only the first neighbour interactions. The further neighbor terms do not change the overall shape (and generality) of the triplet Hamiltonian, only add to the complexity of the coefficients. Though, we keep the second neighbour DM interaction ($D^{\prime\prime}$) to discuss band touching topological transitions as the functions of $D^{\prime}$ and $D^{\prime\prime}$.

In the remainder of this paper we will slowly build up a complete picture of the topological physics of a quantum paramagnet described by Eq. (21), starting in Section IV from a simplified model with only Heisenberg and DM interactions, before progressively restoring the complexity of the full Hamiltonian, including nematic terms. Before doing so, in what follows, we set up the necessary technical framework for evaluating triplon bands.

To describe the dynamics of the triplet excitations at momentum ${\bf k}$ , we rely on the conventional bond wave theorySachdev and Bhatt (1990); Collins et al. (2006) resulting in the Bogoliubov–de Gennes Hamiltonian

$$\mathcal{H}_{k}=\begin{pmatrix}\tilde{\mathbf{t}}^{\dagger}_{\mathbf{k}}\\ \tilde{\mathbf{t}}^{\phantom{\dagger}}_{-\!\bf k}\end{pmatrix}^{T}\begin{pmatrix}\tilde{M}_{\mathbf{k}}&\tilde{N}_{\mathbf{k}}\\ \tilde{N}^{\dagger}_{\mathbf{-k}}&\tilde{M}^{\dagger}_{\mathbf{-k}}\end{pmatrix}\begin{pmatrix}\tilde{\mathbf{t}}^{\phantom{\dagger}}_{\mathbf{k}}\\ \tilde{\mathbf{t}}^{\dagger}_{-\!\bf k}\end{pmatrix}.$$

(27)

The vector $\tilde{\mathbf{t}}^{\dagger}_{\mathbf{k}}$ contains the 9 different triplets excitations, corresponding to the three spin states and the three sublattices;

$$\tilde{\mathbf{t}}^{\dagger}_{\mathbf{k}}=\left(\mathbf{t}_{1,{\bf k}}^{\dagger},\mathbf{t}_{0,{\bf k}}^{\dagger},\mathbf{t}_{-1,{\bf k}}^{\dagger}\right),$$

(28)

where $\mathbf{t}_{m,{\bf k}}^{\dagger}$ ($m=-1,0,1$) is

$$\mathbf{t}_{m,{\bf k}}^{\dagger}=(t_{A,m,{\bf k}}^{\dagger},t_{B,m,{\bf k}}^{\dagger},t_{C,m,{\bf k}}^{\dagger}).$$

(29)

$M_{\mathbf{k}}$ and $N_{\mathbf{k}}$ are 9-by-9 matrices, corresponding to the hopping hamiltonian, and the pairing terms, respectively. As it will turn out below, the matrices $\tilde{M}_{\mathbf{k}}$ and $\tilde{N}_{\mathbf{k}}$ are Hermitian and contain only cosines of the wave vector, so they are even in $\mathbf{k}$: $\tilde{M}^{\phantom{\dagger}}_{\mathbf{k}}=\tilde{M}^{\dagger}_{\mathbf{k}}=\tilde{M}^{\phantom{\dagger}}_{-\mathbf{k}}$ and $\tilde{N}^{\phantom{\dagger}}_{\mathbf{k}}=\tilde{N}^{\dagger}_{\mathbf{k}}=\tilde{N}^{\phantom{\dagger}}_{-\mathbf{k}}$. This will simplify the formulas, also the action of the time-reversal operator.

In the presence of the $\sigma_{h}$ mirror plane, the $\left|t_{0}\right>$ triplet decouples from the time-reversal pair $\left|t_{1}\right>$ and $\left|t_{-1}\right>$ and the most general form of the Bogoliubov–de Gennes Hamiltonian, describing the triplet dynamics, decomposes into two blocks,

	$\displaystyle\mathcal{H}_{\mathbf{k}}^{(1,-1)}$	$\displaystyle=\begin{pmatrix}\mathbf{t}_{{\bf k}}^{\dagger}\\ \mathbf{t}_{-\!{\bf k}}^{\phantom{\dagger}}\end{pmatrix}^{T}\begin{pmatrix}M_{\mathbf{k}}\!+\!M^{\text{Zeeman}}_{\mathbf{k}}&N_{\mathbf{k}}\\ N_{\mathbf{k}}&M_{\mathbf{k}}\!-\!M^{\text{Zeeman}}_{\mathbf{k}}\end{pmatrix}\begin{pmatrix}\mathbf{t}_{{\bf k}}^{\phantom{\dagger}}\\ \mathbf{t}_{-\!{\bf k}}^{\dagger}\end{pmatrix}$		(30a)
	$\displaystyle\mathcal{H}_{\mathbf{k}}^{(0)}$	$\displaystyle=\begin{pmatrix}\mathbf{t}_{0,{\bf k}}^{\dagger}\\ \mathbf{t}_{0,-\!{\bf k}}^{\phantom{\dagger}}\end{pmatrix}^{T}\begin{pmatrix}M_{0,\mathbf{k}}&N_{0,\mathbf{k}}\\ N_{0,\mathbf{k}}&M_{0,\mathbf{k}}\end{pmatrix}\begin{pmatrix}\mathbf{t}_{0,{\bf k}}^{\phantom{\dagger}}\\ \mathbf{t}_{0,-\!{\bf k}}^{\dagger}\end{pmatrix}$		(30b)

where the $M_{\mathbf{k}}$ and $N_{\mathbf{k}}$ are 6-by-6, and $M_{0,\mathbf{k}}$, and $N_{0,\mathbf{k}}$ are 3-by-3 matrices. The spinful subspace in Eq. (30a) is spanned by

$$\mathbf{t}_{{\bf k}}^{\phantom{\dagger}}=\begin{pmatrix}\mathbf{t}_{1,{\bf k}}^{\phantom{\dagger}}\\ \mathbf{t}_{-1,{\bf k}}^{\phantom{\dagger}}\end{pmatrix}\quad\text{and}\quad\mathbf{t}_{-{\bf k}}^{\dagger}=\begin{pmatrix}\mathbf{t}_{-1,-{\bf k}}^{\dagger}\\ \mathbf{t}_{1,-{\bf k}}^{\dagger}\end{pmatrix}.$$

(31)

Using this basis, both the diagonal and off-diagonal matrices are Hermitian, $M^{\dagger}_{\mathbf{k}}=M^{\phantom{\dagger}}_{\mathbf{k}}$, $M^{\dagger}_{0,\mathbf{k}}=M^{\phantom{\dagger}}_{0,\mathbf{k}}$, $N^{\dagger}_{\mathbf{k}}=N^{\phantom{\dagger}}_{\mathbf{k}}$, and $N^{\dagger}_{0,\mathbf{k}}=N^{\phantom{\dagger}}_{0,\mathbf{k}}$. The $\sigma_{h}$ acts on the individual $S=1/2$ spins as $C_{2}=C_{6}^{3}$ combined with swapping the layer indices. As a consequence, the $\left|t_{0}\right>$ tranforms differently from the $\left|t_{\pm 1}\right>$ and $\left|s\right>$ under the $\sigma_{h}$ operation: while $\left|t_{0}\right>\to-\left|t_{0}\right>$, the $\left|t_{\pm 1}\right>\to\left|t_{\pm 1}\right>$ and $\left|s\right>\to\left|s\right>$. This happens for example in the terms containing $S^{x}_{i}S^{z}_{j}$ or $S^{y}_{i}S^{z}_{j}$. Such terms are present in the in-plane DM and in the out-of-plane nematic interactions, resulting in $t^{(\dagger)}_{0,i}t^{(\dagger)}_{\pm 1,j}$ and $t^{(\dagger)}_{\pm 1,i}t^{(\dagger)}_{0,j}$ terms. These terms are odd under the $\sigma_{h}$ reflection, and therefore cancel in the triplet Hamiltonian ¹¹1Let us mention, however, that a magnetic field applied in the $xy$-plane induces mixing between the spinful ($\left|t_{1}\right>$ and $\left|t_{-1}\right>$) and the spinless ($\left|t_{0}\right>$) subspaces.. Due to the cancellation of the in-plane DM terms in the Hamiltonian, the only terms that do not conserve the total $S^{z}_{T}$ are the the in-plane nematic interactions, which change the $S^{z}_{T}$ by $\pm 2$ by creating an $\left|t_{1}\right>$ from $\left|t_{-1}\right>$ and vice versa.

We use the 8 Gell-Mann matrices as the basis for the sublattice degrees of freedom, corresponding to the three dimers, $A$, $B$, and $C$ in the unit cell:

	$$\displaystyle\lambda_{1}\!=\!\!\!\left(\begin{array}[]{ccc}0&1&0\\ 1&0&0\\ 0&0&0\end{array}\right)\!\!,\lambda_{2}\!=\!\!\!\left(\begin{array}[]{ccc}0&-i&0\\ i&0&0\\ 0&0&0\end{array}\right)\!\!,\lambda_{3}=\!\!\!\left(\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right)\!\!,$$		(41)
	$$\displaystyle\lambda_{4}\!=\!\!\!\left(\begin{array}[]{ccc}0&0&1\\ 0&0&0\\ 1&0&0\end{array}\right)\!\!,\lambda_{5}\!=\!\!\!\left(\begin{array}[]{ccc}0&0&i\\ 0&0&0\\ -i&0&0\end{array}\right)\!\!,\lambda_{6}=\!\!\!\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&1&0\end{array}\right)\!\!,$$		(51)
	$$\displaystyle\lambda_{7}\!=\!\!\!\left(\begin{array}[]{ccc}0&0&0\\ 0&0&-i\\ 0&i&0\end{array}\right)\!\!,\lambda_{8}\!=\!\frac{1}{\sqrt{3}}\!\!\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&-2\end{array}\right)\!\!.$$		(58)

The Gell-Mann matrices, and the 3-by-3 identity matrix, $I_{3}$ sufficiently characterize the spinless $m=0$ subspace. The hopping Hamiltonian has the form

	$\displaystyle M_{0,\mathbf{k}}=M^{\text{XXZ}}_{0,\mathbf{k}}$	$\displaystyle=$	$\displaystyle J_{\|}I_{3}+J^{\prime}_{\bot}\left[\cos\frac{\bm{\delta}_{1}\mathbf{k}}{2}\lambda_{4}\right.$		(59)
			$\displaystyle\left.+\cos\frac{\bm{\delta}_{2}\mathbf{k}}{2}\lambda_{1}+\cos\frac{\bm{\delta}_{3}\mathbf{k}}{2}\lambda_{6}\right]\;,$

and the pairing terms are the same as $M_{0,\mathbf{k}}$ without the diagonal elements.

$\displaystyle N_{0,\mathbf{k}}=M_{0,\mathbf{k}}-J_{\|}I_{3}\;.$

(60)

Note that the Hamiltonian in the $m=0$ subspace contains only the Heisenberg terms, the symmetric nematic exchange and the antisymmetric DM interaction do not affect the $\left|t_{0}\right>$ triplet.

The triplet hopping of the spinful subspace,

$$M_{\mathbf{k}}=M^{\text{XXZ}}_{\mathbf{k}}+M^{\text{DM}}_{\mathbf{k}}+M^{\text{Nematic}}_{\mathbf{k}}$$

(61)

in Eq. (30a) is a $6\times 6$ matrix. The spin degree of freedom provided by the $S^{z}=\pm 1$ triplets is represented by the spin operators $s^{x}$, $s^{y}$, and $s^{z}$, corresponding to the Pauli matrices times $\frac{1}{2}$. The 6-dimensional local Hilbert space for the spinful triplets is constructed as the tensor product of the 2-by-2 matrices $\{I_{2},s^{x},s^{y},s^{z}\}$ acting on the spin-space, and the 3-by-3 Gell-Mann matrices extended with the identity $I_{3}$, acting in the sublattice space. We discuss the various contributions separately. The Heisenberg interaction only contains the identity operator, $I_{2}$ in the spin-space, i.e. it does not affect the spin degrees of freedom, and is the same for the $\left|t_{1}\right>$ and $\left|t_{-1}\right>$ triplets.

	$\displaystyle M^{\text{XXZ}}_{\mathbf{k}}$	$\displaystyle\!=\!$	$\displaystyle\frac{J_{\|}\!+\!J_{\bot}}{2}I_{2}\!\otimes\!I_{3}+J^{\prime}_{\|}\cos\frac{\bm{\delta}_{1}\mathbf{k}}{2}I_{2}\!\otimes\!\lambda_{4}$		(62)
			$\displaystyle+J^{\prime}_{\|}\cos\frac{\bm{\delta}_{2}\mathbf{k}}{2}I_{2}\!\otimes\!\lambda_{1}+J^{\prime}_{\|}\cos\frac{\bm{\delta}_{3}\mathbf{k}}{2}I_{2}\!\otimes\!\lambda_{6}.$

The pairing terms from the Heisenberg interaction are similar to $M^{\text{XXZ}}_{\mathbf{k}}$, but have opposite sign and no diagonal elements

$\displaystyle N^{\text{XXZ}}_{\mathbf{k}}$

$\displaystyle=$

$\displaystyle-M^{\text{XXZ}}_{\mathbf{k}}+\frac{J_{\|}+J\bot}{2}I_{2}\otimes I_{3}\;.$

(63)

The DM interaction has the form

	$\displaystyle M^{\text{DM}}_{\mathbf{k}}$	$\displaystyle\!=\!$	$\displaystyle\left[\!D^{\prime}\!\cos\frac{\bm{\delta}_{1}\mathbf{k}}{2}\!+\!D^{\prime\prime}\cos\!\frac{(\bm{\delta}_{2}\!-\!\bm{\delta}_{3})\mathbf{k}}{2}\!\right]\!s^{z}\!\otimes\!\lambda_{5}$		(64)
		$\displaystyle+$	$\displaystyle\left[\!D^{\prime}\!\cos\frac{\bm{\delta}_{2}\mathbf{k}}{2}\!+\!D^{\prime\prime}\cos\!\frac{(\bm{\delta}_{3}\!-\!\bm{\delta}_{1})\mathbf{k}}{2}\!\right]\!s^{z}\!\otimes\!\lambda_{2}$
		$\displaystyle+$	$\displaystyle\left[\!D^{\prime}\!\cos\frac{\bm{\delta}_{3}\mathbf{k}}{2}\!+\!D^{\prime\prime}\cos\!\frac{(\bm{\delta}_{1}\!-\!\bm{\delta}_{2})\mathbf{k}}{2}\!\right]\!s^{z}\!\otimes\!\lambda_{7}\;.$

The intra-dimer DM interaction is forbidden by the bond-inversion of the dimers, thus there are no diagonal elements in $M^{\text{DM}}_{\mathbf{k}}$ and the form of the pairing terms simply correspond to

$\displaystyle N^{\text{DM}}_{\mathbf{k}}$

$\displaystyle=$

$\displaystyle-M^{\text{DM}}_{\mathbf{k}}\;.$

(65)

Here, the only operator acting in the spin-space is $s^{z}$, leaving the spin degrees of freedom unchanged, and introducing a sign difference for the DM interaction in the up and down-spin sector.

Let us make some comments on the time-reversal (TR) properties of the Gell-Mann matrices and the pseudo-spin-half operators. The Gell-Mann matrices act in the sublattice space, i.e. account for changing the dimer indices, $A$, $B$, and $C$. Time-reversal symmetry leaves such indices invariant, therefore the real Gell-Mann matrices are TR invariant. As TR symmetry contains a complex conjugation, the imaginary Gell-Matrices are TR breaking, changing sign under TR. A complete analysis on the TR symmetry of the pseudo-spin-half operators is provided in the Appendix A, where we show that while the $s^{z}$ is TR-breaking, as one would expect from a spin-operator, the $s^{x}$ and $s^{y}$ components are TR invariant operators. This is a consequential difference with respect to the original Kane and Mele model, where the Pauli matrices describe a physical spin-half Kramers doublet and thus all break TR symmetry.

Coming back to the DM terms in our triplet hopping Hamiltonian in Eq. (64), the appearing cross-product operators are all TR invariant, as both the $s^{z}$ spin operator, and the $\lambda_{5}$, $\lambda_{2}$, and $\lambda_{7}$ complex Gell-Mann matrices are odd under TR symmetry.

The nematic interactions couple the subspaces of the different $m$ subspaces. They have the form

	$\displaystyle M_{\mathbf{k}}^{\text{Nematic}}$	$\displaystyle\!=\!$	$\displaystyle\frac{\sqrt{3}K_{\|}}{4}\left[s^{x}\otimes\lambda_{8}-s^{y}\otimes\lambda_{3}\right]$		(66)
			$\displaystyle+K^{\prime}_{\|}\cos\frac{\bm{\delta}_{3}\mathbf{k}}{2}\left(\!\!-\frac{1}{2}s^{x}\!-\!\frac{\sqrt{3}}{2}s^{y}\!\!\right)\!\otimes\!\lambda_{6}$
			$\displaystyle+K^{\prime}_{\|}\cos\frac{\bm{\delta}_{1}\mathbf{k}}{2}\left(\!\!-\frac{1}{2}s^{x}\!+\!\frac{\sqrt{3}}{2}s^{y}\!\!\right)\!\otimes\!\lambda_{4}$
			$\displaystyle+K^{\prime}_{\|}\cos\frac{\bm{\delta}_{2}\mathbf{k}}{2}\cdot s^{x}\!\otimes\!\lambda_{1}\;,$

The Gell-Mann matrices in the nematic interaction are the real ones, preserving TR symmetry. The nematic interactions exclusively consist of spin-mixing operators, $s^{x}$ and $s^{y}$ that are TR invariant terms themselves. (For details see App. A). When the nematic terms are present, the total $S^{z}_{T}$ ceases to be a good quantum number, and the spin up and down components mix.

Such spin-mixing term is present in the Kane and Mele model too, in the form of a Rashba spin-orbit coupling, permitted by the breaking of the mirror-symmetry $\sigma_{h}$. Here, the spin-mixing nematic terms are allowed without breaking $\sigma_{h}$. The TR symmetry does not protect the degeneracy of the spin-up and down triplets at the TR-invariant points in the momentum space, as it would for the Kramers pair electron-spins. Therefore, the nematic term hybridizes the spins, ending the fragile $\mathcal{Z}_{2}$ topology of the bands, as discussed in Sec. VII.

The nematic interaction is allowed on the dimers too. Note that the operators $\lambda_{8}$ and $\lambda_{3}$ are diagonal in the sublattice space. The $s^{x}$ and $s^{y}$ operators, however mix the spins, placing the intra-dimer nematic interaction $K_{\|}$ in the diagonal of the block connecting the $+1$ and $-1$ triplets. The $N^{\text{Nematic}}_{1,\mathbf{k}}$ matrix of the pairing terms corresponds again to the $-M^{\text{Nematic}}_{1,\mathbf{k}}$ minus the ‘diagonal’ elements

$\displaystyle N^{\text{Nematic}}_{\mathbf{k}}\!=\!-M^{\text{Nematic}}_{\mathbf{k}}\!+\!\frac{\sqrt{3}K_{\|}}{4}\left[s^{x}\!\otimes\!\lambda_{8}\!-\!s^{y}\!\otimes\!\lambda_{3}\right]\;.$

(67)

Lastly, the out-of plane magnetic field appears in the diagonal of the Hamiltonian as

$\displaystyle M_{\mathbf{k}}^{\text{Zeeman}}$

$\displaystyle\!=\!$

$\displaystyle-g_{z}h_{z}2s^{z}\otimes I_{3}\;.$

(68)

We conclude this tour of the terms in the quadratic Bogoliubov–de Gennes Hamiltonian with a brief comment on what it neglects, namely interactions between triplon modes occurring at higher order in bond operators. The effect of electron–electron interactions on topological insulators and superconductors remains an open problem. And the effect of triplon–triplon interactions on band topology in quantum paramagnets is even less explored.

None the less, some work has been done the characterize the effect of magnon–magnon interactions in magnetically–ordered systems with topological bands Chernyshev and Zhitomirsky (2015); Chernyshev and Maksimov (2016); McClarty et al. (2018); McClarty and Rau (2019); Kondo et al. (2020); McClarty (2021). One example which has been been quite well characterized in the Kitaev model in high magnetic field, where the non–interacting theory of topological magnon bands can be compared with both interacting spin–wave calculations and DMRG results McClarty et al. (2018). And here the effect of interactions is relatively benign, being chiefly limited to a finite broadening of magnon modes, and renormalisation of band their dispersion.

Its is reasonable to expect the same will be true in quantum paramagnets, since the form of Bogoliubov–de Gennes Hamiltonian is identical. And this particulary true where the potential for triplon decay is restricted by a substantial band gap. It is also plausible that the topological excitations of quantum paramagnets will exhibit some of the same interesting, non–Hermitian features, as topological magnons, a topic reviewed in McClarty (2021). The role of interactions within a $\mathcal{Z}_{2}$ topological phase of a magnetic insulator is clearly worthy of further investigation. However this lies outside the scope of this Article, which has the more limited goal of characterizing band topology in the non–interacting limit.

Let us start with the time reversal symmetric case, when the magnetic field is zero. In the isotropic Heisenberg limit, $J_{\bot}=J_{\|}=J$, and $J^{\prime}_{\bot}=J^{\prime}_{\|}=J^{\prime}$, $D^{\prime}=0$, $D^{\prime\prime}=0$, $K_{\|}=0$, and $K^{\prime}_{\|}=0$ the model has SU(2) symmetry, the $M_{\mathbf{k}}$ and $N_{\mathbf{k}}$ matrix in Eq. (30a) becomes block diagonal,

$$M_{\mathbf{k}}=\begin{pmatrix}M_{1,\mathbf{k}}&0\\ 0&M_{-1,\mathbf{k}}\end{pmatrix}\;,\quad N_{\mathbf{k}}=\begin{pmatrix}N_{1,\mathbf{k}}&0\\ 0&N_{-1,\mathbf{k}}\end{pmatrix}\;,$$

(69)

and the Hamiltonians is identical in each of the $m=-1,0,1$ subspaces (see Eqs. (30a) and (30b)) so that $M_{m,\mathbf{k}}=M^{\text{SU(2)}}_{m,\mathbf{k}}$, and $N_{m,\mathbf{k}}=N^{\text{SU(2)}}_{m,\mathbf{k}}$, with $m$-independent

	$\displaystyle M^{\text{SU(2)}}_{m,\mathbf{k}}$	$\displaystyle\!\!=\!\!$	$\displaystyle JI_{3}\!+\!J^{\prime}\!\!\left[\cos\frac{\bm{\delta}_{1}\mathbf{k}}{2}\lambda_{4}\!+\!\cos\frac{\bm{\delta}_{2}\mathbf{k}}{2}\lambda_{1}\!+\!\cos\frac{\bm{\delta}_{3}\mathbf{k}}{2}\lambda_{6}\right]\!,$
	$\displaystyle N^{\text{SU(2)}}_{m,\mathbf{k}}$	$\displaystyle\!\!=\!\!$	$\displaystyle JI_{3}-M^{\text{SU(2)}}_{\mathbf{k}}\;.$		(70)

matrices (we keep the $m$ index only for bookkeeping purposes). The $m$ subspaces are spanned by the basis

$$(\mathbf{t}_{m,{\bf k}}^{\dagger},\mathbf{t}_{-m,-{\bf k}}^{\phantom{\dagger}}),$$

(71)

defined in Eq. (28). Each of these operators is going to change the total $S^{z}_{T}$ by $m$, either by creating an $m$ triplon with $\mathbf{t}_{m,{\bf k}}^{\dagger}$ or by annihilating a $-m$ triplon with $\mathbf{t}_{-m,-{\bf k}}$. A rotation $e^{-i\varphi S^{z}_{T}}$ by an angle $\varphi$ about the $z$ axis in the spin space manifests in a phase factor

$$(\mathbf{t}_{m,{\bf k}}^{\dagger},\mathbf{t}_{-m,-{\bf k}}^{\phantom{\dagger}})\to e^{-im\varphi}(\mathbf{t}_{m,{\bf k}}^{\dagger},\mathbf{t}_{-m,-{\bf k}}^{\phantom{\dagger}}).$$

(72)

Thereby in the Hamiltonian we encounter normal terms of the form $\mathbf{t}_{m,{\bf k}}^{\dagger}M_{m,\mathbf{k}}^{\phantom{\dagger}}\mathbf{t}_{m,{\bf k}}^{\phantom{\dagger}}$ and $\mathbf{t}_{-m,-{\bf k}}^{\phantom{\dagger}}M_{m,\mathbf{k}}^{\phantom{\dagger}}\mathbf{t}_{-m,-{\bf k}}^{\dagger}$, and anomalous terms in the form of $\mathbf{t}_{m,{\bf k}}^{\dagger}N_{m,\mathbf{k}}^{\phantom{\dagger}}\mathbf{t}_{-m,-{\bf k}}^{\dagger}$ and $\mathbf{t}_{-m,{\bf k}}^{\phantom{\dagger}}N_{m,\mathbf{k}}^{\phantom{\dagger}}\mathbf{t}_{m,-{\bf k}}^{\phantom{\dagger}}$ which are invariant with respect to the phase transformation of Eq. (72). These all commute with the

$$S^{z}_{T}=\mathbf{t}_{1,{\bf k}}^{\dagger}\mathbf{t}_{1,{\bf k}}^{\phantom{\dagger}}-\mathbf{t}_{-1,-{\bf k}}^{\dagger}\mathbf{t}_{-1,-{\bf k}}^{\phantom{\dagger}},$$

(73)

so that the total $S^{z}_{T}$ is conserved, generating a U(1) symmetry. The higher SU(2) symmetry is exemplified by the matrices being independent from $m$.

In this case, each of the three bands, coming from the three sublattices, are threefold degenerate, as the $m=1,0,-1$ have the same energies in the entire Brillouin zone. The dispersion computed from the SU(2) symmetric Bogoliubov–de Gennes Hamiltonian has the form

	$\displaystyle\omega_{1,m}$	$\displaystyle=\sqrt{J(J-2J^{\prime})}$		(74a)
	$\displaystyle\omega_{2,m}$	$\displaystyle=\sqrt{J(J+J^{\prime})-\tilde{J}({\mathbf{k}})}$		(74b)
	$\displaystyle\omega_{3,m}$	$\displaystyle=\sqrt{J(J+J^{\prime})+\tilde{J}({\mathbf{k}})}\;.$		(74c)

with $\tilde{J}({\mathbf{k}})=JJ^{\prime}\sqrt{3+2\sum_{\alpha}\cos{\bm{\delta}_{\alpha}}\cdot\mathbf{k}}$. We show these bands in Fig. 8(a). Additionally, discrete lattice symmetries give rise to degeneracies between the sublattice bands in the form of a linear band touching at the $K$ and $K^{\prime}$ points, and a quadratic band touching at the zone center, $\Gamma$.

The intra-dimer Heisenberg coupling term, $J$, is nothing but the singlet–triplet gap, separating the triplet bands from the ground state, while the inter-dimer Heisenberg interaction, $J^{\prime}$ gives dispersion to the triplets. The form of dispersion is immediately familiar from studies of graphene Haldane (1988); Kane and Mele (2005a), with the additional feature of a flat band just above the singlet–triplet gap, reflecting the frustration of the Kagome lattice.

From the band dispersion, Eq. (74), we can also read off the instabilities of the quantum paramagnet. For the ferromagnetic $J^{\prime}\!=\!-J/4$, the $\omega_{3,m}$ softens at the $\Gamma$ point, signaling a transition to a long-range ordered time-reversal symmetry breaking state, where the spins on each layer are aligned ferromagneticaly, while the two layers are aligned antiferromagneticaly. Similarly, we may notice that the energy of the $\omega_{1,m}$ flat band becomes 0 for antiferromagnetic $J^{\prime}=J/2$, indicating a transition to a time-reversal symmetry breaking state, where the ordering is selected by the DM interactions or quantum fluctuations.