Model Hamiltonian and Time Reversal Breaking Topological Phases of Anti-ferromagnetic Half-Heusler Materials

TL;DR

This paper develops a generalized Kane model to describe various topological phases in anti-ferromagnetic half-Heusler materials, revealing robust anti-ferromagnetic topological insulators and potential for experimental realization.

Contribution

It introduces a new coupling term in the Kane model to capture anti-ferromagnetic effects and identifies multiple topological phases in these materials.

Findings

Identification of diverse topological phases including Dirac, Weyl, and nodal line semimetals.

Discovery that the system becomes an anti-ferromagnetic topological insulator when a bulk gap opens.

The model suggests feasible experimental realization in real materials.

Abstract

In this work, we construct a generalized Kane model with a new coupling term between itinerant electron spins and local magnetic moments of anti-ferromagnetic ordering in order to describe the low energy effective physics in a large family of anti-ferromagnetic half-Heusler materials. Topological properties of this generalized Kane model is studied and a large variety of topological phases, including Dirac semimetal phase, Weyl semimetal phase, nodal line semimetal phase, type-B triple point semimetal phase, topological mirror (or glide) insulating phase and anti-ferromagnetic topological insulating phase, are identified in different parameter regions of our effective models. In particular, we find that the system is always driven into the anti-ferromagnetic topological insulator phase once a bulk band gap is open, irrespective of the magnetic moment direction, thus providing a robust realization of anti-ferromagentic topological insulators. Furthermore, we discuss the possible realization of these topological phases in realistic anti-ferromagnetic half-Heusler materials. Our effective model provides a basis for the future study of physical phenomena in this class of materials.

Figures

Tables

Table 1. Table 1: Expressions for the ξ i subscript 𝜉 𝑖 \xi_{i} parameters ( i = 0 , 1 , 2 , 3 , 4 , 5 𝑖 0 1 2 3 4 5 i=0,1,2,3,4,5 ) in the AFM Hamiltonian (Eq. 7 ). Here M i subscript 𝑀 𝑖 M_{i} ’s are three components of the anti-ferromagnetic magnetic moment, and α i subscript 𝛼 𝑖 \alpha_{i} ’s, β j subscript 𝛽 𝑗 \beta_{j} ’s are parameters independent of each other and momentum 𝐤 𝐤 \mathbf{k} .

Table 2. Table 2: Character table of C 3 ​ v subscript 𝐶 3 𝑣 C_{3v} group. Gelessus  et al. ( 1995 ); Aroyo  et al. ( 2006 )

Table 3. Table 3: This table shows combinations of J i subscript 𝐽 𝑖 J_{i} ’s, their corresponding C 3 ​ v subscript 𝐶 3 𝑣 C_{3v} irreducible representations and their parities under time reversal Θ Θ \Theta and half translation T 𝐚 3 ′ subscript 𝑇 subscript superscript 𝐚 ′ 3 T_{\mathbf{a}^{\prime}_{3}} . The first column shows the corresponding irreducible representations of C 3 ​ v subscript 𝐶 3 𝑣 C_{3}v . The second column shows the expressions of combinations. The third and fourth columns show how signs of those combinations change under Θ Θ \Theta and T 𝐚 3 ′ subscript 𝑇 subscript superscript 𝐚 ′ 3 T_{\mathbf{a}^{\prime}_{3}} , where “ + + ” means no sign change and “ − - ” means the sign should change.

Table 4. Table 4: This table shows combinations of k i subscript 𝑘 𝑖 k_{i} ’s or M i subscript 𝑀 𝑖 M_{i} ’s, their corresponding C 3 ​ v subscript 𝐶 3 𝑣 C_{3v} irreducible representations and their parities under time reversal Θ Θ \Theta and half translation T 𝐚 3 ′ subscript 𝑇 subscript superscript 𝐚 ′ 3 T_{\mathbf{a}^{\prime}_{3}} . The first column shows the corresponding irreducible representations of C 3 ​ v subscript 𝐶 3 𝑣 C_{3v} . The second column shows the expressions of combinations. The third and fourth columns show how signs of those combinations change under Θ Θ \Theta and T 𝐚 3 ′ subscript 𝑇 subscript superscript 𝐚 ′ 3 T_{\mathbf{a}^{\prime}_{3}} , where “ + + ” means no sign change and “ − - ” means the sign should change.

Table 5. Table 5: Character Table of Spin Double Group C 3 ​ v ∗ superscript subscript 𝐶 3 𝑣 C_{3v}^{*} Burns ( 2014 )

$C_{3v}^{\ast }$	E	R	$C_3$	${\sigma }_v$
${\Lambda }_4$	1	-1	-1	i
${\Lambda }_5$	1	-1	-1	-i
${\Lambda }_6$	2	-2	1	0

Table 6. Table 6: Choices of parameters for Fig. 2 a. ξ 4 = ξ 5 subscript 𝜉 4 subscript 𝜉 5 \xi_{4}=\xi_{5} , ξ 3 / ε 0 ∈ [ − 0.3 , 0.3 ] subscript 𝜉 3 subscript 𝜀 0 0.3 0.3 \xi_{3}/\varepsilon_{0}\in[-0.3,0.3] and ξ 4 / ε 0 ∈ [ − 0.3 , 0.3 ] subscript 𝜉 4 subscript 𝜀 0 0.3 0.3 \xi_{4}/\varepsilon_{0}\in[-0.3,0.3] , where a 𝑎 a is a real parameter with the unit of length.

Table 7. Table 7: Choices of parameters for Fig. 3 a. ξ 4 = ξ 5 subscript 𝜉 4 subscript 𝜉 5 \xi_{4}=\xi_{5} , ξ 3 / ε 0 ∈ [ − 0.3 , 0.3 ] subscript 𝜉 3 subscript 𝜀 0 0.3 0.3 \xi_{3}/\varepsilon_{0}\in[-0.3,0.3] and ξ 4 / ε 0 ∈ [ − 0.3 , 0.3 ] subscript 𝜉 4 subscript 𝜀 0 0.3 0.3 \xi_{4}/\varepsilon_{0}\in[-0.3,0.3] .

Table 8. Table 8: Choices of parameters for Fig. 5 . ξ 4 = ξ 5 subscript 𝜉 4 subscript 𝜉 5 \xi_{4}=\xi_{5} .

Table 9. Table 9: Choices of parameters for Fig. 6 . ξ 3 = ξ 4 = ξ 5 ≡ ξ subscript 𝜉 3 subscript 𝜉 4 subscript 𝜉 5 𝜉 \xi_{3}=\xi_{4}=\xi_{5}\equiv\xi .

Table 10. Table 10: Choices of parameters for Fig. 7 a. Here δ ≡ P 2 3 ​ ( E v + ξ 0 − E c ) ​ β c 𝛿 superscript 𝑃 2 3 subscript 𝐸 𝑣 subscript 𝜉 0 subscript 𝐸 𝑐 subscript 𝛽 𝑐 \delta\equiv\frac{P^{2}}{3(E_{v}+\xi_{0}-E_{c})\beta_{c}} , ξ 1 / ε 0 ∈ ( − 0.9 , 0.9 ) subscript 𝜉 1 subscript 𝜀 0 0.9 0.9 \xi_{1}/\varepsilon_{0}\in(-0.9,0.9) , ξ 2 / ε 0 ∈ ( − 0.9 , 0.9 ) subscript 𝜉 2 subscript 𝜀 0 0.9 0.9 \xi_{2}/\varepsilon_{0}\in(-0.9,0.9) .

Table 11. Table 11: Choices of parameters for Fig. 7 b. Here δ ≡ P 2 3 ​ ( E v + ξ 0 − E c ) ​ β c 𝛿 superscript 𝑃 2 3 subscript 𝐸 𝑣 subscript 𝜉 0 subscript 𝐸 𝑐 subscript 𝛽 𝑐 \delta\equiv\frac{P^{2}}{3(E_{v}+\xi_{0}-E_{c})\beta_{c}} , ξ 1 / ε 0 ∈ ( − 0.9 , 0.9 ) subscript 𝜉 1 subscript 𝜀 0 0.9 0.9 \xi_{1}/\varepsilon_{0}\in(-0.9,0.9) , ξ 3 / ε 0 ∈ ( − 0.9 , 0.9 ) subscript 𝜉 3 subscript 𝜀 0 0.9 0.9 \xi_{3}/\varepsilon_{0}\in(-0.9,0.9) .

Table 12. Table 12: Choices of parameters for Fig. 8 a.Here δ ≡ P 2 3 ​ ( E v + ξ 0 − E c ) ​ β c 𝛿 superscript 𝑃 2 3 subscript 𝐸 𝑣 subscript 𝜉 0 subscript 𝐸 𝑐 subscript 𝛽 𝑐 \delta\equiv\frac{P^{2}}{3(E_{v}+\xi_{0}-E_{c})\beta_{c}} , ξ 1 / ε 0 ∈ ( − 0.9 , 0.9 ) subscript 𝜉 1 subscript 𝜀 0 0.9 0.9 \xi_{1}/\varepsilon_{0}\in(-0.9,0.9) , ξ 2 / ε 0 ∈ ( − 0.9 , 0.9 ) subscript 𝜉 2 subscript 𝜀 0 0.9 0.9 \xi_{2}/\varepsilon_{0}\in(-0.9,0.9) and the choice of ( ξ 3 , ξ 4 , ξ 5 ) subscript 𝜉 3 subscript 𝜉 4 subscript 𝜉 5 (\xi_{3},\xi_{4},\xi_{5}) breaks the symmetry group C 3 ​ v subscript 𝐶 3 𝑣 C_{3v} and its combination with T 𝐚 3 ′ subscript 𝑇 subscript superscript 𝐚 ′ 3 T_{\mathbf{a}^{\prime}_{3}} .

Equations

$$H_{Kane}=\left(\begin{array}[]{c|c}H_{\Gamma_{6}}(\mathbf{k})&V\\ \hline\cr V^{\dagger}&H_{\Gamma_{8}}(\mathbf{k})\\ \end{array}\right),$$

$$\begin{array}[]{l}V=\frac{P}{\sqrt{6}}\left(\begin{matrix}-\sqrt{3}k_{+}&2k_{z}&k_{-}&0\\ 0&-k_{+}&2k_{z}&\sqrt{3}k_{-}\end{matrix}\right)\\ +\frac{B^{+}_{8v}}{\sqrt{6}}\left(\begin{matrix}\sqrt{3}k_{-}k_{z}&2ik_{x}k_{y}&k_{+}k_{z}&0\\ 0&k_{-}k_{z}&2ik_{x}k_{y}&\sqrt{3}k_{+}k_{z}\end{matrix}\right)\\ +\frac{B^{-}_{8v}}{3\sqrt{2}}\left(\begin{matrix}0&\sqrt{3}K^{2}&0&k_{\parallel}^{2}-2k_{z}^{2}\\ -k_{\parallel}^{2}+2k_{z}^{2}&0&-\sqrt{3}K^{2}&0\end{matrix}\right)\end{array},$$

$$H_{{\Gamma }_6}(\mathbf{k})=(E_c+{\beta }_c{k}^2)1_2$$

$$H_{{\Gamma }_8}(\mathbf{k})=H_0(\mathbf{k})+H_C(\mathbf{k})$$

$$\begin{array}[]{c}H_{0}(\mathbf{k})=\frac{4}{15}(J_{x}^{2}+J_{y}^{2}+J_{z}^{2})h_{0}+\frac{1}{3}(2J_{z}^{2}-J_{x}^{2}-J_{y}^{2})h_{1}\\ +\frac{1}{\sqrt{3}}(J_{x}^{2}-J_{y}^{2})h_{2}+\frac{2}{\sqrt{3}}J_{xy}h_{3}+\frac{2}{\sqrt{3}}J_{zx}h_{4}+\frac{2}{\sqrt{3}}J_{yz}h_{5}\end{array}$$

$$H_C(\mathbf{k})=\frac{2}{\sqrt{3}}C(k_x{V}_x+k_y{V}_y+k_z{V}_z).$$

$$\begin{array}[]{l}H_{AFM}=\frac{4}{15}(J_{x}^{2}+J_{y}^{2}+J_{z}^{2})\xi_{0}+\frac{1}{3}(2J_{z}^{2}-J_{x}^{2}-J_{y}^{2})\xi_{1}\\ +\frac{1}{\sqrt{3}}(J_{x}^{2}-J_{y}^{2})\xi_{2}+\frac{2}{\sqrt{3}}J_{xy}\xi_{3}+\frac{2}{\sqrt{3}}J_{zx}\xi_{4}+\frac{2}{\sqrt{3}}J_{yz}\xi_{5},\\ \end{array}$$

$$E_{0+AFM,\pm }(\mathbf{k})=(h_0+{\xi }_0)\pm \sqrt{\sum _{i=1}^5(h_i+{\xi }_i{)}^2}.$$

$$\begin{array}[]{l}\mathbf{K}_{1}a=\{0.30,-0.047,-0.47\},\\ \mathbf{K}_{2}a=\{-0.047,0.30,-0.47\},\\ \mathbf{K}_{3}a=\{0.047,-0.30,0.47\},\\ \mathbf{K}_{4}a=\{-0.30,0.047,0.47\}\end{array}$$

$$\begin{array}[]{l}E_{\Lambda_{6}}(k)=E_{v}+\xi_{0}-3\beta_{c}\gamma_{1}k^{2}-(6\beta_{c}\gamma_{3}k^{2}+\sqrt{3}\xi)\\ E_{\Lambda_{5}}(k)=E_{v}+\xi_{0}-3\beta_{c}\gamma_{1}k^{2}+(6\beta_{c}\gamma_{3}k^{2}+\sqrt{3}\xi)+\sqrt{6}Ck\\ E_{\Lambda_{4}}(k)=E_{v}+\xi_{0}-3\beta_{c}\gamma_{1}k^{2}+(6\beta_{c}\gamma_{3}k^{2}+\sqrt{3}\xi)-\sqrt{6}Ck\end{array},$$

$$H_{Kane}^{AFM}=\left(\begin{array}[]{c|c}H_{\Gamma_{6}}(\mathbf{k})&V\\ \hline\cr V^{\dagger}&H_{\Gamma_{8}}(\mathbf{k})+H_{AFM}\\ \end{array}\right),$$

$$\begin{array}[]{c}\left|\Gamma_{6},\frac{1}{2},1\right\rangle=i\left|S,\uparrow,1\right\rangle\\ \left|\Gamma_{6},-\frac{1}{2},1\right\rangle=i\left|S,\downarrow,1\right\rangle\\ \left|\Gamma_{8},\frac{3}{2},1\right\rangle=-\frac{1}{\sqrt{2}}\left|X,\uparrow,1\right\rangle+\frac{i}{\sqrt{2}}\left|Y,\uparrow,1\right\rangle\\ \left|\Gamma_{8},\frac{1}{2},1\right\rangle=-\frac{1}{\sqrt{6}}\left|X,\downarrow,1\right\rangle+\frac{i}{\sqrt{6}}\left|Y,\downarrow,1\right\rangle+\sqrt{\frac{2}{3}}\left|Z,\uparrow,1\right\rangle\\ \left|\Gamma_{8},-\frac{1}{2},1\right\rangle=\frac{1}{\sqrt{6}}\left|X,\uparrow,1\right\rangle+\frac{i}{\sqrt{6}}\left|Y,\uparrow,1\right\rangle+\sqrt{\frac{2}{3}}\left|Z,\downarrow,1\right\rangle\\ \left|\Gamma_{8},-\frac{3}{2},1\right\rangle=\frac{1}{\sqrt{2}}\left|X,\downarrow,1\right\rangle+\frac{i}{\sqrt{2}}\left|Y,\downarrow,1\right\rangle\\ \left|\Gamma_{7},\frac{1}{2},1\right\rangle=-\frac{1}{\sqrt{3}}\left|X,\downarrow,1\right\rangle+\frac{i}{\sqrt{3}}\left|Y,\downarrow,1\right\rangle-\frac{1}{\sqrt{3}}\left|Z,\uparrow,1\right\rangle\\ \left|\Gamma_{7},-\frac{1}{2},1\right\rangle=-\frac{1}{\sqrt{3}}\left|X,\uparrow,1\right\rangle-\frac{i}{\sqrt{3}}\left|Y,\uparrow,1\right\rangle+\frac{1}{\sqrt{3}}\left|Z,\downarrow,1\right\rangle\\ \left|\Gamma_{6},\frac{1}{2},2\right\rangle=i\left|S,\uparrow,2\right\rangle\\ \left|\Gamma_{6},-\frac{1}{2},2\right\rangle=i\left|S,\downarrow,2\right\rangle\\ \left|\Gamma_{8},\frac{3}{2},2\right\rangle=-\frac{1}{\sqrt{2}}\left|X,\uparrow,2\right\rangle+\frac{i}{\sqrt{2}}\left|Y,\uparrow,2\right\rangle\\ \left|\Gamma_{8},\frac{1}{2},2\right\rangle=-\frac{1}{\sqrt{6}}\left|X,\downarrow,2\right\rangle+\frac{i}{\sqrt{6}}\left|Y,\downarrow,2\right\rangle+\sqrt{\frac{2}{3}}\left|Z,\uparrow,2\right\rangle\\ \left|\Gamma_{8},-\frac{1}{2},2\right\rangle=\frac{1}{\sqrt{6}}\left|X,\uparrow,2\right\rangle+\frac{i}{\sqrt{6}}\left|Y,\uparrow,2\right\rangle+\sqrt{\frac{2}{3}}\left|Z,\downarrow,2\right\rangle\\ \left|\Gamma_{8},-\frac{3}{2},2\right\rangle=\frac{1}{\sqrt{2}}\left|X,\downarrow,2\right\rangle+\frac{i}{\sqrt{2}}\left|Y,\downarrow,2\right\rangle\\ \left|\Gamma_{7},\frac{1}{2},2\right\rangle=-\frac{1}{\sqrt{3}}\left|X,\downarrow,2\right\rangle+\frac{i}{\sqrt{3}}\left|Y,\downarrow,2\right\rangle-\frac{1}{\sqrt{3}}\left|Z,\uparrow,2\right\rangle\\ \left|\Gamma_{7},-\frac{1}{2},2\right\rangle=-\frac{1}{\sqrt{3}}\left|X,\uparrow,2\right\rangle-\frac{i}{\sqrt{3}}\left|Y,\uparrow,2\right\rangle+\frac{1}{\sqrt{3}}\left|Z,\downarrow,2\right\rangle\\ \end{array},$$

$$J_{x}\doteq\left(\begin{array}[]{cccc}0&\frac{\sqrt{3}}{2}&0&0\\ \frac{\sqrt{3}}{2}&0&1&0\\ 0&1&0&\frac{\sqrt{3}}{2}\\ 0&0&\frac{\sqrt{3}}{2}&0\\ \end{array}\right)$$

$$J_{y}\doteq\left(\begin{array}[]{cccc}0&-\frac{i\sqrt{3}}{2}&0&0\\ \frac{i\sqrt{3}}{2}&0&-i&0\\ 0&i&0&-\frac{i\sqrt{3}}{2}\\ 0&0&\frac{i\sqrt{3}}{2}&0\\ \end{array}\right)$$

$$J_{z}\doteq\left(\begin{array}[]{cccc}\frac{3}{2}&0&0&0\\ 0&\frac{1}{2}&0&0\\ 0&0&-\frac{1}{2}&0\\ 0&0&0&-\frac{3}{2}\\ \end{array}\right).$$

$$C_3(111)\doteq \exp (-i\frac{J_x+J_y+J_z}{\sqrt{3}}\frac{2\pi }{3})$$

$${\pi }_{1\bar{1}0}\doteq -\exp (-i\frac{J_x-J_y}{\sqrt{2}}\pi ).$$

$$\Theta\doteq\left(\begin{array}[]{cccc}0&0&0&-1\\ 0&0&1&0\\ 0&-1&0&0\\ 1&0&0&0\\ \end{array}\right)\hat{K},$$

$$T_{\mathbf{a}^{\prime}_{3}}\doteq\left(\begin{array}[]{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{array}\right),$$

$$\mathbf{k}a=(0,0,\pm \sqrt{-\frac{{\xi }_1}{2{\beta }_c{\gamma }_2}})$$

$$\mathbf{k}a=\pm (\sqrt{-\frac{{\xi }_2}{\sqrt{3}{\beta }_c{\gamma }_2}},0,0)$$

$$\mathbf{k}a=\pm (0,\sqrt{\frac{{\xi }_2}{\sqrt{3}{\beta }_c{\gamma }_2}},0)$$

$$\mathbf{k}a=\pm (0,\frac{1}{\sqrt[4]{3}}\sqrt{\frac{{\xi }_2}{{\beta }_c{\gamma }_2}},-\frac{1}{\sqrt[4]{3}}\sqrt{\frac{{\gamma }_2{\beta }_c}{{\xi }_2}}\frac{{\xi }_5}{2{\gamma }_3{\beta }_c})$$

$$\mathbf{k}a=\pm (\frac{1}{\sqrt[4]{3}}\sqrt{-\frac{{\xi }_2}{{\beta }_c{\gamma }_2}},0,-\frac{1}{\sqrt[4]{3}}\sqrt{-\frac{{\beta }_c{\gamma }_2}{{\xi }_2}}\frac{{\xi }_4}{2{\gamma }_3{\beta }_c})$$

$$\mathbf{k}a=\pm (-\frac{{\xi }_3}{2\sqrt{3}{\beta }_c{\gamma }_3}\sqrt{\frac{2\sqrt{3}{\beta }_c{\gamma }_2}{\sqrt{3}{\xi }_1+{\xi }_2}},\sqrt{\frac{\sqrt{3}{\xi }_1+{\xi }_2}{2\sqrt{3}{\beta }_c{\gamma }_2}},0)$$

$$\mathbf{k}a=\pm \frac{1}{\sqrt{2}\sqrt[4]{3}}(\sqrt{-\frac{{\xi }_3{\xi }_4}{{\beta }_c{\gamma }_3{\xi }_5}},\sqrt{-\frac{{\xi }_3{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_4}},\sqrt{-\frac{{\xi }_4{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_3}})$$

$$\mathbf{k}a=\pm \frac{1}{\sqrt{2}\sqrt[4]{3}}(\sqrt{-\frac{{\xi }_3{\xi }_4}{{\beta }_c{\gamma }_3{\xi }_5}},\sqrt{-\frac{{\xi }_3{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_4}},-\sqrt{-\frac{{\xi }_4{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_3}})$$

$$\mathbf{k}a=\pm \frac{1}{\sqrt{2}\sqrt[4]{3}}(\sqrt{-\frac{{\xi }_3{\xi }_4}{{\beta }_c{\gamma }_3{\xi }_5}},-\sqrt{-\frac{{\xi }_3{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_4}},\sqrt{-\frac{{\xi }_4{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_3}})$$

$$\mathbf{k}a=\pm \frac{1}{\sqrt{2}\sqrt[4]{3}}(-\sqrt{-\frac{{\xi }_3{\xi }_4}{{\beta }_c{\gamma }_3{\xi }_5}},\sqrt{-\frac{{\xi }_3{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_4}},\sqrt{-\frac{{\xi }_4{\xi }_5}{{\beta }_c{\gamma }_3{\xi }_3}})$$

$$H_D(K_p+q_x,K_p+q_y,K_z+q_z)\approx {\Gamma }_0+q_x{\Gamma }_x+q_y{\Gamma }_y+q_z{\Gamma }_z,$$

Full text

I Introduction

The discovery of time reversal invariant topological insulators (TIs) Qi and Zhang (2011); Hasan and Kane (2010) provides us the first example of a novel topological state that is protected by certain types of symmetry (time reversal symmetry), and greatly deepens our understanding of the role of symmetry and topology in electronic band structures of solid materials. Soon after this discovery, the idea of symmetry protected topological states is generalized to other systems, leading to different topological states, including topological crystalline insulatorsFu (2011); Lin et al. (2010a); Tanaka et al. (2012); Xu et al. (2012); Hsieh et al. (2012) that are protected by crystalline symmetry, topological superconductorsAlicea (2012); Schnyder et al. (2008); Lutchyn et al. (2010); Beenakker (2013) which require particle-hole symmetry, and topolgical semimetals (TSMs)Yan and Felser (2016); Yang et al. (2015); Wan et al. (2011); Weng et al. (2015); Huang et al. (2015); Xu et al. (2015); Lv et al. (2015); Soluyanov et al. (2015); Wang et al. (2012); Liu et al. (2014); Wang et al. (2013); Burkov et al. (2011); Bradlyn et al. (2016). Most current experimental studies of topological states are focused on non-magnetic materials that preserve time reversal symmetryYan and Zhang (2012) and ferromagnetic materials (mainly the quantum anomalous Hall effect)Haldane (1988); Liu et al. (2008); Chang et al. (2013); Yu et al. (2010). Theoretically, a large variety of topological states can also exist in materials with other types of magnetic structures, such as anti-ferromagnetism (AFM) Mong et al. (2010); Fang et al. (2013); Yoshida et al. (2013); Wu et al. (2015); Bègue et al. (2016); Brzezicki and Cuoco (2016); Young and Wieder (2016). Nevertheless, material proposals of these topological states are still rare.

Half-Heusler compounds are a large group of materials consisting of three metal elements and have been widely studied for their flexible electronic properties and functionalities Graf et al. (2011). Around 50 half-Heusler compounds are theoretically predicted to possess inverted band structure and can be driven into the topological insulating phase by applying strains Lin et al. (2010b); Chadov et al. (2010); Xiao et al. (2010); Al-Sawai et al. (2010); Yan and de Visser (2014). Unusual surface states were recently observed experimentally in LnPtBi(Ln=Lu, Y) Liu et al. (2016) and LuPtSb Logan et al. (2016), and serve as an evidence of non-trivial bulk topology. Weyl semimetal (WSM) phase was theoretically discussed in several half-Heusler compounds, including GdPtBi under external magnetic fieldsCano et al. (2016) and LaPtBi with in-plane strainRuan et al. (2016), and the corresponding evidences were found in recent experimentsHirschberger et al. (2016); Shekhar et al. (2016); Suzuki et al. (2016). Our interest in this work is focused on possible topological states in half-Heusler materials with AFM at zero external magnetic field. AFM has been experimentally observed in RPdBi (R $=$ Er, Ho, Gd, Dy, Tb, Nd) and GdPtBi Pan et al. (2013); Gofryk et al. (2011); Müller et al. (2014); Nikitin et al. (2015); Nakajima et al. (2015); Pavlosiuk et al. (2016a, b), with Neel temperature ranging from 1K to 13K. To describe AFM in half-Heusler compounds, we construct a generalized six-band Kane model with anti-ferromagnetic coupling terms based on the symmetry principle and justify this model with the microscopic tight-binding model. With this model, we predicted a large variety of time reversal breaking TSM phases, including Dirac semimetal (DSM) phase (if inversion symmetry breaking is insignificant), WSM phase, nodal line semimetal (NLSM) phase (if AFM preserves mirror or glide symmetry) and Type-B triple point semimetal (TPSM) phaseZhu et al. (2016) (if AFM preserves $C_{3}$ and glide symmetries), and topological insulating phases, including topological mirror (or glide) insulating (TMI) phase and anti-ferromagnetic topological insulating (AFMTI) phase that is protected by the combination of time reversal and half translation, depending on the material parameters. We also discuss phase diagrams of this model and identify the candidate half-Heulser compounds to search for these topological phases experimentally.

This paper is organized as follows. In Sec.II, we first present our generalized six-band Kane model and describe the symmetry aspect of this model. In Sec.III, we focus on the $4\times 4$ block of this model which is relatively simple and includes all the four $\Gamma_{8}$ bands that are close to Fermi energy. A variety of topological phases, including DSM phase, WSM phase, NLSM phase, type-B TPSM phase and TMI phase, were identified for different aligning direction of magnetic moments of AFM. In Sec.IV, we discuss the limitation of the four-band model and extract the phase diagram of the six-band Kane model. We demonstrate a robust realization of AFMTI phase in half-Heusler materials with AFM once anti-ferromagnetic coupling is strong enough. The conclusion is drawn in Sec.V.

II Model Hamiltonian of Anti-ferromagnetic Half-Heusler Materials

We start with a derivation of the model Hamiltonian for half-Heusler materials with AFM. The half-Heusler compound is normally labelled by XYZ Canfield et al. (1991), in which two metal atoms X and Y together play the role of cations and the metal atom Z is regarded as anions. The crystal structure of half-Heusler materials is shown in Fig.1a, in which X and Z atoms form the NaCl-type substructure and Y and Z atoms form the zinc-blende substructure. For certain half-Heusler materials with AFM, including GdPtBi, DyPdBi, HoPdBi and TbPdBi Pavlosiuk et al. (2016a); Nakajima et al. (2015); Müller et al. (2014), magnetic moments come from the X atom and align ferromagnetically in one layer perpendicular to (111) direction and anti-ferromagnetically between two adjacent layers (type-G anti-ferromagnetic ordering), as shown in Fig.1b, while the exact direction of magnetic moments is still not clear and may depend on detailed compound composition.

To construct the effective Hamiltonian, we need to understand the symmetry aspect of this crystal. Without magnetic moments, the space group of half-Heusler compounds is F$\bar{4}3$m Canfield et al. (1991) and the corresponding point group is $T_{d}$, similar to zinc-blende structure. As a result, the lattice vectors should be chosen to be those of a face-centered cubic lattice, labelled as $\mathbf{a}_{1}$, $\mathbf{a}_{2}$ and $\mathbf{a^{\prime}}_{3}$ in Fig.1b. The existence of type-G AFM has two main effects to the symmetry of the system: (1) it doubles the lattice vector and the unit-cell along the $\mathbf{a^{\prime}}_{3}$ direction, and the new lattice vector is labeled as $\mathbf{a}_{3}=2\mathbf{a^{\prime}}_{3}$; (2) in the mean field level, the magnetic moments of AFM give rise to exchange coupling to electron spins. Due to the doubling of unit-cell, the space group and point group of the crystalline structure are reduced to $R3m$Li et al. (2015) and $C_{3v}$, where the latter has two generators: three-fold rotation $C_{3}{(111)}$ along (111) direction and mirror symmetry $\pi_{(1\bar{1}0)}$ with respect to $(1\bar{1}0)$ plane. As a pseudo-vector, magnetic moments of AFM can further lower the symmetry. Besides the above crystal symmetries, another essential symmetry due to AFM is the combination of time reversal $\Theta$ and half tranlation $T_{\mathbf{a}^{\prime}_{3}}$, denoted as $S\equiv T_{\mathbf{a}^{\prime}_{3}}\Theta$.

Similar to the semiconductors with the zinc-blende structure, the low energy physics of half-Heusler materials can be described by two $\Gamma_{6}$ bands with s-orbital nature, labelled by $|\Gamma_{6},S_{z}\rangle$ with z-directional spin component $S_{z}=\pm\frac{1}{2}$, and four $\Gamma_{8}$ bands with p-orbital nature, labelled by $|\Gamma_{8},J_{z}\rangle$ with total angular momentum $J_{z}=\pm\frac{1}{2},\pm\frac{3}{2}$ which is a combination of spin and orbital angular momentum. Here the notations $\Gamma_{6}$ and $\Gamma_{8}$ refer to the irreducible representations of the corresponding bands under the $T_{d}$ group Winkler et al. (2003) and we still keep this notation even though the symmetry is lowered. The form and symmetry of the basis wave functions are also justified with microscopic local atomic orbitals, as discussed in details in Appendix A. It should be emphasized that due to the doubling of the unit-cell, the basis wave functions in the new unit-cell is a bonding or anti-bonding state of the basis functions in the original unit-cell. For the low energy physics, we find both $|\Gamma_{6},S_{z}\rangle$ and $|\Gamma_{8},J_{z}\rangle$ are bonding states in the doubled unit-cell and are still suitable to be bases. To obtain a matrix expansion of the effective Hamiltonian, we need to classify all the matrices according to the irreducible representations of the symmetry group for the above basis. The Hamiltonian should be invariant under symmetry operations of the symmetry group, which means it belongs to the identity representation, and can be constructed by combining the symmetrized matrices and symmetrized tensor components of physical quantities (such as the momentum ${\bf k}$ and the order parameters ${\bf M}$ for AFM) which belong to the same irreducible representation.Winkler et al. (2003) This symmetry approach allows us to systematically construct the full Hamiltonian, including the standard six-band Kane model and an additional AFM term. The standard six-band Kane model is Winkler et al. (2003)

[TABLE]

where

[TABLE]

[TABLE]

with $\mathbb{1}_{2}$ to be the $2\times 2$ identity matrix,

[TABLE]

with

[TABLE]

and

[TABLE]

Here we have $h_{0}=E_{v}-\beta_{c}\gamma_{1}k^{2}$, $h_{1}=\beta_{c}\gamma_{2}(2k_{z}^{2}-k_{\parallel}^{2})$, $h_{2}=\sqrt{3}\beta_{c}\gamma_{2}K^{2}$, $h_{3}=2\sqrt{3}\beta_{c}\gamma_{3}k_{x}k_{y}$, $h_{4}=2\sqrt{3}\beta_{c}\gamma_{3}k_{x}k_{z}$, and $h_{5}=2\sqrt{3}\beta_{c}\gamma_{3}k_{y}k_{z}$. $J_{i}$’s are angular momentum matrices for spin $3/2$ (see Appendix B for details), $J_{ij}=\frac{1}{2}\{J_{i},J_{j}\}$, $V_{x}=\frac{1}{2}\{J_{x},J_{y}^{2}-J_{z}^{2}\}$, $V_{y}=\frac{1}{2}\{J_{y},J_{z}^{2}-J_{x}^{2}\}$, $V_{z}=\frac{1}{2}\{J_{z},J_{x}^{2}-J_{y}^{2}\}$, $\beta_{c}=\hbar^{2}/(2m^{\prime})$, $m^{\prime}$ is the effective mass of $\Gamma_{6}$ bands near $\Gamma$ point, $k^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}$, $k_{\parallel}^{2}=k_{x}^{2}+k_{y}^{2}$, $K^{2}=k_{x}^{2}-k_{y}^{2}$ and $k_{\pm}=k_{x}\pm ik_{y}$. In this work, $\beta_{c}>0$, $\gamma_{2}\neq 0$ and $\gamma_{3}\neq 0$ are always assumed, unless being specified otherwise. Besides the above terms in the standard Kane model, AFM can lead to new terms, which can be constructed in a similar manner with the symmetry group combining point group $C_{3v}$, $T_{\mathbf{a}^{\prime}_{3}}$ and $\Theta$ (see Appendix B for more details). For the materials that we are interested in, the main influence of AFM only occurs for the four $\Gamma_{8}$ bands. Thus, we only consider the anti-ferromagnetic terms in the basis of the four $\Gamma_{8}$ bands and the corresponding Hamiltonian is given by

[TABLE]

where the detailed expressions for $\xi_{i}$ ($i=0,\dots,5$) are listed in Tab.1.

Next we discuss the symmetry properties of this Hamiltonian. For the standard Kane model $H_{Kane}$, if the parameter $C$ in $H_{C}$ is zero, the Hamiltonian possess $O_{h}$ point group. Non-zero $H_{C}$ term breaks inversion symmetry and lowers the point group from $O_{h}$ to $T_{d}$. The existence of AFM results in the doubling of the unit cell along $\mathbf{a}^{\prime}_{3}$ and reduce the point group of the lattice to $C_{3v}$. For a fixed non-zero anti-ferromagnetic order ${\bf M}$, the AFM term $H_{AFM}$ will further reduce the symmetry. If $\mathbf{M}$ is along the (111) direction, $C_{3}(111)$ symmetry is maintained but all mirror symmetries in $C_{3v}$ are broken, whereas the glide symmetries $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$, $T_{\mathbf{a}^{\prime}_{3}}\pi_{(\bar{1}01)}$ and $T_{\mathbf{a}^{\prime}_{3}}\pi_{(01\bar{1})}$ are preserved. If $\mathbf{M}$ lies in a mirror plane $\alpha$ ($\alpha=(1\bar{1}0)$ or $(\bar{1}01)$ or $(01\bar{1})$) but away from the (111) direction, all symmetries in $C_{3v}$ are broken, whereas the glide symmetry $T_{\mathbf{a}^{\prime}_{3}}\pi_{\alpha}$ is preserved. If $\mathbf{M}$ is perpendicular to a mirror plane of the lattice, the only remaining symmetry is that mirror symmetry. For a generic AFM Hamiltonian, $\mathbf{M}$ will break all the symmetries in $C_{3v}$ group, as well as the combination with $T_{\mathbf{a}^{\prime}_{3}}$. Furthermore, we notice that only quadratic terms of $\mathbf{M}$ appear in our Hamiltonian while any linear $\mathbf{M}$ terms vanish. This is because $\mathbf{M}$ reverses its sign under translation $T_{\mathbf{a}^{\prime}_{3}}$, while translation $T_{\mathbf{a}^{\prime}_{3}}$ is just identity matrix for the basis of four $\Gamma_{8}$ bands and thus commutes with any representation matrix (see Appendix B for more details). This suggests that any term with the odd orders of the anti-ferromagnetic order parameter $\mathbf{M}$ cannot exist.

III Topological Phases in the Four-band Model

Since we are interested in the half-Heusler materials with inverted band structures, only four $\Gamma_{8}$ bands appear near the Fermi energy while the $\Gamma_{6}$ bands are far below the Fermi energy. Thus, we first focus on the four $\Gamma_{8}$ bands with the Hamitonian $H_{\Gamma_{8}}(\mathbf{k})+H_{AFM}$. For the inverted band structure, the Fermi energy is between the $\Gamma_{8}$ bands, and the two $\Gamma_{8}$ bands with lower energies are valence bands while the other two $\Gamma_{8}$ bands with higher energies are conduction bands. We emphasize that the $\Gamma_{6}$ bands are important for certain types of topological states even though they are away from the Fermi energy, as discussed in details in the next section. However, for the TSM phases discussed in this section, only the $\Gamma_{8}$ bands are essential. Another advantage of the 4-band Hamiltonian is that it can be solved analytically in certain limit, thus providing us valuable insight into the underlying physics. In this section, we first focus on the case without inversion symmetry breaking term (i.e. choosing $H_{C}(\mathbf{k})=0$) and reveal the occurence of DSM phase due to the coexistence of inversion symmetry and anti-unitary $S$ symmetry. In realistic system, inversion symmetry is broken and DSM phase becomes unstable. Nevertheless, DSM phase can be viewed as the “parent” phase to generate other TSM phases after including $H_{C}(\mathbf{k})$. We further study the situation with non-zero inversion symmetry breaking term $H_{C}(\mathbf{k})$, focusing on the situations with (1) magnetic moments polarized within or perpendicular to the $(1\bar{1}0)$ plane and (2) $\mathbf{M}$ along the (111) direction.

III.1 Dirac Semimetal Phase and Topological Mirror Insulating Phase

In this part, $H_{C}(\mathbf{k})=0$ is assumed and the total Hamiltonian takes the form $H_{0}(\mathbf{k})+H_{AFM}$ with inversion symmetry. In this case, the eigenenergy of Hamiltonian can be solved analytically as

[TABLE]

Without AFM term, $\xi_{i}=0$ for $i=0,\dots,5$ in Eq.7, and there are four-fold degeneracy of the $\Gamma_{8}$ bands at the $\Gamma$ point ($\mathbf{k}=0$) due to the $T_{d}$ group symmetry. The conduction and valence bands touch each other quadratically at $\mathbf{k}=0$, leading to a critical semimetal phase for the four-band Luttinger model $H_{0}(\mathbf{k})$. Early studies have demonstrated various TSM phases induced by strain or external magnetic fields in this system Shekhar et al. (2016); Ruan et al. (2016). The AFM term $H_{AFM}$ can lower the symmetry of the system and remove the four-fold degeneracy at $\Gamma$ point. However, since $H_{AFM}$ preserves the inversion symmetry and $S$ symmetry, all the bands are still doubly degenerate, similar to the Kramer’s degeneracy due to inversion and time reversal symmetriesMurakami (2007).

Next we will study the influence of AFM term on the energy dispersion. The AFM term can lead to a non-zero gap at $\Gamma$ point, given by $2\sqrt{\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2}+\xi_{4}^{2}+\xi_{5}^{2}}$. Thus, the two doubly degenerate bands are split at the $\Gamma$ point, but they may cross each other at some finite momenta ${\bf k}$, giving rise to a semimetal phase. The realization of such semimetal phase requires $h_{i}+\xi_{i}=0$ for all $i=1,\dots,5$ and the details are discussed in Appendix C. Here we focus on the cases where magnetic moments of AFM are perpendicular to $(1\bar{1}0)$ plane ($M_{x}=-M_{y}$ and $M_{z}=0$, where $M_{x,y,z}$ are x,y,z components of AFM magnetic moment $\mathbf{M}$) with mirror symmetry $\pi_{(1\bar{1}0)}$, or lie in the $(1\bar{1}0)$ plane ($M_{x}=M_{y}$) with the glide symmetry $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$. According to the expressions of $\xi_{i}$’s, both conditions imply the same requirement for parameters: $\xi_{2}=0$ and $\xi_{4}=\xi_{5}$, which is reasonable since $\pi_{(1\bar{1}0)}$ and $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$ have the same matrix representation for four basis wave functions of $\Gamma_{8}$ bands. For the existence of gapless points, one of the following additional conditions is required for the values of $\xi_{1}$, $\xi_{3}$ and $\xi_{4}$ (see Appendix C for more details): (i) $\xi_{3}\beta_{c}\gamma_{3}<0$, $\sqrt{3}\gamma_{3}\xi_{3}\xi_{1}+\gamma_{2}\left(\xi_{3}^{2}-\xi_{4}^{2}\right)=0$; (ii) $\xi_{3}=\xi_{4}=0$, $\xi_{1}\beta_{c}\gamma_{2}<0$; (iii) $\xi_{4}=0$, $\xi_{3}\beta_{c}\gamma_{3}>0$, $\sqrt{3}\gamma_{3}\xi_{1}-\gamma_{2}\xi_{3}=0$. Due to $\xi_{2}=0$ and $\xi_{4}=\xi_{5}$, we require $k_{x}^{2}=k_{y}^{2}$ and $k_{x}k_{z}=k_{y}k_{z}$ from $h_{i}=-\xi_{i}$ with $i=2,4,5$ , indicating two possibilities for the locations of gapless point, either on the plane $(1\bar{1}0)$ with the form $(k_{1},k_{1},k_{3})$ for the conditions (i) and (ii), or on the $(1,-1,0)$ axis with the form $(k_{1},-k_{1},0)$ for the condition (iii). Due to the $S$ symmetry, if a gapless point occurs at a finite momentum ${\bf k}_{0}$, there must be another one at $-{\bf k}_{0}$, leading to even number of gapless points. The number of gapless points are confirmed to be 2 by solving for positions of gapless points in each case (see Appendix C for more details).

Based on the above conditions for gapless points, we can further extract the phase diagram of this model as a function of $(\xi_{3},\xi_{4})$. An example of a phase diagram is shown in Fig.2a for the choices of parameters listed in Tab.6 in Appendix F. The blue line in the phase diagram, labelled by II, represents DSM phase. For our choices of the parameters, the condition (i) or (ii) can be satisfied and Dirac points are on the $(1\bar{1}0)$ plane. Fig.2c reveals a typical energy dispersion of the semimetal phase at the point B in Fig.2a with $\xi_{3}/\varepsilon_{0}=-\frac{1}{10}$ and $\xi_{4}/\varepsilon_{0}=\frac{1}{50}\sqrt{8\sqrt{3}+25}$ to satisfy the condition (i), where the energy unit $\varepsilon_{0}$ is defined as $\varepsilon_{0}\equiv\beta_{c}/a^{2}$ and $a$ is a real positive parameter with unit of length. The energy dispersion around the gapless point behaves linearly, thus forming two Dirac cones at ${\bf k}a=\pm\frac{1}{\sqrt[4]{12}}(1,1,-\frac{1}{5}\sqrt{8\sqrt{3}+25})$, given the double degeneracy for each band. Further theoretical analysis of the effective low-energy Hamiltonian expanded around these two gapless points confirms this DSM phase, as shown in details in Appendix D.

The DSM phase separates two insulating phases, labeled by I and III in Fig.2a. To identify the nature of these two insulating phases, we perform a numerical calculation of energy dispersion on the $(001)$ surface of an approximately semi-infinite sample.Sancho et al. (1985) The local density of states at the top surface along $k_{x}=k_{y}$ axis is shown in Fig.2b for the point A with $(\xi_{3},\xi_{4})=(-0.1,0.05)\varepsilon_{0}$ in the phase I and Fig.2d for the point C with $(\xi_{3},\xi_{4})=(-0.1,0.2)\varepsilon_{0}$ in the phase III, respectively. One can see two sets of gapless modes appearing for the phase I while a full gap existing for the phase III. Thus, we expect the phase I is topologically non-trivial while the phase III is trivial. We also perform a calculation of mirror Chern number (MCN) Teo et al. (2008) on the mirror or glide plane ($1\bar{1}0$) for this system and find MCN to be 2 for the phase I and 0 for the phase III. This confirms that two sets of gapless modes in Fig.2b are protected by mirror or glide symmetry and makes phase I to be TMI phase. Thus, DSM phase can be viewed as the topological phase transition point between a TMI phase and a trivial insulating phase.

We emphasize that, in realistic half-Heusler materials, inversion symmetry is absent and the Kramer’s degeneracy at a generic $\mathbf{k}$ is split for both the conduction and valence bands, which means Dirac points are also split. However, DSM phase will evolve into other TSM phases, as discussed in the next section. Therefore, DSM phase can be viewed as the “parent” phase to search for and understand other topological phases.

III.2 Weyl Semimetal Phase and Topological Mirror Insulating Phase

In this part, we include the inversion symmetry breaking term $H_{C}({\bf k})$ into the four-band Hamiltonian and the total Hamiltonian becomes $H_{\Gamma_{8}}(\mathbf{k})+H_{AFM}$. As a consequence, the Kramer’s degeneracy of each band at a generic momentum $\mathbf{k}$ is split. We still consider the magnetic moments of anti-ferromagnetic ordering aligning within or perpendicular to the $(1\bar{1}0)$ plane to preserve either the glide symmetry $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$ or the mirror symmetry $\pi_{(1\bar{1}0)}$, which gives $\xi_{2}=0$ and $\xi_{4}=\xi_{5}$. Due to the existence of $H_{C}$ term, the full Hamiltonian cannot be diagonalized analytically and thus numerical methods are adopted to extract phase diagram. All the parameters are the same as previous choices, except the $C$ parameter which is chosen as $C=0.2(\beta_{c}/a)$, as listed in Tab.7 in Appendix F.

The phase diagram as a function of $\xi_{3}$ and $\xi_{4}$ is shown in Fig.3a. The phases I and III in Fig.3a remain robust due to existence of mirror or glide symmetry. A direct calculation of surface local density of states, as well as MCN, shows four surface modes with MCN being 2 (Fig.3c) for point A in Fig.3a ($\xi_{3}/\varepsilon_{0}=-0.2$ and $\xi_{4}/\varepsilon_{0}=0.01$) and a full gap with zero MCN (Fig.3d) for point C in Fig.3a ($\xi_{3}/\varepsilon_{0}=0.3$ and $\xi_{4}/\varepsilon_{0}=0.1$).

We notice that the phase II is expanded from a line of Dirac semimetal phase in Fig.2a to a region of WSM phase in Fig.3a. The reason is that $H_{C}$ breaks the inversion symmetry and splits each Dirac cone in Dirac semimetal phase into two Weyl points. Since there are two Dirac points in the DSM phase, the phase II in Fig.3a typically has four Weyl points in the whole momentum space, denoted as ${\bf K}_{i}$ ($i=1,2,3,4$), as shown in Fig.3b and Fig.4a. Different Weyl points can be related to each other by symmetries: $\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ (or $\mathbf{K}_{3}$ and $\mathbf{K}_{4}$) are related by $\Pi_{(1\bar{1}0)}\equiv\pi_{(1\bar{1}0)}$ or $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$ and therefore dubbed a ‘$\Pi$ pair’, $\mathbf{K}_{1}$ and $\mathbf{K}_{3}$ (or $\mathbf{K}_{2}$ and $\mathbf{K}_{4}$) are related by $\Pi_{(1\bar{1}0)}S$ and called a ‘$\Pi S$ pair’, and finally $\mathbf{K}_{1}$ and $\mathbf{K}_{4}$ (or $\mathbf{K}_{2}$ and $\mathbf{K}_{3}$) are related by $S$ and named a ‘$S$ pair’. It is known Volovik (2003, 2007, 2013) that a Weyl fermion can carry topological charge or chirality, which can be extracted from Chern number (CN) on a small spherical surface surrounding the Weyl point. Two Weyl points related by mirror symmetry have opposite Chern numbers while time reversal operation leaves Chern number of a Weyl point unchanged. As a result, a $\Pi$ pair carries opposite CNs and so does a $\Pi S$ pair, while a $S$ pair has the same CN. Due to the existence of topological charge, a single Weyl point cannot be gaped and it can only move in the momentum space when tuning the parameters until it merges with another Weyl point with opposite CN. We track the motion of Weyl points through the path $\gamma$ in the inset of Fig.3a, and find $\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ (or $\mathbf{K}_{3}$ and $\mathbf{K}_{4}$) emerge from a point in the mirror (or glide) plane as a $\Pi$ pair and move in the momentum space and finally annihilate at another point on the mirror (or glide) plane, as shown in Fig.3b with its inset.

Now we focus on the energy dispersion of the WSM phase by choosing the B point in Fig.3a with $\xi_{3}=-0.01\varepsilon_{0}$ and $\xi_{4}=0.025\varepsilon_{0}$ as an example. Four Weyl points are approximately located at

[TABLE]

in the momentum space, respectively. By integrating the Berry curvature on the sphere around each point, we found CNs of four Weyl points are $CN_{1}=-1$,$CN_{2}=1$,$CN_{3}=1$ and $CN_{4}=-1$ (see Fig.4a), which is consistent with the symmetry analysis above. The Fermi surface around $\mathbf{K}_{1}$ at the energy of the Weyl point, as shown in Fig.4b, demonstrates the existence of a type II Weyl point in our system. Type-II Weyl fermions are topologically non-trivial and can lead to Fermi arc Soluyanov et al. (2015) on the surface. Thus, we perform a calculation of local density of states at the energy of Weyl points on the $(001)$ surface of an approximately semi-infinite sampleSancho et al. (1985), as shown in Fig.4c, in which a complex surface Fermi arcs overlaps with bulk bands. Weyl points are depicted by white points in the plot and there is one Fermi arc starting from each Weyl point and merging into bulk bands. Additional surface states with same energy exist and they form a circle surrounding the $\Gamma$ point ${\bf k}=0$. The surface energy dispersions along the momentum lines $k_{y}a=0$ and $k_{y}a=0.15$ are shown in Fig.4d and Fig.4e. We notice one chiral edge mode existing along the momentum line $k_{y}a=0.15$ while a helical edge mode along the line $k_{y}a=0$. We may treat $k_{y}$ as a parameter and consider two-dimensional (2D) planes formed by $k_{z}$ and $k_{x}$ for different $k_{y}$. According to the bulk-edge correspondence, the existence of chiral edge mode for $k_{y}a=0.15$ at the surface suggests $CN=-1$ for the corresponding 2D plane. Similarly, CN for the 2D plane at $k_{y}a=0$ should be zero. This is consistent with the fact that two 2D planes at $k_{y}a=0.15$ and $k_{y}a=0$ enclose one Weyl point $\mathbf{K}_{4}$ whose CN is $-1$. However, the helical edge mode along the line $k_{y}a=0$ cannot be explained by CN. The crossing point between two branches of the helical edge mode is protected by S symmetry, thus the 2D plane at $k_{y}a=0$ can be viewed as an AFMTI phase. In addition, since the crossing point at ${\bf k}=0$ also falls into the mirror (or glide) plane, as shown by diagonal line in Fig.4c, that crossing point thus can also be protected by MCN, which is equal to 1 for the phase II. Therefore, two blue lines in Fig.3a can be viewed as transition lines between the phase II with $MCN=1$ and the phase I or III with MCN being 0 or 2 respectively.

Although we focus on the magnetic moments parallel or perpendicular to the mirror plane in this section, the WSM phase can NOT be destroyed immediately when magnetic moments is tilted away from these directions due to the non-zero CNs carried by Weyl points. The Weyl points can only move in the momentum space and should be robust in certain parameter regimes. On the other hand, for the TMI phase, gapless points of helical surface mode at finite non-zero momenta are solely protected by the mirror or glide symmetry and thus sensitively depend on the direction of magnetic moments.

III.3 Nodal Line Semimetal Phase

For magnetic moments of anti-ferromagnetic ordering within or perpendicular to the $(1\bar{1}0)$ plane, another possible TSM phase is NLSM phase. An example of NLSM phase is shown in Fig.5, where the parameters are shown in Tab.8 in Appendix F. Due to differnet AFM parameters, the band sequence is different and two crossing bands at the low energy are from two opposite mirror subspaces, in contrast to the band crossing between two bands with the same mirror parity for the phase boundary in Fig.3a. The energy dispersion of NLSM phase is depicted in Fig.5a and the positions of two nodal rings are shown in Fig.5b. The topological stability of each nodal ring can be extracted by the $\pi$ Berry phase along a small circle (red circle in inset of Fig.5b) around the nodal line.

III.4 Triple Point Semimetal Phase

In this section, we consider the case with anti-ferromagnetic magnetic moments along the $(111)$ direction, where the system has three-fold rotational symmetry $C_{3}(111)$ and glide symmetry $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$. Since the matrix representation of $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$ is equivalent to the mirror symmetry $\pi_{(1\bar{1}0)}$ for the basis of $\Gamma_{8}$ bands, the symmetry group generated by $C_{3}(111)$ and $T_{\mathbf{a}^{\prime}_{3}}\pi_{(1\bar{1}0)}$ is isomorphic to the point group $C_{3v}$. By linearly combining the four basis functions of $\Gamma_{8}$ bands which carry total angular momentum $J=\frac{3}{2}$, we can get a pair of states belonging to two-dimensional representation $\Lambda_{6}$ of the $C_{3v}$ double group, and the other two states belonging to two one-dimensional representations $\Lambda_{4}$ and $\Lambda_{5}$ respectively.Burns (2014) The character table of the $C_{3v}$ double group and the linear combinations of $\Gamma_{8}$ bases are shown in Appendix E.

To confirm this symmetry analysis, we consider the total Hamiltonian $H_{0}(\mathbf{k})+H_{C}(\mathbf{k})+H_{AFM}$ with magnetic moments along the $(111)$ direction ($M_{x}=M_{y}=M_{z}$). This corresponds to the conditions $\xi_{1}=\xi_{2}=0$ and $\xi_{3}=\xi_{4}=\xi_{5}$. Along the $(111)$ direction, we indeed find that four $\Gamma_{8}$ bands are split into one doubly degenerate band, labeled as $\Lambda_{6}$ bands, and another two non-degenerate bands, labeled as $\Lambda_{4}$ and $\Lambda_{5}$ bands respectively. The corresponding energy dispersion can be solved analytically as

[TABLE]

where $\xi_{3}=\xi_{4}=\xi_{5}\equiv\xi$ and $k_{x}=k_{y}=k_{z}=k$. A typical energy dispersion is shown in Fig.6a with the parameters shown in Tab.9 in Appendix F. Under the condition $C^{2}>16\sqrt{3}\beta_{c}\gamma_{3}\xi$ and $C\xi\neq 0$, the $\Lambda_{6}$ bands cross with the $\Lambda_{5}$ ($\Lambda_{4}$) band at two points labeled by 1 and 3 (2 and 4) in Fig.6a. At each crossing point, there is a three-fold degeneracy and the energy dispersion behave linearly along the $(111)$ axis. Points 1 and 2 (or 3 and 4) are connected by four nodal lines, as shown in Fig.6b. Along a circle enclosing any of these four nodal lines, the accumulated Berry phase is found to be $\pi$. This type of semi-metal phase is known as type-B TPSM phase Zhu et al. (2016).

IV Anti-ferromagnetic Topological Insulating Phase in Six-band Kane Model

In the discussion above, we neglected $\Gamma_{6}$ bands and only focused on four $\Gamma_{8}$ bands. This simplification has irrelevant influence on TSM phases since $\Gamma_{6}$ bands are far away from Fermi energy. However, due to the inverted nature between $\Gamma_{6}$ and $\Gamma_{8}$ bands, $\Gamma_{6}$ bands may play an essential role for topological insulating phases. It is well known that the band inversion between the $\Gamma_{6}$ and $\Gamma_{8}$ bands leads to the topological insulating phase in HgTe Bernevig et al. (2006), as well as non-magnetic half-Heusler materials Lin et al. (2010b); Chadov et al. (2010); Xiao et al. (2010); Al-Sawai et al. (2010). Therefore,we study the influence of $\Gamma_{6}$ bands by considering the full six-band Kane model $H_{Kane}$ in this section. With the anti-ferromagnetic ordering, the full Hamitonian takes the form

[TABLE]

which can be diagonalized numerically to extract the energy dispersion.

To systematically understand the AFM term in the Kane model, we consider different $\xi_{i}$ terms ($i=1,\dots,5$) in Eq.7, separately. We notice that applying three-fold rotational symmetry $C_{3}(111)$ to $H_{AFM}$ is equivalent to performing the following transformations: $(\xi_{1},\xi_{2})\rightarrow(-\frac{1}{2}\xi_{1}-\frac{\sqrt{3}}{2}\xi_{2},\frac{\sqrt{3}}{2}\xi_{1}-\frac{1}{2}\xi_{2})$ and $(\xi_{3},\xi_{4},\xi_{5})\rightarrow(\xi_{4},\xi_{5},\xi_{3})$, which means $\xi_{1,2}$ are related with each other by $C_{3}(111)$ and so do $\xi_{3,4,5}$. Therefore, we study the direct band gap as a function of (i) $\xi_{1}$ and $\xi_{2}$ for $\xi_{3,4,5}=0$ in which case the effective Hamiltonian $H_{Kane}^{AFM}$ in Eq.9 preserves two-fold rotation symmetry along the x,y,z axes, or (ii) $\xi_{1}$ and $\xi_{3}$ for $\xi_{2,4,5}=0$ in which case the effective model $H_{Kane}^{AFM}$ preserves $\pi_{1\bar{1}0}$, two-fold rotation along z axis and mirror symmetry perpendicular to (110). The phase diagrams for the case (i) and (ii) are shown in Fig.7a and Fig.7b, respectively, from which one can find both the gapless phases existing in the blue region (the region I) and the gaped phases in the three yellow regions (the region II). Detailed parameters for our calculation can be found in Tab.10 and Tab.11 in Appendix F. We notice that $\xi_{1}$ term takes the same form as the strain term described in Ref.[(46)]. Therefore, on the line $\xi_{2}=0$ in Fig.7a or the line $\xi_{3}=0$ in Fig.7b, we expect the gapless and gaped phases should be equivalent to the corresponding ones studied for strained HgTe and half-Heusler materials Ruan et al. (2016). To verify the nature of gapless and insulating phases, we calculate the energy dispersion for two typical sets of parameters: the point A with $(\xi_{1},\xi_{3})=(-0.8,0.1)\varepsilon_{0}$ and B with $(\xi_{1},\xi_{3})=(-0.1,0.3)\varepsilon_{0}$ in Fig.7b. A non-zero bulk direct gap is found for the point A and thus we consider an approximately semi-infinite configuration and plot the local density of states on (001) surface, as shown in Fig.7c. A helical surface mode is found in the bulk gap and is protected by the $S$ symmetry instead of the time reversal symmetry due to the anti-ferromagnetic ordering, thus giving rise to a realization of AFMTI phase. The gapless phase at point B is found to be WSM phase and the bulk dispersion is shown Fig.7d with the Weyl points located at the momenta $\mathbf{k}a=\pm(0.89,0.35,0)$ (or equivalent $\mathbf{k}a=\pm(-0.35,-0.89,0)$).

Below we emphasize that the AFMTI phase is quite robust in this system. The phase diagram for $\xi_{1}-\xi_{2}$ is shown in Fig.8a for $(\xi_{3},\xi_{4},\xi_{5})=(-0.01,0.023,0.026)\varepsilon_{0}$. Other parameters are listed in Tab.12 in Appendix F. We notice that after introducing non-zero $\xi_{3,4,5}$ which breaks all $C_{3v}$ symmetries and their combinations with half translation, the previous gapless phase region I in Fig.7a shrinks to a smaller region I in Fig.8a, while the region II of AFMTI phase is greatly extended. In the region I, Weyl points are found (not exclusively) at $\mathbf{k}a=\pm(-0.23,-0.21,0.55)$ for point B of Fig.8a, as shown in Fig.8b. The local density of states calculation on (001) surface for point A in Fig.8a with $(\xi_{1},\xi_{2})=(-0.8,0.1)\varepsilon_{0}$ gives very similar graph as Fig.7c, thus demonstrating the AFMTI phase in the region II. Given the large region in the parameter space for the realization of AFMTI phases, we can conclude that anti-ferromagnetic half-Heusler materials provide a robust material realization of the AFMTI phase. Moreover, if mirror or non-symmorphic symmetry exists, anti-ferromagnetic half-Heusler materials can also provide a robust material realization of anti-ferromagnetic mirror or non-symmorphic topological insulator phase, which has not been demonstrated in experiments.

V Conclusion

Based on the above studies of the four-band and six-band Kane models, we can summarize the overall physical picture for electronic structures of anti-ferromagnetic half-Heusler compounds in Fig.9. Without AFM, the band ordering of $\Gamma_{6}$ and $\Gamma_{8}$ bands are inverted and the Fermi energy lies at the four fold degenerate point ($\mathbf{k}=0$) of $\Gamma_{8}$ bands, leading to a critical semimetal phase shown in Fig.9a. With AFM, four-fold degeneracy of $\Gamma_{8}$ bands at $\Gamma$ point is removed. The band structure of four $\Gamma_{8}$ bands can be either normal (Fig.9b) or inverted (Fig.9c), depending on the detailed form and parameters of AFM terms. When the band ordering of $\Gamma_{8}$ bands is normal, it is trivial for the four-band model but non-trivial for the six-band Kane model due to the inversion between the $\Gamma_{6}$ and $\Gamma_{8}$ bands, leading to either AFMTI phase or TMI phase. When the $\Gamma_{8}$ band ordering is inverted, the AFM terms can either lead to a full inverted band gap (AFMTI or TMI phase in Fig.9d) or preserve certain gapless points in the momentum space (the TSM phase in Fig.9e). In either situation, we find that anti-ferromagnetic half-Heusler compounds are topologically non-trivial. Thus, our work demonstrates that half-Heusler materials with AFM provide a platform for a robust realization of anti-ferromagnetic topological phases, either WSM phase or AFMTI phase, in a wide parameter regime.

The AFMTI phase was first proposed in Ref.[(30)] based on a four band toy model and our results have shown this interesting topological phase indeed can exist in anti-ferromagnetic half-Heusler materials. We notice that the first principles calculation in combining with tight-binding model has been adopted for the AFM GdPtBiLi et al. (2015), in which a semi-metal phase is found. However, the topological nature of this semi-metal phase has not been extracted and our results identify the existence of Weyl points in this semimetal phase. In addition, the authors use the representation of crystal symmetry group to label each band, aiming in identifying band inversion. We believe this approach is insufficient for the AFMTI phase since this topological phase is protected by the $S$ symmetry which is not included in the crystal symmetry group. Thus, the AFMTI phase cannot be identified from the crystal symmetry representation. Our results suggest that the AFMTI phase can exist in the G-type anti-ferromagnetic system, which is identified to be topologically trivial in Ref.[(59)]. In the existing experiments, Weyl semi-metal phase has been unveiled in GdPtBi under an external magentic field through the observation of a large anomalous Hall angle Suzuki et al. (2016); Shekhar et al. (2016), large negative magnetoresistanceShekhar et al. (2016); Hirschberger et al. (2016) and the strong suppression of thermopowerHirschberger et al. (2016). Our results suggest Weyl semi-metal phase may already occur even in absence of external magnetic fields. In addition, nodal line fermions, type-B triply degenerate fermions and topological mirror or glide insulators, are also possible in certain parameter regimes when magnetic moments of anti-ferromagnetic ordering are along some specific directions. The topological surface states or surface Fermi arcs in these topological phases can be extracted from angle-resolved photoemission spectroscopy or quasi-particle spectrum from the scanning tunneling microscopy Qi and Zhang (2011); Hasan and Kane (2010); Yan and Felser (2016). Our generalized Kane model also provides a basis for the future study of magnetic, transport or optical phenomena in this class of materials. Furthermore, we notice that superconductivity can coexist with anti-ferromagnetism in RPdBi(R=Tb,Ho,Dy,Er)Nakajima et al. (2015); Pan et al. (2013); Pavlosiuk et al. (2016a). Thus, it is interesting to ask if topological superconductivity can be realized in these materials.

VI Acknowledgment

We would like to thank Rui-Xing Zhang, Jian-Xiao Zhang, Qing-Ze Wang, Yang Ge and Di Xiao for helpful discussion. C.-X.L. acknowledges the support from Office of Naval Research (Grant No. N00014-15-1-2675). B.Y. acknowledges support of the Ruth and Herman Albert Scholars Program for New Scientists in Weizmann Institute of Science, Israel.

Appendix A Tight-Binding Model and its relationship to the Kane model

In this section, we will describe a tight-binding model for our anti-ferromagnetic half-Heusler materials, from which we can justify the extend Kane model that we derived by symmetry principles and used for the low energy physics in Sec.II in the main text. For simplicity, we take the anti-ferromagnetic half-Heusler material ErPdBi as an example and assume it has anti-ferromagnetic structure of GdPtBi. We consider nine orbitals $\left|Bi,s\right\rangle$, $\left|Pd,s\right\rangle$, $\left|Pd,p_{x}\right\rangle$, $\left|Pd,p_{y}\right\rangle$, $\left|Pd,p_{z}\right\rangle$, $\left|Er,s\right\rangle$, $\left|Er,d_{xy}\right\rangle$, $\left|Er,d_{yz}\right\rangle$ and $\left|Er,d_{zx}\right\rangle$ to construct the tight-binding model for this material. Due to the doubling of the unit cell along $\mathbf{a}^{\prime}_{3}$, all the orbitals are labeled as $\left|n,i\right\rangle$, where $n$ stands for atoms and orbitals and $i=1,2$ labels two sub-lattices that are related by $T_{\mathbf{a}^{\prime}_{3}}$. We only considered the hopping terms between the nearest neighbor atoms, including Bi, Pd and Er atoms Slater and Koster (1954), giving rise to a 18 by 18 Hamiltonian $H_{hopping}$. In order to include AFM term and spin-orbit term, we need to consider spin degree of freedom and the hopping term is block diagonal with each block given by $H_{hopping}$ in the spin space. The AFM term is given by $H_{AFM}=(-1)^{i}\mathbf{M}\cdot\mathbf{S}$ with $i=1,2$ for two sets of cells. Here $\mathbf{M}$ is the mean field value for anti-ferromagnetic magnetic moments and $\mathbf{S}$ labels electron spin. The spin-orbit coupling term takes the form $H_{so}=\mathbf{S}\cdot\mathbf{L}$, where $\mathbf{L}$ denotes the angular momentum operator that acts on the oribtal ($s,p,d$ orbitals).

The above 36 by 36 tight-binding Hamiltonian is complicated and we are not interested in using this model for realistic calculation here. Instead, we hope to use this model to justify the form of the extended Kane model that we derived from the symmetry principle in the main text. We need to transform the basis wave functions into the ones used for the Kane model, which can be achieved by the following steps. (1) The basis wave functions of the Kane model only include two $s$ orbital type of bands and 6 $p$ orbital type of bands after taking into account spin degree of freedom. Therefore, we need to first reduce the number of basis wave functions. Since both the $p_{x},p_{y},p_{z}$ and $d_{xy},d_{yz},d_{zx}$ orbitals belong to $T_{2}$ representation of the $T_{d}$ group (The representation table of $T_{d}$ group is shown in Fig.10), the basis wave functions should be a linear combination of the $p$ and $d$ orbitals, which are given by $\left|\gamma,i,\sigma\right\rangle=P_{1}\left|Pd,p_{\alpha},i,\sigma\right\rangle+P_{2}\left|Er,d_{\beta},i,\sigma\right\rangle$, where $i=1,2$, $\sigma=\uparrow,\downarrow$ and $(\gamma,\alpha,\beta)=(X,x,yz),(Y,y,zx),(Z,z,xy)$. For the s orbital type of bands, the basis wave functions take the form $\left|S,i,\sigma\right\rangle=S_{1}\left|Bi,s,i,\sigma\right\rangle+S_{2}\left|Pd,s,i,\sigma\right\rangle+S_{3}\left|Er,s,i,\sigma\right\rangle$, where $i=1,2$ and $\sigma=\uparrow,\downarrow$. Here we still keep the sub-lattice index $i=1,2$ and the number of basis wave functions are reduced to 16. (2) The basis wave functions of the Kane model is chosen to be the eigenstates of total angular momentum operator because of strong spin-orbit coupling. Therefore, we apply a similar transformation to our basis wave functions here, as listed below:

[TABLE]

where $1,2$ are sub-lattice indexes. (3) Since the low energy physics only occurs at the $\Gamma$ point (${\bf k}=0$) for the Kane model, we only need to consider the bonding states between two sub-lattices. The 8 bonding basis wave functions are given by $|\Lambda,\sigma\rangle=\frac{1}{\sqrt{2}}(|\Lambda,\sigma,1\rangle+|\Lambda,\sigma,2\rangle)$, where $\Lambda=\Gamma_{6,7,8}$ and $\sigma=\pm\frac{1}{2}$,or including $\pm\frac{3}{2}$ for $\Gamma_{8}$. By projecting the full tight-binding Hamiltonian into the six basis wave functions, including $|\Gamma_{6},\frac{1}{2}\rangle$, $|\Gamma_{6},-\frac{1}{2}\rangle$, $|\Gamma_{8},\frac{3}{2}\rangle$, $|\Gamma_{8},\frac{1}{2}\rangle$, $|\Gamma_{8},-\frac{1}{2}\rangle$ and $|\Gamma_{8},-\frac{3}{2}\rangle$ and expand the resulting Hamiltonian to the second order in both ${\bf k}$ and ${\bf M}$, we reproduce the extended Kane model Hamiltonian with the anti-ferromagnetic ordering derived by the symmetry principles in our main text.

Appendix B Irreducible Representations of $C_{3v}$ group

In this section, we will use the representation table of the $C_{3v}$ group to classify the polynomials of ${\bf k}$ and ${\bf M}$, as well as all the four by four matrices for the $\Gamma_{8}$ basis wave functions, which is used to construct the effective low energy Hamiltonian for our system in Sec.II.

The $C_{3v}$ group can be generated by two operations, three-fold rotation and mirror operation. For the lattice considered here, the three-fold rotation is along the $(111)$ axis, denoted as $C_{3}(111)$, and mirror operation is with repsect to the $(1,-1,0)$ plane, denoted as $\pi_{1\bar{1}0}$, where $\bar{1}$ means $-1$. It has three irreducible representations $A_{1}$, $A_{2}$ and $E$ and its character table is shown in Tab.2.

Since magnetic moments of the anti-ferromagnetic ordering come from the $d$ orbitals of $Er$ atoms, we only focus on the the $\Gamma_{8}$ bands which contains $d$ orbitals of $Er$ atoms. We need to construct the generator operations of the $C_{3v}$ group on the $\Gamma_{8}$ bands. To achieve that, we notice the angular momentum operators $J_{x,y,z}$ for spin-$\frac{3}{2}$ states can be represented by the four by four matrices

[TABLE]

[TABLE]

[TABLE]

As a result, the $C_{3}(111)$ and $\pi_{1\bar{1}0}$ operations are given by

[TABLE]

[TABLE]

In addition, time reversal operator and half translation operator $T_{\mathbf{a}^{\prime}_{3}}$ are interesting to us since the anti-ferromagnetic ordering preserves the S symmetry which is the combination of these two operators. On the four basis of the $\Gamma_{8}$ bands, we find time reversal operator writeen as

[TABLE]

with the complex conjuate $\hat{K}$ and the half translation $T_{\mathbf{a}^{\prime}_{3}}$ taking the form of an identity matrix

[TABLE]

because of the bonding state nature of two sub-lattices for the four $\Gamma_{8}$ bands. As a consequence, the S symmetry takes the same form as the time reversal symmetry on the four $\Gamma_{8}$ basis.

16 four-by-four matrices can be constructed by three spin $\frac{3}{2}$ matrices $J_{x,y,z}$ as $J_{x}$, $J_{y}$, $J_{z}$, $J_{x}^{2}$, $J_{y}^{2}$, $J_{z}^{2}$, $J_{xy}=\{J_{x},J_{y}\}/2$, $J_{yz}=\{J_{y},J_{z}\}/2$, $J_{zx}=\{J_{z},J_{x}\}/2$, $J_{x}^{3}$, $J_{y}^{3}$, $J_{z}^{3}$, $V_{x}=\{J_{y}^{2}-J_{z}^{2},J_{x}\}/2$, $V_{y}=\{J_{z}^{2}-J_{x}^{2},J_{y}\}/2$, $V_{z}=\{J_{x}^{2}-J_{y}^{2},J_{z}\}/2$ and $J_{xyz}=J_{x}J_{y}J_{z}+J_{z}J_{y}J_{x}$. The linear combinations of these 16 matrices belong to different irreducible representations of the $C_{3v}$ group and we list the corresponding representations in in Tab.3.

Next we will classify all the polynomials of ${\bf k}$ and ${\bf M}$ according to the irreducible representations of the $C_{3v}$ group Winkler et al. (2003). The representations for the polynomials of the momentum ${\bf k}$ and anti-ferromagnetic magnetic moments ${\bf M}$ are listed in Tab.4.

Appendix C Gapless Conditions for the Hamiltonian $H_{0}+H_{AFM}$

In this section, we will present the detailed analysis of the conditions for possible gapless phases for the four-by-four Hamiltonian $H_{0}+H_{AFM}$, which is discussed in Sec.III.1 in the main text to understand the phase diagram of the above Hamiltonian. Without anti-ferromagnetic ordering, the energy dispersion of the Luttinger Hamiltonian $H_{0}$ possesses a quadratic touching at the $\Gamma$ point. With the AFM term (at least one nonzero $\xi_{i}$ for $i\neq 0$), the Hamiltonian $H_{0}+H_{AFM}$ can be analytically solved (Eq.8). The requirement for a gapless point is that $h_{i}({\bf k})+\xi_{i}=0$ for all $i=1,2,\dots,5$ for a certain momentum ${\bf k}$. Below we will list all the possible cases for a gapless point to exist in the momentum space. Here $\beta_{c},\gamma_{2},\gamma_{3}\neq 0$ are always assumed.

Case I: $\xi_{3}=\xi_{4}=\xi_{5}=0$.

In this case, we require $h_{3}=h_{4}=h_{5}=0$. According to the form of $h_{3,4,5}$, we immediately see that two of three components of the momentum $(k_{x},k_{y},k_{z})$ should be zero. Let us assume $k_{x}=k_{y}=0$ and for the remaining non-zero $k_{z}$, we still need to solve two equations $h_{1}=-\xi_{1}$ and $h_{2}=-\xi_{2}$. This suggests that $\xi_{1}$ and $\xi_{2}$ cannot be independent of each other in order to achieve a gapless phase. According to the form of $h_{1,2}$, we find gapless points at

[TABLE]

for $\xi_{3}=\xi_{4}=\xi_{5}=0,\xi_{2}=0,\xi_{1}\beta_{c}\gamma_{2}<0$.

Similarly, if $k_{y}=k_{z}=0$, we have gapless points at

[TABLE]

for $\xi_{3}=\xi_{4}=\xi_{5}=0,\xi_{2}\beta_{c}\gamma_{2}<0,\xi_{2}=-\sqrt{3}\xi_{1}$.

And if $k_{x}=k_{z}=0$, we have gapless points at

[TABLE]

for $\xi_{3}=\xi_{4}=\xi_{5}=0,\xi_{2}\beta_{c}\gamma_{2}>0,\xi_{2}=\sqrt{3}\xi_{1}$.

Case II: For $\xi_{3}$, $\xi_{4}$ and $\xi_{5}$, two of them are zero and one is non-zero.

For this case, let’s take the example of $\xi_{3}=\xi_{4}=0$ and $\xi_{5}\neq 0$ and the analysis for other cases are similar. Since $\xi_{5}\neq 0$, $k_{y,z}$ cannot be zero from $h_{5}=-\xi_{5}$. Therefore, $\xi_{3}=\xi_{4}=0$ gives rise to $k_{x}=0$. For the remaining three equations $h_{1,2,5}=-\xi_{1,2,5}$, we find only two variables $k_{y}$ and $k_{z}$. Thus, one of $\xi_{1,2,5}$ should depend on the other two. By solving the equations $h_{1,2,5}=-\xi_{1,2,5}$, we find gapless points at

[TABLE]

for $\xi_{3}=\xi_{4}=0,\xi_{5}\neq 0,\xi_{2}\gamma_{2}\beta_{c}>0,\xi_{1}=\frac{1}{\sqrt{3}}(\xi_{2}-\frac{\gamma_{2}^{2}\xi_{5}^{2}}{2\gamma_{3}^{2}\xi_{2}})$.

Similarly, if $\xi_{3}=\xi_{5}=0,\xi_{4}\neq 0$, the gapless points are at

[TABLE]

for $\xi_{2}\gamma_{2}\beta_{c}<0,\xi_{1}=-\frac{1}{\sqrt{3}}(\xi_{2}-\frac{\gamma_{2}^{2}\xi_{4}^{2}}{2\gamma_{3}^{2}\xi_{2}})$.

And if $\xi_{4}=\xi_{5}=0,\xi_{3}\neq 0$, the gapless points are at

[TABLE]

for $(\sqrt{3}\xi_{1}-\xi_{2})\beta_{c}\gamma_{2}>0,(\sqrt{3}\xi_{1}+\xi_{2})\beta_{c}\gamma_{2}>0,\gamma_{2}^{2}\xi_{3}^{2}+\gamma_{3}^{2}(\xi_{2}^{2}-3\xi_{1}^{2})=0$.

Case III: For $\xi_{3}$, $\xi_{4}$ and $\xi_{5}$, one of them is zero and the other two are non-zero.

Gapless points cannot exist for this case. This is because once two of $\xi_{3,4,5}$ are nonzero, none of three components $k_{x,y,z}$ of a gapless point can be zero. Thus, the other $\xi$ should also be non-zero for gapless points to exist.

Case IV: All three $\xi_{3,4,5}$ are non-zero.

For this case, we need to solve five equations $h_{i}=-\xi_{i}$ ($i=1,\dots,5$) with three variables $k_{x,y,z}$. Therefore, only three of the five $\xi$’s are independent. Let’s assume $\xi_{3,4,5}$ are independent variables and one can first solve three equations $h_{3,4,5}=-\xi_{3,4,5}$ for the momentum $(k_{x},k_{y},k_{z})$ to get possible positions of gapless points , and then plug them in $\xi_{1,2}=-h_{1,2}$ to get the relation between $\xi_{1}$ and $\xi_{2}$ and the exact possible positions. Results are listed below.

If \xi_{3}\neq 0,\xi_{4}\neq 0,\xi_{5}\neq 0,\xi_{3}\xi_{4}\xi_{5}\beta_{c}\gamma_{3}<0,\xi_{3}\beta_{c}\gamma_{3}<0,\xi_{4}\beta_{c}\gamma_{3}<0,\xi_{5}\beta_{c}\gamma_{3}<0,\xi_{2}=\frac{\gamma_{2}\text{\xi3}(\text{\xi4}-\text{\xi5})(\text{\xi4}+\text{\xi5})}{2\gamma_{3}\text{\xi4}\text{\xi5}},\xi_{1}=-\frac{\gamma_{2}\left(\text{\xi3}^{2}\text{\xi4}^{2}+\text{\xi3}^{2}\text{\xi5}^{2}-2\text{\xi4}^{2}\text{\xi5}^{2}\right)}{2\sqrt{3}\gamma_{3}\text{\xi3}\text{\xi4}\text{\xi5}}, the gapless points are located at

[TABLE]

.

If \xi_{3}\neq 0,\xi_{4}\neq 0,\xi_{5}\neq 0,\xi_{3}\xi_{4}\xi_{5}\beta_{c}\gamma_{3}<0,\xi_{3}\beta_{c}\gamma_{3}<0,\xi_{4}\beta_{c}\gamma_{3}>0,\xi_{5}\beta_{c}\gamma_{3}>0,\xi_{2}=\frac{\gamma_{2}\text{\xi3}(\text{\xi4}-\text{\xi5})(\text{\xi4}+\text{\xi5})}{2\gamma_{3}\text{\xi4}\text{\xi5}},\xi_{1}=-\frac{\gamma_{2}\left(\text{\xi3}^{2}\text{\xi4}^{2}+\text{\xi3}^{2}\text{\xi5}^{2}-2\text{\xi4}^{2}\text{\xi5}^{2}\right)}{2\sqrt{3}\gamma_{3}\text{\xi3}\text{\xi4}\text{\xi5}}, the gapless points are

[TABLE]

.

If \xi_{3}\neq 0,\xi_{4}\neq 0,\xi_{5}\neq 0,\xi_{3}\xi_{4}\xi_{5}\beta_{c}\gamma_{3}<0,\xi_{3}\beta_{c}\gamma_{3}>0,\xi_{4}\beta_{c}\gamma_{3}<0,\xi_{5}\beta_{c}\gamma_{3}>0,\xi_{2}=\frac{\gamma_{2}\text{\xi3}(\text{\xi4}-\text{\xi5})(\text{\xi4}+\text{\xi5})}{2\gamma_{3}\text{\xi4}\text{\xi5}},\xi_{1}=-\frac{\gamma_{2}\left(\text{\xi3}^{2}\text{\xi4}^{2}+\text{\xi3}^{2}\text{\xi5}^{2}-2\text{\xi4}^{2}\text{\xi5}^{2}\right)}{2\sqrt{3}\gamma_{3}\text{\xi3}\text{\xi4}\text{\xi5}}, the gapless points are

[TABLE]

.

If \xi_{3}\neq 0,\xi_{4}\neq 0,\xi_{5}\neq 0,\xi_{3}\xi_{4}\xi_{5}\beta_{c}\gamma_{3}<0,\xi_{3}\beta_{c}\gamma_{3}>0,\xi_{4}\beta_{c}\gamma_{3}>0,\xi_{5}\beta_{c}\gamma_{3}<0,\xi_{2}=\frac{\gamma_{2}\text{\xi3}(\text{\xi4}-\text{\xi5})(\text{\xi4}+\text{\xi5})}{2\gamma_{3}\text{\xi4}\text{\xi5}},\xi_{1}=-\frac{\gamma_{2}\left(\text{\xi3}^{2}\text{\xi4}^{2}+\text{\xi3}^{2}\text{\xi5}^{2}-2\text{\xi4}^{2}\text{\xi5}^{2}\right)}{2\sqrt{3}\gamma_{3}\text{\xi3}\text{\xi4}\text{\xi5}}, the gapless points are

[TABLE]

.

Appendix D Dirac Point With Mirror Symmetry

In this section, we will consider the low energy effective theory around the gapless points of the Hamiltonian $H_{0}+H_{AFM}$ and show it can be described by a Dirac Hamiltonian for the parameter choices discussed in the Sec.III.1 in the main text. Here we consider a generic gapless point $(K_{p},K_{p},K_{z})$ in the momentum space, which exist on a mirror or glide plane. We expand the momentum around these gapless points with $(k_{x},k_{y},k_{z})\equiv(K_{p}+q_{x},K_{p}+q_{y},K_{z}+q_{z})$ where $q_{x,y,z}$ are assumed to take small numbers for perturbation. By expanding $H_{D}\equiv H_{0}+H_{AFM}$ around the gapless points $(K_{p},K_{p},K_{z})$ to the first order of $q_{i}$’s, we obtain

[TABLE]

where $\Gamma_{0}=\frac{4}{15}[E_{v}+\xi_{0}-2\beta_{c}\gamma_{1}K_{p}^{2}-\beta_{c}\gamma_{1}K_{z}^{2}](J_{x}^{2}+J_{y}^{2}+J_{z}^{2})$ ,

$\Gamma_{x}\equiv-\frac{2}{3}\beta_{c}\gamma_{2}K_{p}(2J_{z}^{2}-J_{x}^{2}-J_{y}^{2})+2\beta_{c}\gamma_{2}K_{p}(J_{x}^{2}-J_{y}^{2})+4\beta_{c}\gamma_{3}K_{p}J_{xy}+4\beta_{c}\gamma_{3}K_{z}J_{zx}$,

$\Gamma_{y}\equiv-\frac{2}{3}\beta_{c}\gamma_{2}K_{p}(2J_{z}^{2}-J_{x}^{2}-J_{y}^{2})-2\beta_{c}\gamma_{2}K_{p}(J_{x}^{2}-J_{y}^{2})+4\beta_{c}\gamma_{3}K_{p}J_{xy}+4\beta_{c}\gamma_{3}K_{z}J_{yz}$, and

$\Gamma_{z}\equiv+\frac{4}{3}\beta_{c}\gamma_{2}K_{z}(2J_{z}^{2}-J_{x}^{2}-J_{y}^{2})+4\beta_{c}\gamma_{3}K_{p}J_{zx}+4\beta_{c}\gamma_{3}K_{p}J_{yz}$.

Since $\beta_{c},\gamma_{2},\gamma_{3}\neq 0$ and $K_{p},K_{z}$ cannot be zero at the same time, none of $\Gamma_{x}$, $\Gamma_{y}$ and $\Gamma_{z}$ are vanishing. Let us define $q_{1}\equiv q_{x}+a_{1}q_{y}+a_{2}q_{z}$, $q_{2}\equiv q_{y}+a_{3}q_{z}$, $q_{3}\equiv q_{z}$, $\Gamma_{1}\equiv\Gamma_{x}$, $\Gamma_{2}\equiv-a_{1}\Gamma_{x}+\Gamma_{y}$, $\Gamma_{3}\equiv-a_{3}\Gamma_{x}-a_{3}\Gamma_{y}+\Gamma_{z}$ with $a_{1}\equiv\frac{K_{p}^{2}\left(3\gamma_{3}^{2}-2\gamma_{2}^{2}\right)}{4\gamma_{2}^{2}K_{p}^{2}+3\gamma_{3}^{2}\left(K_{p}^{2}+K_{z}^{2}\right)}$, $a_{3}\equiv\frac{K_{p}K_{z}\left(-2\gamma_{2}^{2}+3\gamma_{3}^{2}\right)}{2K_{p}^{2}\left(\gamma_{2}^{2}+3\gamma_{3}^{2}\right)+3\gamma_{3}^{2}K_{z}^{2}}$ and $a_{2}\equiv a_{3}(1+a_{1})$. We find that the Hamiltonian $H_{D}$ can be re-written as

[TABLE]

where $\{\Gamma_{i},\Gamma_{j}\}=0,i\neq j$ and $\{\Gamma_{i},\Gamma_{i}\}\neq 0$ for $i,j=1,2,3$. Thus, we conclude that all gapless points, if exist, are Dirac points with the low energy effective theory described by Dirac fermions, for the parameter choices discussed in the Sec.III.1 in the main text.

Appendix E $C_{3v}^{*}$ spin double group

This section shows how four bases of $\Gamma_{8}$ bands can be constructed into $\Lambda_{4,5,6}$ irreducible representations of $C_{3v}^{*}$ spin double group, as discussed in Sec.III.4. The character table of $C_{3v}^{*}$ spin double group is shown in Tab.5. Wave functions for each irreducible representations in bases $|\Gamma_{8},3/2\rangle$, $|\Gamma_{8},1/2\rangle$, $|\Gamma_{8},-1/2\rangle$ and $|\Gamma_{8},-3/2\rangle$ are shown below:

$\Lambda_{4}$:

[TABLE]

$\Lambda_{5}$:

[TABLE]

$\Lambda_{6}$:

[TABLE]

,where $N_{4}$ and $N_{5}$ are normalization factors.

Appendix F Tables for Parameters

This section is devoted to a summary of all the parameters for the four-band model and the extended Kane model used throughout the main text of the manuscript.

Appendix G Determination of Critical Lines in Fig.3a

The section describes how to determine critical lines in Fig.3a in the main text.

First, we hope to demonstrate that the Weyl points are from the $\Pi$ pairs on the $(1\bar{1}0)$ plane in our case. The creation or annihilation of Weyl points requires two Weyl points with opposite Chern numbers, thus only occuring between a $\Pi$ pair(like along path $\gamma$) or a $\Pi S$ pair for the case described in Sec.III(b). If a $\Pi S$ pair merges, it can only happen on the $(1,-1,0)$ axis where the Hamiltonian is invariant under $\pi_{1\bar{1}0}S$ and each band is doubly degenerate. If Hamiltonian is gapless at some point on the $(1,-1,0)$ axis, say $(k^{\prime},-k^{\prime},0)$, it requires the Hamiltonian to be an identity at that point, and implies that $\xi_{4}=0$, $Ck^{\prime}=0$, $\xi_{3}=2\sqrt{3}\beta_{c}\gamma_{3}k^{\prime 2}$ and $\xi_{1}=2\beta_{c}\gamma_{2}k^{\prime 2}$. Since $C\neq 0$ and $\sum_{i=1,,,5}\xi_{i}^{2}\neq 0$, the Hamiltonian cannot be gapless on the $(1,-1,0)$ axis, and therefore Weyl points cannot exist on the $(1,-1,0)$ axis. We conclude Weyl points can only originate from the $\Pi$ pairs on the $(1\bar{1}0)$ plane for the parameter regions that we are interested in.

When Weyl points are away from the mirror $(1\bar{1}0)$ plane, energy bands are gapped in the $(1\bar{1}0)$ plane, where the mirror Chern number is well-defined and can be computed to be 1. Thus, the mirror Chern number provides additional characterization of topological property in our system. The mirror Chern number always changed by 1 between the transition from the phase I to II or from II to III in Fig. 3(a). This suggests that band gap closing at the transition lines should occur between two bands in one mirror (or glide) subspace. This allows us to determine the phase transition lines (blue lines in Fig. 3(a)) by solving gapless condition of a mirror (or glide) subspace:

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