Majorana multipole response of topological superconductors

TL;DR

This paper develops a theory linking the electromagnetic multipole responses of Majorana fermions on superconductor surfaces to crystal symmetry, enabling identification of unconventional pairing and high-spin superconductivity.

Contribution

The authors establish a general framework connecting Majorana fermion multipole responses to crystal symmetry and pairing, revealing unique signatures of high-spin superconductors.

Findings

Most helical Majorana fermions exhibit magnetic-dipole responses.

High-spin superconductors can show magnetic-octupole responses in surface Majorana fermions.

Magnetic response signatures can identify unconventional pairing and high-spin states.

Abstract

In contrast to elementary Majorana particles, emergent Majorana fermions (MFs) in condensed-matter systems may have electromagnetic multipoles. We developed a general theory of magnetic multipoles for surface helical MFs on time-reversal-invariant superconductors. The results show that the multipole response is governed by crystal symmetry, and that a one-to-one correspondence exists between the symmetry of Cooper pairs and the representation of magnetic multipoles under crystal symmetry. The latter property provides a way to identify nonconventional pairing symmetry via the magnetic response of surface MFs. We also find that most helical MFs exhibit a magnetic-dipole response, but those on superconductors with spin-3/2 electrons may display a magnetic-octupole response in leading order, which uniquely characterizes high-spin superconductors. Detection of such an octupole response provides direct evidence of high-spin superconductivity, such as in half-Heusler superconductors.

Tables

Table 1. Table 1: Magnetic multipole of MFs. From left to right, each column shows two-dimensional point groups (PGs), IRs of Δ Γ subscript Δ Γ \Delta_{\Gamma} with w M1D ≠ 0 subscript 𝑤 M1D 0 w_{\rm M1D}\neq 0 , the basis of Δ Γ subscript Δ Γ \Delta_{\Gamma} , 𝒰 𝒰 \mathcal{U} associated with Γ M subscript Γ M \Gamma_{\rm M} , IRs of 𝒪 Γ subscript 𝒪 Γ \mathcal{O}_{\Gamma} , and the basis of 𝒪 Γ subscript 𝒪 Γ \mathcal{O}_{\Gamma} . Here, J i subscript 𝐽 𝑖 J_{i} are the spin matrices, “ − - ” means the absence of IRs, and 𝒪 Γ subscript 𝒪 Γ \mathcal{O}_{\Gamma} describes the leading order of the magnetic multipoles.

PG	${\Delta }_{\Gamma }$	basis of ${\Delta }_{\Gamma }$ $(\times e^{-i\pi J_y})$	$𝒰$	${𝒪}_{\Gamma }$	basis of ${𝒪}_{\Gamma }$
$C_2,C_4,C_6$	A	$𝒌\cdot 𝑱$	$C_2$	A	$J_z$
$C_3$	$-$	$-$	$-$	$-$	$-$
$C_s$	A	$k_x{J}_z,k_x{J}_y,k_y{J}_x,k_z{J}_x$	${\sigma }_v\left(yz\right)$	A	$J_x$
$C_{2v}$	$A_2$	$k_z{J}_z$	$C_2$	$A_2$	$J_z$
	$B_1$	$k_x{J}_z,k_z{J}_x$	${\sigma }_v\left(yz\right)$	$B_1$	$J_x$
	$B_2$	$k_y{J}_z,k_z{J}_y$	${\sigma }_v\left(xz\right)$	$B_2$	$J_y$
$C_{3v}$	$A_1$	$k_z\left(J_x^3-J_x{J}_y{J}_y-J_y{J}_x{J}_y-J_y{J}_y{J}_x\right)$	${\sigma }_v\left(yz\right)$	$A_1$	$J_x^3-J_x{J}_y{J}_y-J_y{J}_x{J}_y-J_y{J}_y{J}_x$
$C_{4v}$	$A_2$	$k_z{J}_z$	$C_2$	$A_2$	$J_z$
$C_{6v}$	$A_2$	$k_z{J}_z$	$C_2$	$A_2$	$J_z$
	$B_1$	$k_z\left(J_x^3-J_x{J}_y{J}_y-J_y{J}_x{J}_y-J_y{J}_y{J}_x\right)$	${\sigma }_v\left(yz\right)$	$B_1$	$J_x^3-J_x{J}_y{J}_y-J_y{J}_x{J}_y-J_y{J}_y{J}_x$
	$B_2$	$k_z\left(J_y^3-J_y{J}_x{J}_x-J_x{J}_y{J}_x-J_x{J}_x{J}_y\right)$	${\sigma }_d\left(xz\right)$	$B_2$	$J_y^3-J_y{J}_x{J}_x-J_x{J}_y{J}_x-J_x{J}_x{J}_y$

Table 2. Table S1: Possible matrices O = s μ ​ σ ν 𝑂 subscript 𝑠 𝜇 subscript 𝜎 𝜈 O=s_{\mu}\sigma_{\nu} for local operators of the two-orbital model in S4. We classify O 𝑂 O by IRs of D 3 ​ h subscript 𝐷 3 ℎ D_{3h} .

IR of $D_{3h}$	TR-even	TR-odd	basis by magnetic fields
$A_1^{\prime }$	${\sigma }_0{s}_0,{\sigma }_y{s}_z$
$A_2^{\prime }$		${\sigma }_0{s}_z,{\sigma }_y{s}_0$	$B_z$
$E^{\prime }$	$\{{\sigma }_x{s}_0,{\sigma }_z{s}_0\}$	$\{{\sigma }_x{s}_z,{\sigma }_z{s}_z\}$
$A_1^{\prime \prime }$		${\sigma }_z{s}_x-{\sigma }_x{s}_y$	$B_x^3-3B_x{B}_y^2$
$A_2^{\prime \prime }$		${\sigma }_z{s}_y+{\sigma }_x{s}_x$	$B_y^3-3B_y{B}_x^2$
$E^{\prime \prime }$	$\{{\sigma }_y{s}_x,{\sigma }_y{s}_y\}$	$\{{\sigma }_0{s}_x,{\sigma }_0{s}_y\},\{{\sigma }_z{s}_x+{\sigma }_x{s}_y,{\sigma }_x{s}_x-{\sigma }_z{s}_y\}$	$\{B_x,B_y\}$

Equations

$${\hat{O}}_{\mathrm{MF}}$$

$$=-\frac{i}{4}\sum _{ab}{\hat{\gamma }}^{(a)}{\hat{\gamma }}^{(b)}\mathrm{tr}\left[\mathcal{O}{\rho }^{(ab)}\right],$$

$${\Gamma }_{\mathrm{M}}|u_0^{(a)}⟩=|u_0^{(a)}⟩.$$

$$H_{\mathrm{LK}}(\mathbf{k})=$$

$$+\delta \sum _i{k}_i(J_{i+1}{J}_i{J}_{i+1}-J_{i+2}{J}_i{J}_{i+2}),$$

$$w_{M1D}=\frac{i}{4\pi }\int d{k}_{\perp }\mathrm{tr}\left[{\Gamma }_{\mathrm{M}}{\mathcal{H}}^{-1}(k_{\perp },{\mathbf{k}}_{\parallel }){\partial }_{k_{\perp }}\mathcal{H}(k_{\perp },{\mathbf{k}}_{\parallel })\right],$$

$${\Gamma }_{\mathrm{M}}|u_0^{(a)}⟩=|u_0^{(a)}⟩,$$

$${\Gamma }_{\mathrm{M}}{\mathcal{O}}_{\Gamma }{\Gamma }_{\mathrm{M}}^{†}={\mathcal{O}}_{\Gamma }.$$

$$C_2{\mathcal{O}}_{\Gamma }{C}_2^{†}={\mathcal{O}}_{\Gamma },$$

$$\mathcal{C}|u_0^{(a)}⟩=|u_0^{(a)}⟩,(a=1,2),$$

$$\mathcal{T}|u_0^{(1)}⟩=|u_0^{(2)}⟩,$$

$$\sigma |u_0^{(a)}⟩=(-{)}^{a}i|u_0^{(a)}⟩,(a=1,2).$$

$$\sigma {\mathcal{O}}_{\Gamma }{\sigma }^{-1}=-{\mathcal{O}}_{\Gamma }.$$

$$C_n|u_0^{(a)}⟩={\lambda }^{(a)}|u_0^{(a)}⟩,$$

$$C_n{\mathcal{O}}_{\Gamma }{C}_n^{†}={\mathcal{O}}_{\Gamma }.$$

$$H_{\mathrm{TI}}(\mathbf{k})=c(\mathbf{k})+m(\mathbf{k}){\sigma }_x+v_z{k}_z{\sigma }_y+v(k_x{s}_y-k_y{s}_x){\sigma }_z+\lambda (k_x^3-3k_x{k}_y^2)s_z,$$

$${\mathcal{H}}_{\mathrm{TI}}(\mathbf{k})=\left(\begin{array}{cc}{H}_{\mathrm{TI}}(\mathbf{k})-\mu & \Delta (\mathbf{k})\\ \Delta (\mathbf{k}{)}^{†}& -H_{\mathrm{TI}}(-\mathbf{k}{)}^t+\mu \end{array}\right),$$

$$H(\mathbf{k})=a_0(\mathbf{k})s_0+a_x(\mathbf{k})s_x+a_y(\mathbf{k})s_y+a_z(\mathbf{k})s_z,$$

$$C_3^{†}H(0,0,k_z)C_3=H(0,0,k_z),$$

$${{\sigma }_v}^{†}H(0,0,k_z){\sigma }_v=H(0,0,k_z),$$

$$C_3=\mathrm{diag}(e^{-i\frac{2\pi }{3}{j}_z},e^{i\frac{2\pi }{3}{j}_z}),$$

$${\sigma }_v=i{s}_x,$$

$$C_{\infty }^{†}H(0,0,k_z)C_{\infty }=H(0,0,k_z),$$

$$C_3=e^{-i\frac{\pi }{3}{s}_z}$$

$${\sigma }_v=i{s}_x.$$

$$C_3=-s_0$$

$${\sigma }_v=i{s}_x.$$

$$H(0,0,k_z)=a_0(k_z)s_0+b{k}_z{s}_x,$$

$$H(\mathbf{k})=C(\mathbf{k}){\sigma }_0{s}_0+{\lambda }_1{\sigma }_y{s}_z+{\lambda }_2\sin k_z({\sigma }_x{s}_x+{\sigma }_z{s}_y),$$

$${\sigma }_v=-i{\sigma }_z{s}_y,$$

$${\sigma }_h=-i{\sigma }_0{s}_z,$$

$$C_3={\left(\begin{array}{cc}-\frac{1}{2}& -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& -\frac{1}{2}\end{array}\right)}_{\sigma }\otimes e^{-i\frac{\pi }{3}{s}_z},$$

$$\displaystyle\mathcal{H}({\bm{k}})=\left(\begin{array}[]{cc}H({\bm{k}})-\mu&\Delta({\bm{k}})\\ \Delta({\bm{k}})^{\dagger}&-H({\bm{k}})^{t}+\mu\end{array}\right),$$

$$\Delta (\mathbf{k})=i{\Delta }_0{s}_y+{\Delta }_1{\sigma }_y{s}_x+{\Delta }_2\sin k_z(-{\sigma }_x{s}_z+i{s}_0{\sigma }_z).$$

$${\tilde{\sigma }}_v=\mathrm{diag}({\sigma }_v,{\sigma }_v^{\ast })=-i{\sigma }_x{s}_y{\tau }_0,$$

Full text

Introduction.

The emergence of Majorana fermions (MFs) in electron systems has led to intense interest in searching for such exotic new excitations in condensed-matter physics. Particularly, recent developments have shown that emergent MFs appear as gapless Andreev bound states in topological superconductors (TSCs) Hu (1994); Kashiwaya and Tanaka (2000); Volovik (2003); Schnyder et al. (2008); Sato (2009); Wilczek (2009); Hasan and Kane (2010); Qi and Zhang (2011); Tanaka et al. (2012); Alicea (2012); Sato and Fujimoto (2016); Mizushima et al. (2016); Chiu et al. (2016); Sato and Ando (2017), which provide a potential candidate for fault-tolerant qubits for topological quantum computation Nayak et al. (2008). The increased interest in topological materials has led to a proposal of versatile three-dimensional (3D) time-reversal-invariant (TR-invariant) TSCs, such as superconducting doped topological insulators (TIs) Hor et al. (2010); Fu and Berg (2010); Sasaki et al. (2011, 2012); Hashimoto et al. (2015); Fu (2014); Matano et al. (2016); Yonezawa et al. (2016) and Dirac semimetals Aggarwal et al. (2015); Wang et al. (2015); Kobayashi and Sato (2015); Hashimoto et al. (2016); Oudah et al. (2016); Kawakami et al. (2018), which commonly host helical MFs forming Kramers pairs on their surfaces.

Emergent MFs share some properties with elementary Majorana particles Majorana (1937); Avignone et al. (2008). For example, both obey Dirac equations with charge-conjugation symmetry. Furthermore, a pair of MF zero modes are required to define the fermionic creation and annihilation operators, from which zero modes exhibit non-Abelian anyon statistics. However, compared with elementary Majorana particles, emergent MFs respond very differently to electric and magnetic fields. Contrarily, neither electric nor magnetic multipoles are possible for elementary MFs  Kayser and Goldhaber (1983); Radescu (1985); Boudjema et al. (1989): $\mathcal{CPT}$ invariance, where $\mathcal{C}$ is charge conjugation, $\mathcal{P}$ is space inversion, and $\mathcal{T}$ is time reversal, is a fundamental symmetry that any relativistic elementary particles is expected to respect. This symmetry forbids intrinsic electric and magnetic multipoles for elementary Majorana particles because they are their own antiparticles under $\mathcal{CPT}$. Contrarily, in superconductors, fundamental symmetry is just charge-conjugation (namely, particle-hole (PH) symmetry), and the emergent MFs are self-conjugate under $\mathcal{C}$. Therefore, MFs in condensed-matter physics are not subject to such a strong constraint, and no systematic study on their electromagnetic multipoles has yet been attempted.

In this Letter, we develop a theory describing the electric and magnetic response of MFs in superconductors. For clarity, we focus here on surface helical MFs on 3D TR-invariant TSCs. A key ingredient specific to emergent MFs is crystalline symmetry. In analogy with $\mathcal{CPT}$ invariance for elementary MFs, crystal symmetry provides additional symmetry constraints on electromagnetic structures of emergent MFs. Considering the constraints, we establish a response theory for helical MFs in a low-energy limit, in which the problem reduces to the selection rule for crystal-symmetry groups. Applying our theory to possible crystal-symmetry groups, we find that helical MFs can host magnetic-multipole structures of dipole or octupole orders as the leading contribution. Additionally, the results predict a one-to-one correspondence between irreducible representation (IR) of Cooper pairs and magnetic multipoles, which helps to determine the pairing symmetry experimentally through the magnetic response of MFs.

Particularly, the proposed theory provides a unique way to identify topological superconductivity of spin-3/2 electrons. Although research interest has recently focused on high-spin topological superconductivity Goll et al. (2008); Butch et al. (2011); Tafti et al. (2013); Xu et al. (2014); Bay et al. (2012); Kim et al. (2018); Boettcher and Herbut (2016); Brydon et al. (2016); Savary et al. (2017); Roy et al. (2017); Timm et al. (2017); Venderbos et al. (2018); Boettcher and Herbut (2018); Yu and Liu (2018); Kuzmenko et al. (2018); Kawakami et al. (2018), little is known about distinguishing TSCs of spin-$3/2$ electrons from those of spin-$1/2$ electrons. Thus, we clarify that magnetic responses of helical MFs can unambiguously distinguish between these two types of SCs because the magnetic-octupole response is unique to higher-spin TSCs. To illustrate this, we apply the proposed theory to superconducting TIs of ordinary spin-1/2 electrons Fu and Berg (2010) and parity-mixed half-Heusler superconductors of spin-$3/2$ electrons Brydon et al. (2016); Kim et al. (2018). The results of both numerical and analytical analyses show that only the latter exhibits the octupole response under the same crystalline symmetry.

Majorana multipole.

Helical MFs are a superconducting analogue of surface Dirac fermions of TIs and can be realized in 3D TR-invariant TSCs. From the bulk-boundary correspondence, the existence of helical MFs is ensured by the so-called 3D winding number Grinevich and Volovik (1988); Schnyder et al. (2008); Sato (2009, 2010). Whereas the 3D winding number is defined only for fully gapped TSCs, its parity is well defined even for nodal superconductors Sasaki et al. (2011). Provided TR symmetry is maintained, these invariants are well defined and protect surface helical MFs for both nodal and nodeless superconductors.

We consider the quantum response of helical MFs when exposed to external electric or magnetic fields. First, note that electric fields only elicit a moderate response from helical MFs because electric fields maintain TR symmetry, helical MFs remains gapless so their response is weak. Conversely, magnetic fields may substantially affect them. Magnetic fields break TR symmetry, so the 3D winding number and its parity become invalid. However, this does not mean that helical MFs are not immune to some magnetic fields because actual TSCs have their own crystalline symmetry. Depending on the direction of the applied magnetic field, TR symmetry may be partially preserved by combining it with crystalline symmetry. Such magnetic crystalline symmetry determines the stability of helical MFs under magnetic fields Note (1).

As relevant point group operations, we now consider mirror reflections and rotations that are compatible with the surface in question. The mirror plane and the rotation axis should be normal to the surface (see Fig. 1). We consider all two-dimensional point groups formed by them: $C_{2}$, $C_{3}$, $C_{4}$, $C_{6}$, $C_{s}$, $C_{2v}$, $C_{3v}$, $C_{4v}$, and $C_{6v}$, in addition to TR symmetry. Under a magnetic field, we retain only magnetic mirror reflection (or magnetic two-fold rotation). Note that the retained magnetic symmetry is selected by the direction of an applied magnetic field: Only for a magnetic field parallel (normal) to the mirror plane (rotation axis) is magnetic mirror reflection (magnetic two-fold rotation) preserved. The above magnetic field is easily seen to flip under TR, but it points back to the original when we simultaneously do a mirror reflection (two-fold rotation).

The retained magnetic symmetry enables TSCs to host an additional topological number that is valid even when the TSC is exposed to a magnetic field. Combining magnetic symmetry with PH symmetry, which is intrinsic to superconductors, one can introduce the magnetic one-dimensional (1D) winding number Mizushima et al. (2012); Shiozaki and Sato (2014); Dumitrescu et al. (2014); Xiong et al. (2017): $w_{\rm M1D}=\frac{i}{4\pi}\int dk_{\perp}{\rm tr}\left[\Gamma_{\rm M}{\cal H}^{-1}(k_{\perp},\bm{k}_{\parallel})\partial_{k_{\perp}}{\cal H}(k_{\perp},\bm{k}_{\parallel})\right]$ where ${\cal H}(\bm{k})$ is the Bogoliubov-de Gennes (BdG) Hamiltonian, $(k_{\perp},\bm{k}_{\parallel})$ are the momentum normal and parallel to the surface, respectively, and $\Gamma_{\rm M}\equiv\mathcal{U}\mathcal{T}\mathcal{C}$ is the magnetic chiral operator. Here, $\mathcal{U}$ is a mirror reflection or two-fold rotation. If $w_{\rm M1D}$ for magnetic two-fold rotation (magnetic mirror reflection) is nonzero in the absence of magnetic fields, then helical MFs remain gapless even under a magnetic field normal (parallel) to the rotation axis (mirror plane), provided the system maintains the bulk gap. Conversely, helical MFs do not necessarily remain gapless under other magnetic fields. This direction dependence results in an anisotropic magnetic response of helical MFs. Note that $w_{\rm M1D}$ for magnetic two-fold rotation (magnetic mirror reflection) is defined only on the symmetric axis (plane), so it protects the gapless point (line) of helical MFs at the symmetry axis (plane) in the surface Brillouin zone (see Fig. 1). From the bulk-boundary correspondence, the gapless points or lines are obtained as zero modes $|u_{0}^{(a)}\rangle$ ($a=1,2,\dots$) of the BdG equation.

To systematically study the magnetic response of MFs, we examine possible contributions of MFs to local operators $\hat{O}(x)=\hat{c}_{\sigma}^{\dagger}(x)O_{\sigma,\sigma^{\prime}}\hat{c}_{\sigma^{\prime}}(x)$ of electrons, where $\hat{c}_{\sigma}^{\dagger}(x)$ and $\hat{c}_{\sigma}(x)$ are the electron operators and $\sigma$ is the internal degrees of freedom such as spin, orbital, and so on. To obtain a physical response, the matrix $O_{\sigma,\sigma^{\prime}}$ should be Hermitian. The MFs have a nonzero response to external fields through local operators $\hat{O}(x)$. For instance, if MFs make a nonzero contribution to the electron-spin operator $\hat{S}_{i}(x)=\hat{c}_{\sigma}^{\dagger}(x)[s_{i}/2]_{\sigma,\sigma^{\prime}}\hat{c}_{\sigma^{\prime}}(x)$ with a Pauli matrix $s_{i}$, then the MF shows a nonzero magnetic response through the Zeeman term of electrons.

In the Nambu space with $\hat{\Psi}^{\dagger}(x)=(\hat{c}^{\dagger}_{\sigma}(x),\hat{c}_{\sigma}(x))$, $\hat{O}(x)$ is recast into $\hat{O}(x)=(1/2)\hat{\Psi}^{\dagger}(x){\cal O}\hat{\Psi}(x),$ with ${\cal O}={\rm diag}(O,-O^{T})={\rm diag}(O,-O^{*})$, where we have used the Hermiticity of $O$. Next, by expanding the mode of the quantum field $\hat{\Psi}(x)=\sum_{a}\hat{\gamma}^{(a)}|u^{(a)}_{0}\rangle+(\text{nonzero modes}),$ we obtain the coupling between $\hat{O}(x)$ and the MFs $\hat{\gamma}^{(a)}$ in the low-energy limit,

[TABLE]

where $\rho^{(ab)}\equiv i\left(|u_{0}^{(b)}\rangle\langle u_{0}^{(a)}|-|u_{0}^{(a)}\rangle\langle u_{0}^{(b)}|\right)$. In this formalism, crystalline symmetry is properly considered by the irreducible decomposition of ${\cal O}$ as ${\cal O}=\sum_{\Gamma}{\cal O}_{\Gamma},$ where ${\cal O}_{\Gamma}$ is an IR of the point group on the surface. As shown below, $|u_{0}^{(a)}\rangle$ obeys several symmetry constraints, so only a few representations of ${\cal O}_{\Gamma}$ can provide nonzero contributions in Eq. (1).

We now discuss the symmetry constraints. First, to have a nonzero $w_{\rm M1D}$, we need a bulk superconducting gap at the high-symmetry line or plane on which $w_{\rm M1D}$ is defined. This requirement restricts the possible pairing symmetry of Cooper pairs. The pairing symmetry must also maintain TR symmetry because we consider 3D TR-invariant TSCs. Moreover, if the bulk system has inversion symmetry, the pairing symmetry must be odd under inversion Xiong et al. (2017); sup . Second, the zero modes $|u_{0}^{(a)}\rangle$ should be a representation of the point group that is compatible with both the surface and the pairing symmetries because the BdG Hamiltonian respects these symmetries. Note that nonconventional Cooper pairs spontaneously break part of the crystalline symmetry, so the zero modes respect only the unbroken part. Third, a much stronger symmetry constraint is obtained from the index theorem of $w_{\rm M1D}$ Sato et al. (2011). The index theorem says that zero modes $|u^{(a)}_{0}\rangle$ should be eigenstates of $\Gamma_{\rm M}$; for example,

[TABLE]

Here all stable zero modes should have the same eigenvalue $\Gamma_{\rm M}$, otherwise zero modes with opposite eigenvalues are easily gapped in pairs, even by a symmetry-preserving perturbation. In fact, this important property can be rigorously proven for generic lattice systems Xiong et al. (2017). Finally, for the zero modes to exist, any surface-preserving point-group operation for the BdG Hamiltonian should not anticommute with $\Gamma_{\rm M}$. The last claim is proven by contradiction. If such a point group operation exists, one can generate another zero mode whose eigenvalue is the opposite of $\Gamma_{\rm M}$ by operating on a zero mode with the point group. This contradicts the above property of the zero modes, so the claim holds.

Note that the last constraint also restricts any possible pairing symmetry of Cooper pairs. In a nonconventional superconductor, depending on the pairing symmetry, crystalline symmetry can be realized projectively as a combination with a $U(1)$ gauge rotation, which changes their commutation to that of the chiral operator $\Gamma_{\rm M}$. For instance, if the gap function (Cooper pair) is odd under a mirror reflection, then the mirror reflection of the BdG Hamiltonian is the original reflection combined with a $U(1)$ $\pi$ rotation, so it anticommutes with the PH operator. Therefore, its commutation with $\Gamma_{\rm M}$ changes. As discussed above, stable zero modes only exist when any surface-preserving point-group operation does not anticommute with $\Gamma_{\rm M}$, which restricts any possible pairing symmetry between Cooper pairs sup .

Based on these arguments, we determine the possible pairing symmetry of Cooper pairs and IRs of $\rho^{(ab)}$ sup . We also find that only the same IRs of ${\cal O}_{\Gamma}$ give nonzero contributions in Eq. (1). Table 1 summarizes the IRs of ${\cal O}_{\Gamma}$ and pairing symmetry that satisfy the above constraints. Remarkably, the results show that the IRs of $\Delta_{\Gamma}$ and those of $\mathcal{O}_{\Gamma}$ coincide with each other up to leading order. This notable property allows us to determine the pairing symmetry through the magnetic response of MFs. The results also show that a magnetic-octupole response is possible when the surface has $C_{\rm 3v}$ or $C_{\rm 6v}$ symmetry. As shown below, the octupole response only appears for MFs in high-spin TSCs of spin-3/2 electrons. The order of magnetic multipoles reflects a difference between TSCs with spin $1/2$ and $3/2$.

Majorana octupole in spin-3/2 superconductors.

The results presented in Table 1 indicate that helical MFs on a surface-preserving $C_{3v}$ or $C_{6v}$ host the magnetic octupole. This unique behavior is intrinsic to high-spin TSCs of spin-3/2 electrons for the following reasons.

First, the base of $O_{\Gamma}$ for the magnetic octupole vanishes if the $J_{i}$ are given by the Pauli matrices of spin-1/2 electrons. In fact, we have $J_{x}^{3}-J_{x}J_{y}J_{y}-J_{y}J_{x}J_{y}-J_{y}J_{y}J_{x}=J_{y}^{3}-J_{y}J_{x}J_{x}-J_{x}J_{y}J_{x}-J_{x}J_{x}J_{y}=0$ for $J_{i}=\sigma_{i}/2$. Furthermore, if the pairing symmetry is $A_{1}$ ($B_{i=1,2}$) for $C_{\rm 3v}$ ($C_{\rm 6v}$), which is required for the octupole response, the spin-1/2 superconductor hosts a superconducting node at a high-symmetry line, so it cannot support well-defined helical MFs with magnetic octupoles because $C_{3}$ symmetry for spin-1/2 electrons is enhanced to $C_{\infty}$ on the axis of rotation in the Brillouin zone  sup .

Contrastingly, for spin-3/2 superconductors, helical MFs exhibit an octupole response. To illustrate this, we calculate the magnetic response of MFs in half-Heusler compounds. In these compounds Goll et al. (2008); Butch et al. (2011); Tafti et al. (2013); Xu et al. (2014); Bay et al. (2012); Kim et al. (2018), a strong spin-orbit interaction (SOI) and high crystal symmetry provide a fourfold degenerate band at the $\Gamma$ point, which is well described by spin-$3/2$ fermions Brydon et al. (2016). Additionally, recent experiments have suggested the existence of parity-mixed superconductivity with line nodes Bay et al. (2012); Kim et al. (2018). We show here that the parity-mixed superconductor exhibits a magnetic-octupole response. Consider the low-energy model with $T_{d}$ symmetry Brydon et al. (2016):

[TABLE]

where $i=x,y,z$ and $i+1=y$ if $i=x$, etc., and $J_{i}$ are the $4\times 4$ spin matrices of spin-$3/2$ fermions. Because inversion symmetry is absent, the Hamiltonian includes the antisymmetric SOI, which is proportional to $\delta$ and causes spin splitting at the Fermis surface Kim et al. (2018). In their superconducting states, Cooper pairs form between spin-$3/2$ electrons, which allows quintet and septet parings in addition to the conventional singlet and triplet pairings Ho and Yip (1999); Yang et al. (2016); Brydon et al. (2016). Furthermore, the antisymmetric SOI generally mixes the parity of the gap function, so the even- and odd-parity components coexist in the gap function Gor’kov and Rashba (2001); Frigeri et al. (2004); Fujimoto (2007a, b); Bauer et al. (2012) and the odd-parity component is aligned with the antisymmetric SOI Frigeri et al. (2004), providing the spin-septet pairing Brydon et al. (2016); Kim et al. (2018). Based on this insight, the gap function must include the spin-septet component, $\Delta(\bm{k})=\Delta/{\sqrt{1+\eta^{2}}}[\eta 1_{4}+\sum_{i}k_{i}(J_{i+1}J_{i}J_{i+1}-J_{i+2}J_{i}J_{i+2})](e^{-iJ_{y}\pi})$, in addition to an $s$-wave singlet state, even when we choose the conventional $A_{1}$ state of $T_{d}$, where $\eta$ parametrizes the mixing between the $s$-wave and spin-septet components and $1_{n}$ is the $n\times n$ identity matrix. Here, the PH, TR, and $T_{d}$ symmetry operations hosted by the BdG Hamiltonian are $\mathcal{C}=\tau_{x}K$, $\mathcal{T}=e^{-iJ_{y}\pi}K$, and $\operatorname{diag}[e^{-i\frac{i2\pi}{q}\bm{J}\cdot\bm{n}},e^{i\frac{i2\pi}{q}\bm{J}\cdot\bm{n}}]$, respectively.

The superconducting state hosts six line nodes encircling the $k_{x}$, $k_{y}$, and $k_{z}$ axis Brydon et al. (2016), in analogy with other parity-mixed superconductors Tanaka et al. (2010); Yada et al. (2011); Sato et al. (2011); Brydon et al. (2011); Schnyder et al. (2012); Matsuura et al. (2013); Schnyder and Brydon (2015). Here, we focus on the (111) surface because the magnetic-octupole response requires $C_{\rm 3v}$ symmetry. To verify the existence of helical MFs, we numerically diagonalize the BdG Hamiltonian with the surface normal to the $[111]$ direction and find a helical MF with three flat dispersion curves (see Fig. S2 in the Supplemental Material sup ), as schematically depicted in Fig. 2(c). Each flat dispersion curve lies on the mirror planes with mirror-reflection symmetries, $\sigma$, $C_{3}^{\dagger}\sigma C_{3}$, and $(C_{3}^{\dagger})^{2}\sigma(C_{3})^{2}$, where $\sigma$ is mirror-reflection with respect to the $(1\bar{1}0)$ plane and $C_{3}$ is a threefold rotation around the $[111]$ direction. Combining these mirror reflections with PH and TR operations, we obtain three $\Gamma_{\rm M}$ and the associated $w_{\rm M1D}$, which protects zero modes on each flat dispersion curve. In particular, the three flat dispersion curves meet at a $C_{\rm 3v}$ symmetry point.

Based on the constraint (2), the zero modes can be simultaneous eigenstates of $\Gamma_{\rm M}$ and of $C_{3}$. In this case, we have $\Gamma_{\rm M}|u_{0}^{(a)}\rangle=|u_{0}^{(a)}\rangle$ and $C_{3}|u_{0}^{(a)}\rangle=-|u_{0}^{(a)}\rangle$ with $a=1,2$ being the label for a Kramers pair sup , which lead to $\Gamma_{\rm M}\rho^{(12)}\Gamma_{\rm M}^{-1}=C_{3}\rho^{(12)}C_{3}^{-1}=\rho^{(12)}$. Thus, $\mathcal{O}_{\Gamma}$ needs to be the trivial representation $A_{1}$ in $C_{\rm 3v}$, as shown in Table 1. To demonstrate magnetic response, we add a Zeeman magnetic term $\mu\bm{B}\cdot\bm{J}$ in Eq. (3), which leads to an anisotropic response with $C_{3}$ symmetry in Fig. 2(d). The Zeeman magnetic term contributes to the energy gap of the MFs on the order of $3\sqrt{2}\mu^{3}B^{3}/32E_{F}^{2}$, where $E_{F}$ is the Fermi energy sup , implying a magnetic-octupole response.

Another high-spin superconductor of spin-3/2 electrons was recently proposed for antiperovskite materials with $O_{h}$ group Oudah et al. (2016); Kawakami et al. (2018). We obtain a similar magnetic-octupole response of MFs on the $(111)$ surface when its pairing symmetry is $A_{2u}$ of $O_{h}$.

For comparison, we also examine magnetic response of helical MFs in the doped superconducting TI, AxBi2Se3 ($A$=Cu, Sr, Nb), which becomes a TSC when an odd-parity Cooper pair is realized Sato (2009, 2010); Fu and Berg (2010); Fu (2014); Matano et al. (2016); Yonezawa et al. (2016). Since Bi2Se3 has $D_{3d}$ symmetry, the surface normal to the $c$ axis (i.e., (111) surface) hosts $C_{\rm 3v}$ symmetry like the half-Heusler case. However, the doped TI merely exhibits the magnetic-dipole response of MFs to leading order, or it cannot host a well-define helical MFs on the (111) surface sup ; Sumita and Yanase (2018), since it is a conventional spin-1/2 TSC.

Conclusions.

In this paper, we develop a theory of Majorana multipoles for 3D TR-invariant TSCs, which provide novel experimental means to identify bulk pairing symmetry and high-spin superconductivity. The Majorana multipoles may be observed through spin-sensitive measurements such as spatially resolved NMR measurements Chung and Zhang (2009) or the surface tunneling spectroscopy under magnetic fields Fogelström et al. (1997); Tanaka and Kashiwaya (1995); Tanaka et al. (2002); Tanuma et al. (2002); Tanaka et al. (2009); Tamura et al. (2017).

This work was supported by the Grants-in-Aid for Scientific Research on Innovative Areas “Topological Material Science” (Grant Nos. JP15H05855, JP15H05851, JP15H05853, JP15K21717, and JP18H04224) from JSPS of Japan, and by JSPS KAKENHI Grant Nos. JP17H02922 and JP18H01176. S.K. was supported by the CREST project (JPMJCR16F2) from Japan Science and Technology Agency (JST), and the Building of Consortia for the Development of Human Resources in Science and Technology.