$Z_4$ Topological Superconductivity in UCoGe

TL;DR

This paper proposes UCoGe under pressure as a candidate for $Z_4$ topological nonsymmorphic crystalline superconductivity, analyzing its topological invariants and surface states, and discusses potential realization in related materials.

Contribution

It introduces a method to evaluate $Z_4$ invariants in UCoGe and identifies three odd-parity representations as $Z_4$-nontrivial TNCS, advancing the understanding of topological superconductivity.

Findings

UCoGe under pressure is a promising $Z_4$ TNCS candidate.

Three odd-parity representations in UCoGe are $Z_4$-nontrivial.

Discussion of potential $Z_4$ TNCS in CrAs and related materials.

Abstract

Topological nonsymmorphic crystalline superconductivity (TNCS) is an intriguing phase of matter, offering a platform to study the interplay between topology, superconductivity, and nonsymmorphic crystalline symmetries. Interestingly, some of TNCS are classified into $Z_4$ topological phases, which have unique surface states referred to as a M\"obius strip or an hourglass, and have not been achieved in symmorphic superconductors. However, material realization of $Z_4$ TNCS has never been known, to the best of our knowledge. Here we propose that the paramagnetic superconducting phase of UCoGe under pressure is a promising candidate of $Z_4$-nontrivial TNCS enriched by glide symmetry. We evaluate $Z_4$ invariants of UCoGe by deriving the formulas relating $Z_4$ invariants to the topology of Fermi surfaces. Applying the formulas and previous ab-initio calculations, we clarify that three odd-parity representations, out of four, are $Z_4$-nontrivial TNCS, while the other is also $Z_2$-nontrivial TNCS. We also discuss possible $Z_4$ TNCS in CrAs and related materials.

Tables

Table 1. Table 1: Topological invariants of TNCS in the space group P ​ n ​ m ​ a 𝑃 𝑛 𝑚 𝑎 Pnma . Here, “IR” means irreducible representations of D 2 ​ h subscript 𝐷 2 ℎ D_{2h} . θ 4 ( a , n ) ​ ( π ) superscript subscript 𝜃 4 𝑎 𝑛 𝜋 \theta_{4}^{(a,n)}(\pi) are given by modulo four, and ν ± ( a , n ) ​ ( π ) superscript subscript 𝜈 plus-or-minus 𝑎 𝑛 𝜋 \nu_{\pm}^{(a,n)}(\pi) are by modulo two.

glide parity & IR		topological invariants
${\hat{G}}_a$	odd $A_u$, $B_{1u}$	${\theta }_4^{\left(a\right)}\left(\pi \right)=\#{\text{FS}}_{R\to U}/2+\Delta M\left(R\right)$
	even $B_{2u}$, $B_{3u}$	${\nu }_{\pm }^{\left(a\right)}\left(\pi \right)=\#{\text{FS}}_{R\to U}/4$
${\hat{G}}_n$	odd $A_u$, $B_{3u}$	${\theta }_4^{\left(n\right)}\left(\pi \right)=\#{\text{FS}}_{S\to U}/2$
	even $B_{1u}$, $B_{2u}$	${\nu }_{\pm }^{\left(n\right)}\left(\pi \right)=\#{\text{FS}}_{S\to U}/4+\Delta M\left(S\right)/2$

Table 2. Table 2: Characters of the irreducible representations of D 2 ​ h subscript 𝐷 2 ℎ D_{2h} , into which gap functions of P ​ n ​ m ​ a 𝑃 𝑛 𝑚 𝑎 Pnma system are classified  Sigrist and Ueda ( 1991 ); Nomoto  et al. ( 2016 ); Bradley and Cracknell ( 1972 ) . We show all the odd-parity irreducible representations and the simplest form of d 𝑑 d -vector compatible with the pairing symmetry.

	$\hat{I}$	${\hat{G}}_a$	${\hat{M}}_y$	${\hat{G}}_n$	$d$-vector
$A_u$	$-1$	$-1$	$-1$	$-1$	$k_x\hat{x},k_y\hat{y},k_z\hat{z}$
$B_{1u}$	$-1$	$-1$	$+1$	$+1$	$k_y\hat{x},k_x\hat{y}$
$B_{2u}$	$-1$	$+1$	$-1$	$+1$	$k_z\hat{x},k_x\hat{z}$
$B_{3u}$	$-1$	$+1$	$+1$	$-1$	$k_z\hat{y},k_y\hat{z}$

Table 3. Table 3: Summary of the topological invariants of UCoGe obtained by formulas in Table  1 and the ab-initio calculations  Fujimori  et al. ( 2015 , 2016 ) .

IR	topological invariants at ZF
	${\hat{G}}_a$	${\hat{G}}_n$
$A_u$	${\theta }_4^{\left(a\right)}\left(\pi \right)=2$	${\theta }_4^{\left(n\right)}\left(\pi \right)=2$
$B_{1u}$	${\theta }_4^{\left(a\right)}\left(\pi \right)=2$	${\nu }_{\pm }^{\left(n\right)}\left(\pi \right)=1$
$B_{2u}$	${\nu }_{\pm }^{\left(a\right)}\left(\pi \right)=1$	${\nu }_{\pm }^{\left(n\right)}\left(\pi \right)=1$
$B_{3u}$	${\nu }_{\pm }^{\left(a\right)}\left(\pi \right)=1$	${\theta }_4^{\left(n\right)}\left(\pi \right)=2$

Table 4. Table S1: Irreducible representations of ℳ ℳ \mathcal{M} .

	$\hat{E}$	$\hat{I}$	$\hat{G}$
${\Gamma }_{+g}$	$+1$	$+1$	$+1$
${\Gamma }_{-g}$	$+1$	$+1$	$-1$
${\Gamma }_{+u}$	$+1$	$-1$	$+1$
${\Gamma }_{-u}$	$+1$	$-1$	$-1$

Table 5. Table S2: Irreducible representations of ℳ R superscript ℳ 𝑅 \mathcal{M}^{R} and compatibility relations with those of ℳ a subscript ℳ 𝑎 \mathcal{M}_{a} .

irreps. of ${ℳ}^R$	irreps. of ${ℳ}_a$	$\hat{I}$	${\hat{G}}_a$	${\hat{M}}_y$
$E_1$	${\Gamma }_{+g}$	$+1$	$+1$
	${\Gamma }_{-u}$	$-1$	$-1$
$E_2$	${\Gamma }_{+u}$	$-1$	$+1$
	${\Gamma }_{-g}$	$+1$	$-1$

Table 6. Table S3: Irreducible representations of ℳ S superscript ℳ 𝑆 \mathcal{M}^{S} and compatible relations with those of ℳ n subscript ℳ 𝑛 \mathcal{M}_{n} .

irreps. of ${ℳ}^S$	irreps. of ${ℳ}_n$	$\hat{I}$	${\hat{G}}_n$	${\hat{M}}_y$
$E_{+}$	${\Gamma }_{+g}$	$+1$	$+1$
	${\Gamma }_{+u}$	$-1$	$+1$
$E_{-}$	${\Gamma }_{-g}$	$+1$	$-1$
	${\Gamma }_{-u}$	$-1$	$-1$

Table 7. Table S4: Irreducible representations of ℳ U superscript ℳ 𝑈 \mathcal{M}^{U} and compatible relations with those of ℳ a subscript ℳ 𝑎 \mathcal{M}_{a} .

irreps. of ${ℳ}^U$	irreps. of ${ℳ}_a$	$\hat{I}$	${\hat{G}}_a$	${\hat{M}}_y$
$E_g$	${\Gamma }_{+g}$	$+1$	$+1$
	${\Gamma }_{-g}$	$+1$	$-1$
$E_u$	${\Gamma }_{+u}$	$-1$	$+1$
	${\Gamma }_{-u}$	$-1$	$-1$

Table 8. Table S5: Irreducible representations of ℳ U superscript ℳ 𝑈 \mathcal{M}^{U} and compatibility relations with those of ℳ n subscript ℳ 𝑛 \mathcal{M}_{n} . The same results have been obtained for ℳ a subscript ℳ 𝑎 \mathcal{M}_{a} in Table  S4 .

irreps. of ${ℳ}^U$	irreps. of ${ℳ}_n$	$\hat{I}$	${\hat{G}}_n$	${\hat{M}}_y$
$E_g$	${\Gamma }_{+g}$	$+1$	$+1$
	${\Gamma }_{-g}$	$+1$	$-1$
$E_u$	${\Gamma }_{+u}$	$-1$	$+1$
	${\Gamma }_{-u}$	$-1$	$-1$

Table 9. Table S6: List of order parameters used in the numerical calculation. We take δ = 1 𝛿 1 \delta=1 in B 2 ​ u subscript 𝐵 2 𝑢 B_{2u} .

irrep.	$\tilde{\Delta }\left(𝒌\right)/d_0$
$A_u$	$\sin k_x{s}_x+\sin k_y{s}_y+\sin k_z{s}_x{\sigma }_z$
$B_{1u}$	$\sin k_x{s}_y+\sin k_y{s}_x+\sin k_z{s}_y{\sigma }_z$
$B_{2u}$	$\sin k_x{s}_x{\sigma }_z+\sin k_y{s}_y{\sigma }_z+\sin k_z{s}_x+\delta \left\{\left(1+\cos k_x\right)s_x{\sigma }_y{\eta }_z-\sin k_x{s}_x{\sigma }_x\right\}/2$
$B_{3u}$	$\sin k_x{s}_y{\sigma }_z+\sin k_y{s}_x{\sigma }_z+\sin k_z{s}_y$

Table 10. Table S7: Table of the generators of the glide-odd topological phases from LABEL:Yanase2017_SM .

	$W$	${\theta }_4\left(0\right)$	${\theta }_4\left(\pi \right)$
${\mathscr{H}}_1$	$1$	$1$	$0$
${\mathscr{H}}_2$	$0$	$0$	$2$
${\mathscr{H}}_3$	$0$	$1$	$1$

Table 11. Table S8: Table of the generators of the glide-even topological phases from Ref.  S ( 14 ) .

	${\nu }_{+}\left(0\right)$	${\nu }_{-}\left(0\right)$	${\nu }_{+}\left(\pi \right)$	${\nu }_{-}\left(\pi \right)$
${\mathscr{H}}_1$	$1$	$0$	$1$	$0$
${\mathscr{H}}_2$	$0$	$0$	$1$	$1$
${\mathscr{H}}_3$	$1$	$1$	$1$	$1$

Table 12. Table S9: Topological indices of UCoGe. The first and the second row correspond to the two possibilities of the winding number, W = 4 ​ n 𝑊 4 𝑛 W=4n and W = 4 ​ n + 2 𝑊 4 𝑛 2 W=4n+2 , respectively. Curly braces represent glide-odd indices { ℤ W , ℤ 2 strong , ℤ 4 weak } superscript ℤ 𝑊 superscript subscript ℤ 2 strong superscript subscript ℤ 4 weak \{\mathbb{Z}^{W},\mathbb{Z}_{2}^{\mathrm{strong}},\mathbb{Z}_{4}^{\mathrm{weak}}\} , while square braces represent glide-even indices [ ℤ 2 CS T , ℤ 2 strong , ℤ 2 weak ] superscript subscript ℤ 2 subscript CS 𝑇 superscript subscript ℤ 2 strong superscript subscript ℤ 2 weak [\mathbb{Z}_{2}^{\mathrm{CS}_{T}},\mathbb{Z}_{2}^{\mathrm{strong}},\mathbb{Z}_{2}^{\mathrm{weak}}] .

	${\hat{G}}_a$	${\hat{G}}_n$
$A_u$	$\{4n,0,2\}$	$\{4n,1,0\}$
	$\{4n+2,1,0\}$	$\{4n+2,0,2\}$
$B_{1u}$	$\{0,0,2\}$	$[0,1,0]$
$B_{2u}$	$[0,0,1]$	$[0,1,0]$
$B_{3u}$	$[0,0,1]$	$\{0,1,0\}$

Equations

$$\hat{G}\hat{I}=\left\{E\right\}\hat{c}+\mathbf{a}\hat{I}\hat{G}.$$

$$\hat{G}{|\mathbf{k}⟩}_{\pm }=\pm i{e}^{-i\mathbf{k}\cdot \mathbf{a}/2}{|\mathbf{k}⟩}_{\pm }.$$

$$\hat{G}\left(\hat{\Theta }\hat{I}{|\mathbf{k}⟩}_{\pm }\right)$$

$$=\pm i{e}^{-i{k}_a/2}\left(\hat{\Theta }\hat{I}{|\mathbf{k}⟩}_{\pm }\right),({\Gamma }_c=\pi )$$

$${\theta }_4({\Gamma }_c)=2{\int }_{-\pi }^{\pi }$$

$$-{\int }_{0\le k_a\le \pi }\frac{d^{2}k}{\pi i}{F}_{+}(k_a,k_b,{\Gamma }_c),$$

$${\theta }_4(\pi )=\sum _{i=1,2}\{\begin{array}{l}{M}_{+u}^{<}({\Gamma }_i)+M_{-g}^{<}({\Gamma }_i)\\ M_u^{<}({\Gamma }_i),\end{array}(\text{mod}4)$$

$${\nu }_{\pm }({\Gamma }_c)={\int }_{-\pi }^{\pi }\frac{d{k}_b}{\pi i}{A}_{\pm }^{\mathrm{I}}(\pi ,k_b,{\Gamma }_c)(\text{mod}2).$$

$${\nu }_{\pm }(\pi )=\sum _{i=1,2}\{\begin{array}{l}{M}_{\pm }^{<}({\Gamma }_i)/2\\ 0,\end{array}(\text{mod}2)$$

$${\hat{G}}_a=\left\{M_z\right\}\hat{x}/2+\hat{z}/2,{\hat{G}}_n=\left\{M_x\right\}\hat{x}/2+\hat{y}/2+\hat{z}/2,$$

$$\#{\text{FS}}_{{\Gamma }_1\to {\Gamma }_2}\equiv {\int }_{{\Gamma }_1}^{{\Gamma }_2}d\mathbf{k}\cdot {\nabla }_{\mathbf{k}}{M}^{<}(\mathbf{k})\in 4\mathbb{Z},$$

$$\Delta M({\Gamma }_i)\equiv M_{+u}^{<}({\Gamma }_i)-M_{-u}^{<}({\Gamma }_i)\in 2\mathbb{Z},$$

$$\#{\text{FS}}_{R\to U}\in 4(2\mathbb{Z}+1),$$

$$\#{\text{FS}}_{S\to U}\in 4(2\mathbb{Z}+1),$$

$$\hat{g}\mathbf{x}=p\mathbf{x}+\mathbf{a}.$$

$$\hat{g}{c}_l^{†}(\mathbf{R}+{\mathbf{r}}_n){\hat{g}}^{-1}=c_{l^{\prime }}^{†}(p\mathbf{R}+\Delta {\mathbf{R}}_n^g+{\mathbf{r}}_{n^{\prime }})D_{n^{\prime }n}(\hat{g}){\mathcal{D}}_{l^{\prime }l}(\hat{g}),$$

$$\hat{g}({\mathbf{r}}_n)=p{\mathbf{r}}_n+\mathbf{a}={\mathbf{r}}_{n^{\prime }}+\Delta {\mathbf{R}}_n^g,$$

$$c_{\mathbf{k}ln}^{†}\equiv \frac{1}{\sqrt{V}}\sum _{\mathbf{R}}{e}^{i\mathbf{k}\cdot \mathbf{R}}{c}_l^{†}(\mathbf{R}+{\mathbf{r}}_n),$$

$$c_{\mathbf{k}+\mathbf{G}ln}^{†}=c_{\mathbf{k}ln}^{†},$$

$$c_{ln}^{†}(\mathbf{k})\equiv \frac{1}{\sqrt{V}}\sum _{\mathbf{R}}{e}^{i\mathbf{k}\cdot (\mathbf{R}+{\mathbf{r}}_n)}{c}_l^{†}(\mathbf{R}+{\mathbf{r}}_n)=c_{\mathbf{k}l{n}^{\prime }}^{†}{V}_{n^{\prime }n}(\mathbf{k}),V_{n^{\prime }n}(\mathbf{k})=e^{i\mathbf{k}\cdot {\mathbf{r}}_n}{\delta }_{n^{\prime }n},$$

$$c_{ln}^{†}(\mathbf{k}+\mathbf{G})=c_{l{n}^{\prime }}^{†}(\mathbf{k})V_{n^{\prime }n}(\mathbf{G}).$$

$$\hat{g}{c}_{ln}^{†}(\mathbf{k}){\hat{g}}^{-1}$$

$$=\frac{1}{\sqrt{V}}\sum _{{\mathbf{R}}^{\prime }=p\mathbf{R}+\Delta {\mathbf{R}}_n^g}{e}^{i\mathbf{k}\cdot (p^{-1}({\mathbf{R}}^{\prime }-\Delta {\mathbf{R}}_n^g)+{\mathbf{r}}_n)}{c}_{l^{\prime }}^{†}({\mathbf{R}}^{\prime }+{\mathbf{r}}_{n^{\prime }})D_{n^{\prime }n}(\hat{g}){\mathcal{D}}_{l^{\prime }l}(\hat{g})$$

$$=\frac{1}{\sqrt{V}}\sum _{{\mathbf{R}}^{\prime }}{e}^{ip\mathbf{k}\cdot ({\mathbf{R}}^{\prime }-\Delta {\mathbf{R}}_n^g+p{\mathbf{r}}_n)}{c}_{l^{\prime }}^{†}({\mathbf{R}}^{\prime }+{\mathbf{r}}_{n^{\prime }})D_{n^{\prime }n}(\hat{g}){\mathcal{D}}_{l^{\prime }l}(\hat{g})$$

$$=\frac{1}{\sqrt{V}}\sum _{{\mathbf{R}}^{\prime }}{e}^{ip\mathbf{k}\cdot ({\mathbf{R}}^{\prime }-\mathbf{a}+{\mathbf{r}}_{n^{\prime }})}{c}_{l^{\prime }}^{†}({\mathbf{R}}^{\prime }+{\mathbf{r}}_{n^{\prime }})D_{n^{\prime }n}(\hat{g}){\mathcal{D}}_{l^{\prime }l}(\hat{g})$$

$$=e^{-ip\mathbf{k}\cdot \mathbf{a}}{c}_{l^{\prime }{n}^{\prime }}^{†}(p\mathbf{k})D_{n^{\prime }n}(\hat{g}){\mathcal{D}}_{l^{\prime }l}(\hat{g})\equiv c_{l^{\prime }{n}^{\prime }}^{†}(p\mathbf{k})U_{(l^{\prime }{n}^{\prime })(ln)}^g(\mathbf{k}),$$

$$\hat{g}{c}_{\mathbf{k}ln}^{†}{\hat{g}}^{-1}=c_{p\mathbf{k}{l}^{\prime }{n}^{\prime }}^{†}{\tilde{U}}_{(l^{\prime }{n}^{\prime })(ln)}^g(\mathbf{k}),$$

$${\tilde{U}}_{(l^{\prime }{n}^{\prime })(ln)}^g(\mathbf{k})=(V(p\mathbf{k})U^g(\mathbf{k})V(\mathbf{k}{)}^{†}{)}_{(l^{\prime }{n}^{\prime })(ln)}=e^{i(p\mathbf{k}\cdot {\mathbf{r}}_{n^{\prime }}-p\mathbf{k}\cdot \mathbf{a}-\mathbf{k}\cdot {\mathbf{r}}_n)}{D}_{n^{\prime }n}(\hat{g}){\mathcal{D}}_{l^{\prime }l}(\hat{g})=e^{-ip\mathbf{k}\cdot \Delta {\mathbf{R}}_n^g}{D}_{n^{\prime }n}(\hat{g}){\mathcal{D}}_{l^{\prime }l}(\hat{g}).$$

$$C{H}_{\mathrm{BdG}}(\mathbf{k})C^{-1}=-H_{\mathrm{BdG}}(-\mathbf{k}),$$

$$C{\tilde{U}}_{\mathrm{BdG}}^G(\mathbf{k})=\pm {\tilde{U}}_{\mathrm{BdG}}^G(-\mathbf{k})C,\text{or equivalently,}\hat{C}\hat{G}=\pm \hat{G}\hat{C}.$$

$$\hat{G}=\left\{M_c\right\}\hat{c}/2+\mathbf{a}/2.$$

$$\hat{S}=\left\{C_{2c}\right\}\hat{c}/2+\mathbf{a}/2,\hat{I}=\left\{I\right\}0.$$

$$\mathbf{k}=k_a\mathbf{\alpha }+k_b\mathbf{\beta }+k_c\hat{c},$$

$$\mathbf{\alpha }=\frac{\mathbf{b}\times \hat{c}}{\mathbf{a}\cdot \mathbf{b}\times \hat{c}},\mathbf{\beta }=\frac{\mathbf{a}\times \hat{c}}{\mathbf{b}\cdot \mathbf{a}\times \hat{c}}.$$

$$\mathbf{\alpha }=\hat{y},\mathbf{\beta }=\hat{z}-\hat{y},\hat{c}=\hat{x},$$

$$k_a=k_{}$$