Three-dimensional Majorana fermions in chiral superconductors

Majorana nodes on rotation axis. The type of chiral pairing that gives rise to the Majorana nodal quasiparticles—the focus of this work—corresponds to J = 2j mod n. This implies odd J, and we list all these cases in Table 1. Except for two cases (n,j)=(2,12) and (6,32) (to be addressed separately later), we have 2j ≠ −2j mod n. Under this condition, the spin ↑ states that carry angular momentum j are allowed to (and generally will) pair up and form Cooper pairs that carry total angular momentum 2j, whereas the spin ↓ states remain gapless at ±K because of the angular momentum mismatch. The resulting nodal quasiparticles are therefore spin-nondegenerate Majorana fermions.

The low-energy Hamiltonian for these quasiparticles is given byH=∑qξq(cq1†cq1+c−q2†c−q2)+(Δqcq1†c−q2†+H.c.)(5)where we have defined c_q1,2 ≡ c_±K+q↓ and Δ_q ≡ Δ_2,q. In addition, ξ_q ≡ ε_K+q − μ, where ε_k is the single-particle energy-momentum relation and μ is the chemical potential. For small q, we have ξ_q = v_Fq_z, where v_F = k_F/m is the Fermi velocity in the ẑ direction.

It is instructive to write H in Nambu space by introducing the four-component fermion operator Ψq†Ψq†=(cq1†,cq2†,c−q1,c−q2)(6)so that H can be expressed asH=12∑qΨq†H(q)Ψq(7)with the 4 × 4 matrix H(q) taking the general formH(q)=(ξq00Δq0ξ−q−Δ−q00−Δ−q*−ξ−q0Δq*00−ξq)(8)

The four-component quantum field Ψ satisfies the same reality condition as Majorana fermions in high-energy physics, which reads as Ψq†=(τxΨ−q)T in momentum space or, equivalently, Ψr†=(τxΨr)T in real space, where the Pauli matrix τ_x acts on Nambu space and Ψ^T is the transpose of Ψ. This reality condition demonstrates that the low-energy quasiparticles can be regarded as Majorana fermions in three dimensions.

At small q, the pairing term Δ_q in Eq. 8 can be expanded in powers of q₊ or q₋. The exponent is determined by the mismatch between the angular momentum of the Cooper pair J = 2j and that of the spin ↓ pairing operator Γ₂ at q = 0, which is equal to − 2j. Hence, one finds thatΔq∝[qx+isgn(l)qy]|l| with l=4j mod n(9)

The smallest allowed integer |l| gives the form of Δ_q to the leading order. For any given (n,j) and with J = 2j fixed, the corresponding l is listed in Table 1 (additional details can be found in the Supplementary Materials).

From Table 1, we find three types of pairing terms Δ_q with different l’s, which give rise to two types of Majorana fermions with different energy-momentum relations. First, for (n,j)=(3,12), one has l = 1 mod n; hence, |Δ_q| ∝ q_⊥, where we defined q⊥=(qx2+qy2)1/2. This implies that the quasiparticles near the nodes ±K disperse linearly with q in all directions, as governed by the following effective Hamiltonian to first order in qH(q)=vFqzσz+vΔσx(qyτx−qxτy)(10)where σ_z = ±1 denotes the two nodes ±K, and v_Δ is defined via |Δ_q| = v_Δq_⊥ + O(q²). Except for the velocity anisotropy, H(q) is identical to the relativistic Hamiltonian for Majorana fermions in particle physics.

Second, we find several cases for which l = ±2 mod n. According to Eq. 9, this implies that the gapless quasiparticles disperse quadratically in q_x, q_y and linearly in q_z (see Table 1), as governed by the following effective Hamiltonian H(q) to second order in qH(q)=vFqzσz+12mΔσy[(qx2−qy2)τy+2qxqyτx](11)where m_Δ is an effective mass defined by |Δq|=q⊥2/(2mΔ).

In the case of fourfold rotational symmetry, that is, n = 4, both q+2 and q−2 terms, with angular momenta of l = 2 and −2, respectively, are allowed in Δ_q. As a result, the Hamiltonian H(q) takes a more involved form, which is discussed in the Supplementary Materials.

The above cases of chiral pairing with |l| = 1 and 2 both give rise to gapless Majorana quasiparticles at ±K. According to Table 1, there are two remaining cases that both have l = 0 mod n: (n,j)=(2,12) and (6,32). The property l = 0 mod n implies that spin ↓ states at ±K are allowed to pair and form a Cooper pair Γq=02=cK↓†c−K↓† carrying the same angular momentum 2j = −2j mod n as the spin ↑ Cooper pair Γq=01=cK↑†c−K↑†. As a result, both Cooper pairs coexist in the superconducting state and generate a full gap at ±K.

Spin-orbit coupling and nonunitary pairing.. It is clear from our derivation of the Majorana nodal quasiparticles that these can only be present in chiral superconductors with nonunitary gap structures, that is, with a spin-nondegenerate quasiparticle spectrum, such that spin ↑ and ↓ states have different gaps (9). So far, nonunitary superconductors have received much less attention than their unitary counterparts. Although nonunitary pairing states have been discussed in relation to UPt₃ (10–13), to Sr₂RuO₄ (2, 14, 15), and recently to LaNiGa₂ (16, 17), the only established example of nonunitary pairing is superfluid ³He in high magnetic fields (18), known as the A₁ phase. However, from a symmetry point of view, nonunitary pairing is generic and more natural (in a theoretical sense) in chiral superconductors with strong spin-orbit coupling. This is a consequence of the lack of spin-rotation symmetry, replaced by the symmetry of combined spin and momentum rotation under the crystal point group. In these cases, there are typically more than one pairing components with different spin S or orbital angular momentum L but the same total angular momentum J = L + S. As a result, the full bulk gap function Δ_k of the chiral superconductor, defined with HΔ=∑k(iΔksy)αβckα†c−kβ†+H.c., is generally a mixture of these pairing components, which all belong to the same irreducible representation of the point group. Specifically, Δ_k can be written asΔk=Δ0∑tλtFtJ(k)(12)where FtJ(k) are pairing components (that is, crystal harmonics) with total angular momentum J but different L and S, and λ_t are dimensionless coefficients describing the admixture of these different components. For each pairing channel J, the set of allowed pairing components FtJ(k) depends on both the point group symmetry of the crystal and the spin angular momentum j. In Table 2, we present a full list of gap function components FtJ(k) for trigonal (C₃), tetragonal (C₄), and hexagonal (C₆) superconductors and for general spin angular momentum j. Table 2 thus generalizes standard gap function classifications for j=12 Bloch electrons (19) and applies to energy bands of, for instance, j=32 electrons, such as reported in the half-Heusler superconductors YPtBi and LuPtBi (20, 21). In addition, the heavy fermion superconductor UPt₃ has recently been proposed to have j=52 bands (22).

As an example of Δ_k in Eq. 12, consider the following gap function of a J = 1 superconductor of j=12 electrons, consisting of two pairing components with (L, S) = (1, 0) and (L, S) = (0, 1), respectivelyΔk=Δ0kF(λak+sz+λbkzs+)(13)where we defined k_± = k_x ± ik_y and s_± = s_x ± is_y. It is straightforward to verify that the pairing is nonunitary as a result of the second (L, S) = (0, 1) term. The first term of Eq. 13 corresponds to the gap function of the A phase of superfluid ³He (3). The spin-degenerate quasiparticle spectrum of the ³He-A phase hosts Weyl fermions at low energies (23, 24), which can be viewed as a complex quantum field made up of two degenerate Majorana fields. The admixture of the (L, S) = (0, 1) component, which is enabled by spin-orbit coupling, gaps out the spin ↑ states at ±K and gives rise to gapless spin ↓ excitations governed by Hamiltonian (Eq. 11) (Supplementary Materials).

This example illustrates a general feature of spin-orbit–coupled chiral superconductors: The lack of spin-rotation symmetry naturally leads to nonunitary pairing, which serves as a spin-selective gapping mechanism and creates spin-nondegenerate nodal excitations, obeying the Majorana reality condition.

As we pointed out earlier, the Majorana condition makes the quantum field Ψ_q a four-component real field. One may be tempted to rewrite Eq. 5 in terms of a two-component complex quantum field, fq†≡(cq1†,c−q2): H=fq†(ξqσz+Δqσ++Δq*σ−)fq, which is invariant under the U(1) transformation, f^† → f^†e^iφ. However, this U(1) symmetry is only present in the presence of translational symmetry and broken by impurity-induced potential scattering between the nodes. To see this, consider the spin-conserving internode scattering termHs=∑qM(cq1†cq2+cq2†cq1)(14)where M is the scattering amplitude at the momentum 2k_F. In terms of the complex field fq†, H_s involves fq†f−q† terms and thus removes the emergent U(1) symmetry in the clean limit. In terms of the four-component Majorana field Ψ, H_s is given byHs=M∑qΨq†σxτzΨq(15)

Including Eq. 14 in the Majorana Hamiltonian (Eq. 8), the energy E of the Majorana nodes with linear dispersion, Eq. 10, is given by E2=(vFqz)2+vΔ2(qx2+qy2)+M2. Thus, the effect of internode scattering is to generate a mass term without U(1) symmetry, that is, a Majorana mass term. As a result, the fundamental quantum field describing the gapless quasiparticles is a four-component real field, that is, a field obeying the Majorana condition.

We note that spin-nondegenerate point nodes also occur when the Fermi surface in the normal state is already spin-split due to magnetism—as theoretically shown in magnetic topological insulator-superconductor heterostructures (25) and ferromagnetic p-wave superconductors (26) and in the mixing of chiral d- and p-waves (27)—or spin-orbit coupling in noncentrosymmetric superconductors (28–37). This is different from our case, where the spin-selective point nodes occur via nonunitary pairing in a spin-degenerate normal state. Also, the present case of nonunitary pairing does not assume any special feature in the band structure and should be distinguished from theoretical surveys of possible pairing states in Weyl and Dirac semimetals (38–45).

Off-axis point nodes. A key result of our symmetry analysis presented in the first part of this section is the presence of nodal Majorana excitations at the rotationally invariant Fermi surface momentum K on the principal rotation axis for nonzero l mod n (see Eq. 9). Rotational symmetry further dictates the form of the energy-momentum dispersion of these on-axis Majorana quasiparticles. We find that spin-orbit–coupled odd-parity chiral superconductors can have additional point nodes located at generic Fermi surface momenta away from the north and south poles, that is, off-axis Majorana nodes. We will first illustrate the presence of these point nodes using examples and then explain their topological origin.

As a first example, let us consider a chiral superconductor with C₃ symmetry and angular momentum J = 1. We now show that its gap structure can exhibit nodes at Fermi surface momenta other than ±K. The full pairing potential Δ_k of Eq. 12 is given to leading p-wave order by (see the Supplementary Materials for details)Δk=Δ0kF(λak+sz+λbkzs++λcik−s−)(16)where λ_a,b,c are three real admixture coefficients. In addition to the on-axis nodes at k = ±K, the quasiparticle spectrum corresponding to Δ_k of Eq. 16 exhibits six nodes located at off-axis Fermi surface momenta. Writing the Fermi momenta as kF=kF(cosφkF sin θkF, sin φkFsinθkF,cosθkF), the location of these nodes can be expressed as the relations cos θkF=±λa2/λa4+16λb2λc2 and sin 3φkF=∓1 (here, we have assumed λ_bλ_c > 0. If λ_bλ_c < 0, one should substitute φkF→−φkF). There are three nodes on the northern Fermi surface hemisphere, which are related by threefold rotation C₃, and each of these nodes has a partner on the southern hemisphere related by inversion P, as shown in Fig. 1C.

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As a second example, consider a J = 1 superconductor with a twofold rotational symmetry C₂. The pairing potential Δ_k is composed of all terms that are odd under C₂, and to p-wave order in spherical harmonics is given byΔk=Δ0kF[(λak++λbk−)sz+kz(λcs++λds−)](17)

At the Fermi surface momentum K, one has Δ_K = Δ₀(λ_cs₊ + λ_ds₋), which implies a pairing gap for both c_±K+q↑ and c_±K+q↓, in agreement with the result shown in Table 1. Although no nodes exist on the twofold z axis, it is straightforward to verify that for general nonzero admixture coefficients, two pairs of nodes are located on the Fermi surface, each pair related by twofold rotation with the partners of a pair related by the inversion P. For instance, taking λ_b = 0 (λ_a = 0), the nodes are located on the intersection of the Fermi surface with the yz (xz) plane, as shown in Fig. 1D.

In both these examples, all point nodes located at generic rotation noninvariant Fermi surface momenta are spin-nondegenerate, and the gapless quasiparticles are therefore Majorana fermions. We call these nodes off-axis Majorana nodes. The presence of the off-axis Majorana nodes in these examples motivates the question of whether these are accidental or whether the off-axis nodes are related to the Majorana nodes on the rotation axis at a deep level.

We now address this question by focusing on the topological nature of the Majorana point nodes. We show that the classification of different types of low-energy Majorana quasiparticles in terms of location on the Fermi surface (that is, on- or off-axis) and energy-momentum dispersion (that is, linear or quadratic) is linked to a topological property of point nodes in momentum space.

In band theory, crossing points of (quasiparticle) energy bands are endowed with an integer topological quantum number given by the Chern number C defined asC=14π∫ΩF⋅dS(18)

Here, F = ∇_k × A(k) is the Berry curvature, that is, the field strength of the momentum space gauge Berry connection A(k), which is integrated over a surface Ω, enclosing the point node. Hence, C quantifies the Berry curvature monopole strength of the point node. Because monopoles cannot be removed unless they annihilate with a monopole of equal but opposite strength, point nodes, which carry a nonzero monopole charge, are topologically protected.

It is straightforward to show that the Majorana nodes with linear dispersion described in Eq. 10 have a monopole charge C = ∓1 at ±K, similar to Weyl fermions in topological semimetals (46, 47), and that the Majorana nodes in Eq. 11, which disperse quadratically tangential to the Fermi surface, have a monopole charge C = ±2, similar to double-Weyl fermions (48). Therefore, the former may be called single Majorana nodes and the latter double Majorana nodes. These single and double on-axis Majorana nodes, corresponding to C₃ and C_4,6 symmetry, respectively, are schematically shown in Fig. 1, with the monopole charge C explicitly indicated.

From the perspective of topology, the presence of the off-axis Majorana nodes in superconductors with C₃ or C₂ can be understood from monopole charge conservation. For instance, the C₃-symmetric superconductor with the gap function 16 can be viewed as a descendent of the C₆-symmetric superconductor with the gap function 13, obtained from lowering the symmetry. As a result of lower symmetry, additional gap functions can mix in with the J = 1 channel, and according to our symmetry analysis, this transforms the C = 2 double Majorana node at K of a C₆ superconductor into a C = −1 Majorana node at K of a C₃ superconductor. Because the total monopole charge must be conserved and the gap structure must preserve the C₃ symmetry, three additional point nodes with a monopole charge of C = +1 must exist. This effective “splitting” of a C = 2 Majorana node into four single nodes agrees with the explicit analysis in Eq. 16 and is schematically shown in Fig. 1. An analogous argument explains the existence of two C = +1 off-axis Majorana nodes in the case of the C₂-symmetric superconductor, which has a full pairing gap at K. The nodal structure of the C₂ superconductor is depicted in Fig. 1. It is therefore natural to think of the off-axis Majorana nodes as originating from the on-axis double Majorana nodes in a J = 1 superconductor, obtained by lowering C₆ or full rotational symmetry to trigonal (C₃) and orthorhombic (C₂) symmetry.

These topological arguments demonstrate that the chiral superconductors discussed here are topological nodal superconductors. Topological nodal superconductors are superconductors with topologically protected gapless quasiparticle excitations in the bulk, in analogy with the protection of Weyl fermions in Weyl semimetals (see also the “Surface Andreev bound states: Majorana arcs” section) (46, 47). In particular, the gapless Majorana quasiparticles are topologically protected by monopole charge conservation.

Signatures of nonunitary chiral superconductors. The nonunitary gap structure of chiral spin-orbit–coupled superconductors gives rise to a number of distinctive experimental signatures. The most prominent characteristic of nonunitary pairing, the nondegenerate quasiparticle excitation spectrum, leads to a spin-dependent density of states: The densities of states N_↑,↓(E) for (pseudo)spin ↑,↓ excitations are unequal [that is, N_↑(E) ≠ N_↓(E)], and in particular, the low-energy branch of the spectrum consists of fully spin-polarized states near the point nodes. As a consequence of the nondegenerate spectrum, the total density of states ∑_αN_α(E) can exhibit two distinct peaks, rather than a single peak at Δ₀, which is characteristic of conventional s-wave superconductors. This can lead to a two-gap–like feature in the specific heat, as is demonstrated more explicitly in simple examples in the Supplementary Materials. Therefore, experimental signatures that are commonly attributed to multiband superconductivity may actually originate from nonunitary pairing in a single band.

A well-known and discriminating property of nodal superconductors is the characteristic temperature dependence of a diverse set of dynamic and thermodynamic quantities, such as the electronic part of the specific heat, the London penetration depth, and the NMR spin relaxation rate (T1−1) (49). The low-energy branch of the quasiparticle spectrum of chiral nonunitary superconductors consists of C = ±1 or C = ±2 point nodes, which implies that the density of states N(E) in the regime E ≪ Δ₀ takes the form N(E)/N₀ ~ (E/Δ₀)ⁿ, where N₀ is the normal-state density of states. The exponent n depends on the nature of the point node: It equals n = 2 for C = ±1 and n = 1 for C = ±2. The form of the density of states at the nodes is responsible for the typical power law temperature dependence of the specific heat, the penetration depth, and the spin relaxation, which probe the density of quasiparticle states, at temperatures T ≪ T_c ~ Δ₀.

In the next section, we consider the NMR spin relaxation rate in more detail. In the case of chiral superconductors with low-energy Majorana quasiparticles, not only should the spin relaxation rate T1−1 exhibit power law temperature dependence at low temperatures, but the temperature dependence of T1−1 is also expected to crucially depend on the direction of the nuclear spin polarization. This follows from the fact that the only quasiparticle excitations available at low energy are spin ↓ states, which is intimately related to the nonunitary nature of the superconducting state. Therefore, we derive below the theory of NMR spin relaxation in nonunitary superconductors, which serves as a powerful tool to identify Majorana nodal quasiparticles.

NMR: Spin relaxation rate.. The measurement of the NMR spin-lattice relaxation rate 1/T₁ at low temperature is a well-established experimental technique to probe the gap structure of superconductors. The temperature dependence of 1/T₁ at temperatures T ≪ T_c ~ Δ₀ can be used as a measure of the density of low-energy quasiparticle states and allows to distinguish fully gapped, point nodal gap, and line nodal gap structures (9).

The coupling of quasiparticle states to the nuclear spin originates from the hyperfine interaction between the nuclear spin and itinerant electrons. The hyperfine coupling Hamiltonian H_hf is given byHhf=γNAhf∑kk′gij(k,k′)S^ickα†sαβjck′β(19)where Ŝⁱ are the components of the nuclear spin operator and sⁱ are Pauli matrices representing the electron pseudospin (summation of repeated spin indices α, β is implied). Furthermore, γ_N is the gyromagnetic ratio of the nuclear spin, A_hf is the hyperfine coupling constant, and g_ij(k, k′) is a momentum-dependent tensor describing the coupling of the nuclear spin to the pseudospin of electrons in spin-orbit–coupled materials. The form of this tensor can be complicated and material-specific. However, because only quasiparticles around the nodes contribute to the spin relaxation at low temperature, it suffices to consider g_ij(k, k′) at the nodes, a key simplification that enables us to find universal features of the spin relaxation rate below.

Our aim is to derive 1/T₁ for chiral nonunitary superconductors hosting 3D Majorana fermions. Because the Majorana nodes are nondegenerate with definite pseudospin, the low-energy quasiparticles couple anisotropically to the nuclear spin. To demonstrate this, consider the case of gapless Majorana particles pinned to ±K. Projecting the Hamiltonian (Eq. 19) into the space of Bloch states near ±K (P projects onto the low-energy Hilbert space), one findsPHhfP=γNAhf∑qq′S^z[gzz1cq1†cq′1+gzz2cq2†cq′2+gzz3cq1†cq′2+gzz3*cq2†cq′1](20)where g¹ = g(K, K), g² = g(−K, −K), and g³ = g(K, −K) are the g tensors evaluated at the nodes. Only the z component of the nuclear spin enters the low-energy Hamiltonian due to the rotational symmetry of the crystal around the z axis. As a result, we expect that the nuclear spin relaxation time 1/T₁ is highly direction-dependent and, to leading-order approximation, diverges when the nuclear spin is initially polarized along the z direction.

As we show below, the strongly anisotropic spin relaxation rate is a generic consequence of spin-selective nodes, whereas the divergence for nuclear spin polarization along the nodal direction is an artifact of the leading-order Hamiltonian (Eq. 20). We now go beyond this leading-order approximation and expand the full form factor g_ij(k, k′) into crystal spherical harmonics. Keeping the lowest-order s-wave component, we have g_ij(k, k′) ~ δ_ij, reducing the hyperfine coupling toHhf=γNAhf∑kk′S^ickα†sαβick′β(21)

Using Eq. 21, we proceed to explicitly calculate the NMR relaxation rate 1/T₁ for a nonunitary superconductor. For simplicity, we consider a nuclear spin of S = 1/2. The spin relaxation rate 1/T₁ is expressed through the transverse spin susceptibility χ⁻⁺(k, ω) (that is, transverse to nuclear spin direction) and reads as (50, 51)1T1=γN2Ahf2Tlimω→0∑kImχ−+(kω)ω(22)

When evaluating 1/T₁, it is important to distinguish unitary and nonunitary pairing states. In the case of former, it has been shown that the spin relaxation rate does not depend on the direction of polarization of the nuclear spin (reviewed in the Supplementary Materials). The case of nonunitary pairing requires separate and more careful treatment. For concreteness, we first consider the C₆-symmetric J = 1 superconductor, which only has double nodes at ±K. Other nodal nonunitary superconductors are discussed toward the end of this section.

The gap structure of the J = 1 superconductor with C₆ symmetry is given by Eq. 13. The full BdG Hamiltonian is diagonalized in terms of Bogoliubov quasiparticle operators, which we define as a_kα. Writing the hyperfine interaction Hamiltonian (Eq. 21) in terms of the Bogoliubov quasiparticle operators, the spin susceptibility can be readily evaluated. Because only low-energy excitations contribute to the relaxation rate at low temperatures (that is, ~T ≪ Δ₀ min{λ_a,λ_b}), we can restrict to Bogoliubov quasiparticle states with small momenta q relative to ±K and only keep the low-energy gapless states a_q corresponding to energies Eq=[ξq2+(q⊥2/2mΔ)2]1/2 (see Eq. 11), where q⊥=(qx2+qy2)1/2. Then, to the leading order in small momentum q, the Hamiltonian (Eq. 21) reads asHhf≅γNAhf∑qq′pp′aqp†aq′p′[S^zFpp′(q,q′)+p′S^−G(q,q′)+pS^+G*(q′,q)](23)with the form factors Fpp′(q,q′) and G(q, q′) defined asFpp′(q,q′)=12q⊥q⊥′(q+q−′P−−q−q+′pp′P+)G(q,q′)=18mΔΔ∼0(q⊥q+′q⊥′P+−q+q⊥′q⊥P−)(24)and the momentum-dependent factors P±=(1±ξq/Eq)(1±ξq′/Eq′). Here, p, p′ = +1 (−1) for the north (south) node.

In Eq. 23, the anomalous terms a_qa_q′ and aq†aq′† have been omitted because they do not contribute to 1/T₁ due to energy conservation. The effective mass m_Δ of the Bloch states c_±K↓ and energy Δ∼0 associated with the Bloch states c_±K↑ are defined in terms of gap function parameters in the Supplementary Materials. Equation 23, which is the projection of Hamiltonian (Eq. 19) into the space of low-energy Bogoliubov quasiparticles, should be compared to the projection into the space of low-energy Bloch states given in Eq. 20. The former contains a coupling to Ŝ_± = Ŝ_x ± iŜ_y, originating from the nonzero support of the Bogoliubov quasiparticle states on the Bloch electron states c_±K+q↑. The support is vanishingly small near the nodes as a result of q±/(mΔΔ∼0)1/2∝q±/kF≪1, indicating a suppression of the spin relaxation rate for a nuclear spin initially polarized along the z direction. In the limit that this coupling is strictly absent, as in Eq. 20, the spin initially polarized along the z direction does not relax.

Using the projection of the hyperfine coupling into the space of low-energy Bogoliubov quasiparticles given by Eq. 23, it is straightforward to calculate the NMR relaxation rate given by Eq. 22 in the low-temperature limit T ≪ Δ₀ min{λ_a,λ_b} (Supplementary Materials). We find 1/T₁ to be given by1T1=D⊥T3S⊥2+DzT4Δ∼0Sz2(25)where S_⊥ and S_z are the projections of the nuclear spin polarization on the xy plane and z axis, respectively, and the coefficients D_⊥,z are given byD⊥=γN2Ahf2π96(mΔvF)2, Dz=γN2Ahf29ζ(3)4π2(mΔvF)2(26)

Equation 25 proves that there is a strong anisotropy of spin relaxation depending on the polarization of the nuclear spin. For a nuclear spin polarized perpendicular to the z axis, the spin relaxation behaves as 1/T₁ ~ T³, as expected for C = ±2 point nodes with quadratic dispersion and linear dependence of density of states on energy. Instead, for a nuclear spin polarized along the z axis, the spin relaxation rate is suppressed by a factor of T/Δ∼0, that is, the zeroth-order term in an expansion in T/Δ∼0 is absent.

The strong relaxation rate anisotropy can be intuitively understood from a simple physical picture. The hyperfine interaction leading to nuclear spin relaxation is given by Eq. 21, and consequently, nuclear spin relaxation occurs simultaneously with an electron spin flip. However, at low energies, only the c_±K+q↓ Bloch states are available, implying that a nuclear spin initially polarized along z is vanishingly improbable to relax. This physical picture is captured by Eq. 20. The qualitative difference of this result compared to the case of unitary pairing (9) can be understood in a similar way. In the case of the latter, the quasiparticle energy spectrum is doubly degenerate, implying that both spin species are present and, consequently, leading to the same temperature dependence of 1/T₁ for all nuclear spin polarizations.

A qualitatively similar anisotropic spin relaxation rate has been predicted for 2D Majorana fermions, which live on the surface of a 3D topological superfluid, that is, the ³He-B phase (52, 53). In this case, where the 2D surface Majorana modes are described by a two-component real quantum field, the anisotropy in the spin relaxation arises because one can only construct the Ising spin operator, which points perpendicular to the surface and takes the form of a Majorana mass term. This is different from our case, that is, with the 3D Majorana fermions, where the spin operator constructed from the low-energy quasiparticles does not correspond to a mass term.

On the basis of the result for the J = 1 superconductor with C₆ symmetry and quadratic point nodes at ±K (Eq. 25), we now comment on other chiral nonunitary superconductors. First, consider the J = 1 superconductor, with C₄ symmetry shown in Fig. 1 and discussed in more detail in the Supplementary Materials. Because the C₄-symmetric superconductor only has on-axis Majorana nodes with C = ±2, Eq. 25 remains valid, and a significant suppression of 1/T₁ for the nuclear spin polarized along the z axis is expected.

Next, consider the J = 1 superconductor with C₃ symmetry with gap function 16. In this case, the low-energy gap structure consists of eight single Majorana nodes (that is, linear dispersion), two of which are located at ±K, and six at off-axis Fermi surface momenta (see Fig. 1). This nodal structure complicates the explicit derivation of an analytical expression for 1/T₁, but the final result, however, can be inferred from the C₆ symmetric case. For a weak trigonal anisotropy and intermediate temperatures given by the condition Δ₀λ_c < T < Δ₀ min{λ_a, λ_b}, the trigonal λ_c term can be neglected, and one can expect the same behavior as in the hexagonal case with the relaxation rate given by Eq. 25.

However, at the lowest temperatures given by T ≪ Δ₀ min{λ_a, λ_b, λ_c}, the linear dispersion of the nodes comes into play: The density of states becomes a quadratic function of energy, leading to 1/T₁ ~ T⁵. Whereas at the on-axis nodes only quasiparticles with definite spin ↓ are available at low energies, the low-energy quasiparticles at the off-axis nodes are mixtures of spin ↑ and ↓. As a result, the temperature dependence of 1/T₁ will exhibit the same power law behavior, that is, ~T⁵ for all nuclear spin polarizations. However, the numerical prefactors will reflect a directional anisotropy, which may be large.

Superconductors with C₂ symmetry, due to the four off-axis linear nodes, are expected to show behavior similar to C₃ crystals, exhibiting strong anisotropy in the functional temperature dependence at intermediate temperatures and point node power law behavior at the lowest temperatures for all nuclear spin polarizations. Again, numerical prefactors will generically be different.