Topological surface states in nodal superconductors

We consider two different phenomenological models of superconductors/superfluids with point nodes. The first one is the Anderson-Brinkman-Morel (ABM) state [52–54], which is believed to occur in the A phase of superfluid ³He as well as in cold atomic and polar molecular gases [55]. The ABM state is a three-dimensional superfluid of spinless (or fully spin-polarized) fermions with (p_x ± ip_y)-wave pairing and is described by the Bogoliubov-de Gennes (BdG) Hamiltonian $\mathcal{H} = \frac{1}{2} \sum_{\bf k} \phi^{\dagger}_{\bf k} \mathcal{H} ( {\bf k} ) \phi_{\bf k}$ with

$$\begin{eqnarray} &&\mathcal{H} ( {\bf k} ) =h({\bf k}) \tau_z + \frac{\Delta_0}{k_{\rm F}} {\bf k} \cdot \left( \hat{\bf e}_1 \tau_x + \hat{\bf e}_2 \tau_y \right) , \end{eqnarray} \tag{ 1 }$$

where $\phi_{\bf k} = ( c_{\bf k} , c^{\dagger}_{\bf k} )^{{\rm T}}$ and $c^{\dagger}_{\bf k}(c_{\bf k})$ is the creation (annihilation) operator of spinless fermions with momentum k. Here, τ_i denotes the three Pauli matrices which act in particle–hole space, h (k) is the normal-state dispersion, k_F is the Fermi wave vector, and Δ₀ denotes the amplitude of the superconducting order parameter. The direction of the orbital momentum of the Cooper pair is defined by $\hat{\bf I} = \hat{\bf e}_1 \times \hat{\bf e}_2$. In the following we assume a spherical Fermi surface centered at the Γ point and a momentum independent vector $\hat{\bf l}$ pointing along the k_z axis in the Brillouin zone. Thus, the eigenvalues of Hamiltonian (1) are $\pm \lambda ({\bf k} ) = \pm \sqrt{[ h ({\bf k}) ]^2 + ( \Delta_0 / k_{\rm F} )^2 ( k_x^2 + k_y^2)}$, and the spectrum λ(k) exhibits two Weyl nodes at the north and south poles of the Fermi sphere K_± = (0, 0, ±k_F). The low-energy physics in the vicinity of the point nodes K_± can be captured by the anisotropic Weyl Hamiltonian

$$\begin{equation} \mathcal{H}_{\pm} ( {\bf p} ) =(\Delta_0 / k_{\rm F} ) ( p_x \tau_x + p_y \tau_y ) \pm \hbar v_{\rm F} p_z \tau_z , \end{equation} \tag{ 2 }$$

where p = k − K_± and v_F = ℏk_F/m denotes the Fermi velocity. Hence, equation (1) defines the simplest version of a Weyl superconductor with two point nodes. We note that, in contrast to ordinary point nodes, the bands crossing at a Weyl node are individually nondegenerate. Moreover, Weyl nodes are impossible to gap out, since there exists no 'fourth Pauli matrix' that anticommutes with the three Pauli matrices τ_i of equation (2). The dispersion close to the Weyl points is linear in all three directions, leading to a density of states that increases quadratically with energy. As opposed to Weyl points in semimetals, the particle-hole symmetry dictates that the point nodes K_± remain pinned at zero energy even as the chemical potential is varied. Each Weyl node can be characterized in terms of a chirality index, which measures the relative handedness of the three momenta p with respect to the Pauli matrices in equation (2). For Hamiltonian $\mathcal{H}_\pm ( {\bf p} )$ the chirality of the Weyl point is ±1. We note that while arbitrary (small) perturbations of Hamiltonian (2) in general lead to a shift of the point nodes, they cannot open up a gap in the spectrum. We will see in section 3.3.1 that the stability of the Weyl nodes is protected by a Chern number (or Skyrmion number), which takes on the values ±1.

As a second example of a superconductor with point nodes, we consider the three-dimensional spin-singlet chiral $(d_{x^2-y^2} \pm {\rm i} d_{xy})$-wave state, which is likely realized in the heavy fermion system URu₂Si₂ [41–44] and the pnictide material SrPtAs [47, 48]. The BdG Hamiltonian of this chiral superconducting phase is given by $\mathcal{H}=\frac{1}{2}\sum_{\bf k}\Phi^\dagger_{\bf k} \mathcal{H} ( {\bf k} ) \Phi_{\bf k}$, with

$$\begin{equation} \mathcal{H} ( {\bf k} ) =h ( {\bf k} ) \tau_z + \frac{\Delta_0}{k_{\rm F}^2} \left[ (k_x^2 - k_y^2 ) \tau_x + 2 k_x k_y \tau_y \right] \end{equation} \tag{ 3 }$$

and the Nambu spinor $\Phi_{\bf k} = ( c_{{\bf k} \uparrow} , c^{\dagger}_{-{\bf k} \downarrow} )^{{\rm T}}$, where $c^{\dagger}_{{\bf k} \sigma} (c_{{\bf k} \sigma})$ represents the electron creation (annihilation) operator with momentum k and spin σ. Observe that for this pairing symmetry the orbital angular momentum projected along the k_z axis is ±2, whereas for the (p_x±ip_y)-wave state discussed above it is ±1. Similar to equation (1), we find that for a spherical Fermi surface centered at the Γ point, Hamiltonian (3) exhibits two point nodes located at K_± = (0, 0, ±k_F), which are double Weyl points pinned at zero energy by the particle-hole symmetry. Expanding equation (3) around K_± to second order in the momentum, we obtain the following low-energy description of the double Weyl nodes

$$\begin{eqnarray} &&\fl \mathcal{H}_{\pm} ( {\bf p} ) = \frac{\Delta_0}{k_{\rm F}^2} \left[ (p_x^2 - p_y^2 ) \tau_x + 2 p_x p_y \tau_y \right]\nonumber\\ &&+ \left[ \frac{\hbar^2}{2 m} \left| {\bf p} \right|^2 \pm \hbar v_{\rm F} p_z \right] \tau_z , \end{eqnarray} \tag{ 4 }$$

where p = k − K_±. Hence, the low-energy nodal quasiparticles near K_± exhibit linear and quadratic dispersions along the p_z direction and in the p_xp_y plane, respectively. All in all, this anisotropic dispersion leads to a density of states which increases linearly with energy. As explained in section 3.3.2, the double Weyl points (4) are protected against gap opening by the conservation of a Chern or Skyrmion number that takes on the values ±2.

The prototype of a topological superconductor with line nodes are time-reversal-symmetric three-dimensional noncentrosymmetric superconductors (NCS). In these systems, the crystal structure breaks inversion symmetry leading to strong electric field gradients within the unit cell, which in turn generates strong SOC. Furthermore, the absence of inversion symmetry means that parity is no longer a good quantum number, and so a mixed pairing state with both singlet and triplet gaps is possible.

The minimal description for such a system has the BdG Hamiltonian $\mathcal{H} = \frac{1}{2} \sum_{\bf{k}} \psi^{\dagger}_{\bf{k}} \mathcal{H} (\bf{k}) \psi_{\bf{k}}$, with

$$\begin{equation} \mathcal{H} ( \bf{k} ) = \left(\begin{array}{@{}cc@{}} h ( \bf{k} ) & \Delta ( \bf{k} ) \\ \Delta^{\dagger} ( \bf{k} ) & - h^{T} ( - \bf{k} ) \end{array}\right) \end{equation} \tag{ 5 }$$

and $\psi_{\bf{k}} = ( c_{{\bf k} \uparrow}, c_{\bf{k} \downarrow}, c^{\dagger}_{- \bf{k} \uparrow}, c^{\dagger}_{- \bf{k} \downarrow} )^{{\rm T}}$, where $c^{\dagger}_{\bf{k}\sigma}$ $(c_{\bf{k}\sigma})$ denotes the electron creation (annihilation) operator with momentum k and spin σ. The normal-state dispersion of the electrons in the spin basis is given by

$$\begin{equation} h ( {{\bf k}} ) =\varepsilon_{{\bf k}} \sigma_0 + {\bf g}_{{\bf k}} \cdot {\boldsymbol {\sigma}}, \end{equation} \tag{ 6 }$$

where ε_k is the nonmagnetic dispersion while g_k denotes the SOC potential. σ = (σ_x, σ_y, σ_z) denotes the vector of the Pauli matrices, while σ₀ stands for the 2 × 2 unit matrix. By time-reversal symmetry, we require that ε_k and g_k are symmetric and antisymmetric in k, respectively. The normal-state Hamiltonian (6) is diagonalized in the so-called helicity basis, taking the form $\tilde{h} ( {\bf k}) = {\rm diag} ( \xi^+_{\bf k} , \xi^-_{\bf k} )$, where $\xi^{\pm}_{\bf k} = \varepsilon_{\bf k} \pm \left| {\bf g}_{\bf k} \right|$ is the dispersion of the positive-helicity and negative-helicity bands, respectively.

The superconducting gap contains both even-parity spin-singlet and odd-parity spin-triplet pairing potentials

$$\begin{equation} \Delta ( {{\bf k}} ) = \left( \psi_{\bf k} \sigma_0 + {\bf d}_{{\bf k}} \cdot {\boldsymbol {\sigma}} \right) \left( {\rm i} \sigma_y \right) , \end{equation} \tag{ 7 }$$

where ψ_k and d_k represent the spin-singlet and spin-triplet components, respectively. In a time-reversal-symmetric system they have the same global phase, and without loss of generality we assume them to be real. It is well known that in the absence of interband pairing, the superconducting transition temperature is maximized when the spin-triplet pairing vector d_k is aligned with the polarization vector g_k of the SOC [56]. This implies that there is only pairing between states on the same helicity Fermi surface, with positive and negative helicity gaps $\Delta^{\pm}_{\bf k} = \psi_{\bf k} \pm |{\bf d}_{\bf k}|$. Assuming an s-wave like order parameter for the singlet pairing, we observe that while the positive-helicity Fermi surface is fully gapped, the negative-helicity gap may possess nodes. We parametrize the singlet and triplet components of the superconducting gap function as

$$\begin{equation} \psi_{\bf k} =\Delta_s = r\, \Delta_0 , \end{equation} \tag{ 8 }$$

$$\begin{equation} {\bf d}_{\bf k} = \Delta_t {\bf l}_{\bf k} = (1-r)\, \Delta_0 {\bf l}_{\bf k} , \end{equation} \tag{ 9 }$$

where l_k = g_k/λ, with λ the SOC strength. Here, r parametrizes the mixed pairing state, tuning the system between purely triplet (r = 0) and purely singlet (r = 1) gaps.

Let us start by discussing the ten-fold classification of superconducting nodes located at high symmetry points of the Brillouin zone [57–61]. For this case the classification of point nodes can be derived in a heuristic manner from the ten-fold classification of fully gapped superconductors through a dimensional reduction procedure [57]. That is, the surface state of a fully gapped d_BZ-dimensional topological superconductor realizes a topologically stable point node in d_BZ − 1 dimensions. Indeed, the bulk topological invariant of the d_BZ-dimensional fully gapped superconductor is directly related to the topological charge of the nodal point at the boundary [60, 64]. From these observations it follows that the classification of global-symmetry-invariant nodal points with d_n = 0 is obtained from the original ten-fold classification of fully gapped superconductors by the dimensional shift d_BZ → d_BZ − 1. This reasoning also holds for nodal structures with codimension p < d_BZ (i.e. d_n > 0), provided that they are protected by a $\mathbb{Z}$ invariant or $2\mathbb{Z}$ invariant. $\mathbb{Z}_2$ numbers, on the other hand, guarantee only the stability of nodes with d_n = 0, i.e. point nodes. These findings are confirmed by more rigorous derivations based on K theory [58–60] and minimal Dirac Hamiltonians [61]. The latter approach uses Clifford algebra to classify all possible symmetry-preserving mass terms that can be added to the Hamiltonian. The classification of global-symmetry-invariant nodal structures (and Fermi surfaces) is summarized in table 1, where the first row indicates the codimension p of the superconducting nodes. For any codimension p there are three symmetry classes for which stable superconducting nodes (or Fermi surfaces) exist that are protected by a $\mathbb{Z}$ invariant or $2\mathbb{Z}$ invariant, where the prefix '2' indicates that the topological number only takes on even values. Furthermore, in each spatial dimension d_BZ there exist two symmetry classes that allow for topologically stable point nodes (Fermi points) which are protected by a binary $\mathbb{Z}_2$ number.

Second, we review the topological classification of superconductors with nodes that are located away from high-symmetry points of the Brillouin zone [57, 61]. These point or line nodes are pairwise mapped onto each other by the global antiunitary symmetries, which relate k to −k. An analysis based on the minimal-Dirac-Hamiltonian method [61] shows that only $\mathbb{Z}$-type invariants can guarantee the stability of superconducting nodes off high-symmetry points, whereas $\mathbb{Z}_2$ numbers do not lead to stable nodes. However, as illustrated in terms of the example of section 3.3.3, $\mathbb{Z}_2$ numbers may nevertheless give rise to zero-energy surface states at time-reversal-invariant momenta of the surface Brillouin zone. The complete classification of superconducting nodes that are located away from high-symmetry points is presented in table 1, where the second row gives the codimension p of the superconducting node. Observe that this classification scheme is related to the ten-fold classification of fully gapped superconductors and insulators by the dimensional shift d_BZ → d_BZ + 1.

For the phenomenological model Hamiltonians given in section 2, we derive in this subsection explicit expressions for the topological invariants that protect the superconducting nodes against gap opening. We also use these examples to illustrate the bulk-boundary correspondence [64, 65], which links the topological characteristics of the nodal gap structure to the appearance of zero-energy states at the boundary. Depending on the case, these zero-energy surface states are either linearly dispersing Majorana cones, Majorana flat bands, or arc surface states (see figure 1). We note that in real superconducting materials the gap nodes are usually positioned away from the high-symmetry points of the Brillouin zone. Indeed, this is the case for the three examples of section 2, which are therefore classified according to section 3.2. Note that the topological invariants introduced here can be straightforwardly generalized to more complicated systems.

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3.3.1. The A phase of ³He.

The A phase of ³He is phenomenologically described by Hamiltonian (1), which satisfies particle-hole symmetry $C^{-1} \mathcal{H} (-{\bf k} ) C = - \mathcal{H} ( {\bf k} )$ with $C= \mathcal{K} \tau_x$. Time-reversal symmetry, however, is broken, because the superconducting order parameter of equation (1) is complex. Hence, since $C^2 = + 1\hspace{-3.8pt}1$, Hamiltonian (1) belongs to symmetry class D. We infer from table 1 that the Weyl nodes of the Hamiltonian (1), which have codimension p = 3 and occur off high symmetry points, are protected by a $\mathbb{Z}$ topological number. In order to derive a formula for this topological number it is convenient to rewrite Hamiltonian (1) as

$$\begin{equation} \mathcal{H} ( {\bf k} )= {\bf N} ( {\bf k} ) \cdot {\boldsymbol \tau}, \end{equation} \tag{ 13 }$$

i.e. a dot product between the pseudospin vector N(k) = (Δ₀k_x/k_F, Δ₀k_y/k_F, h (k)) and the vector of Pauli matrices τ = (τ_x, τ_y, τ_z). The unit vector field n_k = N(k)/|N(k)| exhibits singular points at the Weyl nodes K_± of $\mathcal{H} ({\bf k})$. These point singularities realize (anti)hedgehog defects in momentum space and are characterized by the Chern number [25–28]

$$\begin{eqnarray} &&N_{\mathcal{C}} =\frac{1}{4 \pi} \int_{\mathcal{C}} d^2 \widetilde{\bf k} \; {\bf n}_{\bf k} \cdot \left[ \partial_{k_1} {\bf n}_{\bf k} \times \partial_{k_2} {\bf n}_{\bf k} \right] , \end{eqnarray} \tag{ 14 }$$

where $\mathcal{C}$ denotes a two-dimensional surface parametrized by $\widetilde{\bf k} = (k_1, k_2)$, which encloses one of the two Weyl nodes. Choosing $\mathcal{C}$ to be a sphere S² centered at one of the two Weyl points, the integral (14) can be straightforwardly evaluated. The nodes at K_± = (0, 0, ±k_F) have topological charge $N_{S^2} = \pm 1$. We note that n_k restricted to the sphere S² in momentum space defines a map from k ∈ S² to the space n_k ∈ S². The Chern number $N_{S^2}$, equation (14), distinguishes between different homotopy classes of these maps [67]. Since the homotopy group π₂(S²) equals the group of the integer numbers $\mathbb{Z}$, there is an infinite number of homotopy classes of maps k ∈ S² ↦ n_k and $N_{S^2}$ can in general take on any integer value.

Alternatively, instead of considering a sphere S² one can take $\mathcal{C}$ to be a plane perpendicular to the k_z axis in the Brillouin zone. This yields $N_{k_z} = +1$ for |k_z| < k_F and zero otherwise. That is, the unit vector n_k for fixed |k_z| < k_F possesses a unit skyrmion texture in momentum space and $N_{k_z}$ measures its skyrmion number. In this picture, we can give the topological invariant (14) a more physical meaning by reexpressing it in terms of the Berry curvature F(k) = ∇_k × A(k), i.e.

$$\begin{equation} N_{k_z} = \frac{1}{2 \pi} \int {\rm d} k_x {\rm d} k_y \, F_z ( {\bf k} ). \end{equation} \tag{ 15 }$$

Here, A(k) = i〈u₋ (k)|∇_k| u₋ (k)〉 denotes the Berry connection (also known as Berry vector potential), which is obtained from the negative-energy wavefunctions u₋ (k) of the BdG Hamiltonian. For Hamiltonian (1), the three components of the Berry curvature are given by

$$\begin{eqnarray} &&F_x ( {\bf k} ) =\frac{\Delta^2_0 k_{\rm F}^2 k_x k_z} {\mu^2 \left[ ( k^2 - k_{\rm F}^2 )^2 + \frac{\Delta^2_0}{\mu^2} k_{\rm F}^2 ( k_x^2 + k_y^2) \right]^{\frac{3}{2}}} , \nonumber\\ &&F_y ( {\bf k} )= \frac{\Delta^2_0 k_{\rm F}^2 k_y k_z} {\mu^2 \left[ ( k^2 - k_{\rm F}^2 )^2 + \frac{\Delta^2_0}{\mu^2} k_{\rm F}^2 ( k_x^2 + k_y^2) \right]^{\frac{3}{2}}} ,\\ &&F_z ( {\bf k} )= \frac{\Delta_0^2 k_{\rm F}^2 ( k_z^2 - k_x^2 - k_y^2 - k_{\rm F}^2 )} {2 \mu^2 \left[ (k^2 -k_{\rm F}^2 )^2 + \frac{\Delta^2_0}{\mu^2} k_{\rm F}^2 (k_x^2 + k_y^2) \right]^{\frac{3}{2}}} . \nonumber \end{eqnarray} \tag{ 16 }$$

The Berry curvature is plotted in figure 2(a). We observe from equation (16) that the Weyl nodes are sources and drains of Berry flux. That is, they act as unit (anti-)monopoles of the Berry curvature F(k) and there is a Berry flux of 2π flowing from one Weyl node to the other along the z direction. Along the x and y directions, on the other hand, the Berry flux is vanishing since F_x(y)(k) is odd in k_x(k_y) and k_z. We also note that the Berry curvature is sharply peaked within a shell of width Δ₀ about the Fermi surface.

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Due to the bulk-boundary correspondence, the band topology of Weyl superconductors gives rise to arc surface states. Consider for example Hamiltonian (1) in a three-dimensional slab geometry with a (1 0 0) surface. Because of translation invariance along the surface, the three-dimensional (1 0 0) slab can be decomposed into a family of two-dimensional (1 0) ribbons parametrized by k_z. Ribbons with a k_z in between the pair of Weyl nodes have a non-zero Chern number, i.e. $N_{k_z} = +1$ for |k_z| < k_F, while for |k_z| > k_F the topology is trivial. Each ribbon with |k_z| < k_F can be interpreted as a two-dimensional fully gapped (p_x + ip_y)-wave superconductor with a chiral Majorana edge mode. Hence, the Majorana surface states of the Weyl superconductor form a one-dimensional open arc in the surface Brillouin zone, connecting the projected Weyl nodes of the bulk gap. (See figure 4(a) and section 4.2). These chiral surface states give rise to anomalous spin and thermal Hall effects which are proportional to the separation of the Weyl nodes in momentum space. (see section 5.2).

3.3.2. Chiral $(d_{x^2-y^2} \pm {\rm i} d_{xy})$-wave superconductor.

The spin-singlet chiral $(d_{x^2-y^2} \pm {\rm i} d_{xy})$-wave superconductor (3) satisfies both particle-hole symmetry and SU(2) spin-rotation symmetry and therefore belongs to symmetry class C in table 1 (or alternatively to class A if one considers only a U(1) part of the spin-rotation symmetry). Hence, as indicated by the classification table, the Weyl nodes of Hamiltonian (3) (which occur off high symmetry points) are protected by a $2 \mathbb{Z}$ invariant, which takes on only even values. This topological number is given by the same expression as equation (14) but with ${\bf N} ( {\bf k} ) = \left( \Delta_0 ( k^2_x - k^2_y ) / k^2_{\rm F} , 2 \Delta_0 k_x k_y / k^2_{\rm F}, h ( {\bf k}) \right)$. As opposed to the previous example, the unit vector n_k = N(k)/|N(k)| for fixed |k_z| < k_F now describes a double Skyrmion texture with $N_{k_z} = \pm 2$. That is, the double Weyl points at K_± = (0, 0, ±k_F) correspond to double (anti-)monopoles of the Berry curvature F(k), which is given by [44]

$$\begin{eqnarray} &&F_x ( {\bf k} ) =\frac{2 \Delta_0^2 k_x k_z ( k_x^2 + k_y^2)} {\mu^2 \left[ ( k^2 - k_{\rm F}^2 )^2 + \frac{\Delta_0^2}{\mu^2} (k_x^2 + k_y^2)^2 \right]^{\frac{3}{2}}} , \nonumber\\ &&F_y ( {\bf k} ) =\frac{2 \Delta_0^2 k_y k_z ( k_x^2 + k_y^2)} {\mu^2 \left[ ( k^2 - k_{\rm F}^2 )^2 + \frac{\Delta_0^2}{\mu^2} (k_x^2 + k_y^2)^2 \right]^{\frac{3}{2}}} ,\\ &&F_z ( {\bf k} )= \frac{2 \Delta_0^2 ( k_z^2 - k_{\rm F}^2 ) (k_x^2 + k_y^2 )} {\mu^2 \left[ (k^2 -k_{\rm F}^2 )^2 + \frac{\Delta_0^2}{\mu^2}( k_x^2 + k_y^2)^2 \right]^{\frac{3}{2}}} . \nonumber \end{eqnarray} \tag{ 17 }$$

The Berry curvature is plotted in figure 2(b). As a result of the topological non-triviality, the d-wave Weyl superconductor (3) supports two spin-degenerate, chirally dispersing arc surface states. (See figure 4(b). These surface states carry anomalous spin and thermal Hall currents.

3.3.3. Nodal noncentrosymmetric superconductor.

Nodal NCSs defined by the BdG Hamiltonian $\mathcal{H} ( {\bf k} )$ in equation (5) satisfy time-reversal symmetry $T^{-1} \mathcal{H} ( - {\bf k} ) T = + \mathcal{H} ( {\bf k} )$, with $T = \mathcal{K} \tau_0 \otimes {\rm i} \sigma_y$, and particle-hole symmetry $C^{-1} \mathcal{H} (- {\bf k} ) C = - \mathcal{H} ( {\bf k} )$, with $C = \mathcal{K} \tau_x \otimes \sigma_0$. Here, τ_i and σ_i are the Pauli matrices operating in particle-hole and spin space, respectively. Since $T^{2} = - 1\hspace{-3.8pt}1$ and $C^2 = + 1\hspace{-3.8pt}1$, Hamiltonian (5) belongs to symmetry class DIII. According to table 1, the topological properties of class DIII Hamiltonians are characterized by $\mathbb{Z}_2$ topological numbers. While these binary invariants can give rise to protected Majorana cone surface states or arc surface states, they do not guarantee the stability of the superconducting nodes [61]. Hence, one might conclude that the superconducting nodes of the NCS (5) are unstable against gap opening. In addition to TRS and PHS, however, $\mathcal{H} ( {\bf k} )$ in Equation (5) also possesses the chiral symmetry $S^{-1} \mathcal{H} ( {\bf k} ) S = - \mathcal{H} ({\bf k} )$, with S = −τ_x ⊗ σ_y. Therefore, $\mathcal{H} ( {\bf k} )$ can be viewed as a member of symmetry class AIII and, as indicated by table 1, its line nodes are protected by a $\mathbb{Z}$ invariant, i.e. a winding number.

The $\mathbb{Z}_2$ and $\mathbb{Z}$ invariants of symmetry class DIII and AIII, respectively, can be conveniently expressed in terms of the off-diagonal block q (k) of the spectral projector. For Hamiltonian (5), the matrix q (k) takes the following form [21, 22]

$$\begin{eqnarray} &&\fl q ( {\bf k} ) = \frac{1}{2 \Lambda_{1, {\bf k}} \Lambda_{2, {\bf k}}} \bigg[ \left( \Lambda^+_{\bf k} B^{\phantom{+}}_{\bf k} - \Lambda^-_{\bf k} A^{\phantom{+}}_{\bf k} \left| {\bf l}_{\bf k} \right| \right) \sigma_0 \nonumber\\ &&+ \left( \Lambda^+_{\bf k} A^{\phantom{+}}_{\bf k}\left| {\bf l}_{\bf k} \right| - \Lambda^-_{\bf k} B^{\phantom{+}}_{\bf k} \right) \frac{{\bf l}_{\bf k}}{\left| {\bf l}_{\bf k} \right|} \cdot {\boldsymbol \sigma} \bigg] . \end{eqnarray} \tag{ 18 }$$

In equation (18) we have introduced the short-hand notation A_k = λ + iΔ_t, B_k = ε_k + iΔ_s, and $\Lambda^{\pm}_{\bf k} = \Lambda_{1, {\bf k}} \pm \Lambda_{2, {\bf k}}$, where

$$\begin{eqnarray} &&\fl \Lambda_{1, {\bf k}} = \sqrt{\left( \xi^+_{\bf k} \right)^2 + \left( \Delta^+_{\bf k} \right)^2} \quad {\rm and} \quad \Lambda_{2, {\bf k}} = \sqrt{\left( \xi^-_{\bf k} \right)^2 + \left( \Delta^-_{\bf k} \right)^2}\nonumber\\ \end{eqnarray} \tag{ 19 }$$

are the positive eigenvalues of $\mathcal{H} ( {\bf k} )$. The unitary matrix q (k) ∈ U(2) is constrained by time-reversal symmetry as iσ_yq^T (−k) = q (k) iσ_y.

Winding number.

The stability of the line nodes of NCS (5) is ensured by the conservation of the winding number [19, 21, 57, 68, 69]

$$\begin{equation} W_{\mathcal{L}}=\frac{1}{2\pi i} \oint_\mathcal{L} {\rm d}k_l \, {\rm Tr} [q^{\dagger} ( {\bf k} ) \partial_{k_l} q ( {\bf k} ) ], \end{equation} \tag{ 20 }$$

where $\mathcal{L}$ is a closed one-dimensional contour parametrized by k_l, which interlinks with a line node. Mathematically speaking, equation (20) labels the homotopy classes of mappings from $\mathcal{L} \simeq S^1 \to q ( {\bf k} ) \in U(2)$. Since the first homotopy group of U(2) equals the group of the integer numbers $\mathbb{Z}$ [67], i.e. $\pi_1 [ U (2) ] = \mathbb{Z}$, the winding number $W_{\mathcal{L}}$ can in principle take on any integer value. Using equation (18), we find that $W_{\mathcal{L}}$ for the BdG Hamiltonian (5) can be rewritten in the simpler form

$$\begin{eqnarray} &&\fl W_{\mathcal{L}} =\frac{1}{2 \pi} \oint_{\mathcal{L}} {\rm d} k_l \, \partial_{k_l} \left[ {\rm arg} \left( \xi^+_{\bf k} + {\rm i} \Delta^+_{\bf k} \right) + {\rm arg} \left( \xi^-_{\bf k} + {\rm i} \Delta^-_{\bf k} \right) \right]. \end{eqnarray} \tag{ 21 }$$

Assuming that the superconducting gaps $\Delta^{\pm}_{\bf k}$ of Hamiltonian (5) are nonzero only in the vicinity of the Fermi surfaces (which is the case for weak-pairing superconductors), equation (21) can be further simplified to [22]

$$\begin{eqnarray} &&\fl W_{\mathcal{L}} = - \frac{1}{2} \sum_{\nu = \pm} \, \sum_{{\bf k}_0 \in S^\nu_{\mathcal{L}}} {\rm sgn} \left( \left. \partial_{k_l} \xi^{\nu}_{\bf k} \right|_{{\bf k} = {\bf k}_0} \right) {\rm sgn} \left( \Delta^{\nu}_{{\bf k}_0} \right) . \end{eqnarray} \tag{ 22 }$$

The second sum in equation (22) is over the set of points $S^\nu_{\mathcal{L}}$ given by the intersection of the ν-helicity Fermi surface with the one-dimensional contour $\mathcal{L}$. Equation (22) demonstrates that for a weak-pairing nodal superconductor of the form (5), the topological characteristics of the nodal structure is entirely determined by the phases of the gaps in the neighborhood of the positive- and negative-helicity Fermi surfaces. This is in complete agreement with the discussion of section 4.3.

By the bulk-boundary correspondence [64], a non-zero value of $W_{\mathcal{L}}$ signals the appearance of zero-energy states at the surface of the NCS. To make this more explicit, let us consider a contour $\mathcal{L}$ along a line parallel to the (lmn) direction in the bulk Brillouin zone. (See figure 3). With this choice we obtain for equation (22)

$$\begin{eqnarray} &&\fl W_{(lmn)} ( {\bf k}_{\parallel} ) = - \frac{1}{2} \sum_{\nu = \pm} \left[ {\rm sgn} \left( \Delta^{\nu}_{{\bf k}_{\nu}} \right) - {\rm sgn} \left( \Delta^{\nu}_{{\bf k}^{\prime}_{\nu}} \right) \right] , \end{eqnarray} \tag{ 23 }$$

where k_∥ denotes the momentum perpendicular to the (lmn) direction, and k_± = (k_∥, k_⊥,±) and ${\bf k}^{\prime}_{\pm} = ( {\bf k}_{\parallel}, k^{\prime}_{\perp, \pm} )$ are momenta on the positive- and negative-helicity Fermi surfaces which project onto k_∥, where k_⊥,± and $k^{\prime}_{\perp, \pm}$ have opposite signs. With this, it follows from the bulk-boundary correspondence that zero-energy states appear at the (lmn) face of the NCS whenever W_(lmn) (k_∥)≠ 0. This corresponds to regions of the surface Brillouin zone that are bounded by the projected nodal rings of the bulk gap. (See figure 3). In other words, the non-trivial topology of the nodal rings in the bulk leads to protected flat-band surface states (see figure 1(c) and section 4.3.3).

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$\mathbb{Z}_2$ topological invariant.

The one-dimensional (two- dimensional) $\mathbb{Z}_2$ invariant of symmetry class DIII is given by [4, 20, 21, 70, 71]

$$\begin{equation} N_{\mathcal{L}}= \prod_{{\bf K} \in \mathcal{L}} \frac{{\rm Pf}\, \left[ {\rm i} \sigma_y q^{{\rm T}}({\bf K})\right]} {\sqrt{\det \left[ {\rm i} \sigma_y q^{{\rm T}} ( {\bf K} ) \right]}}, \end{equation} \tag{ 24 }$$

where Pf is the Pfaffian and the product is over the two (four) time-reversal invariant momenta K of the one-dimensional (two-dimensional) centrosymmetric contour $\mathcal{L}$. In order for $N_{\mathcal{L}}$ to be well defined, the contour $\mathcal{L}$ needs to be centrosymmetric (i.e. mapped onto itself under k → −k) and must not intersect with the nodes of $\mathcal{H} ({\bf k} )$. Inspection of equation (24) reveals that $N_{\mathcal{L}}$ can only take on the values ±1, where −1 (+1) indicates a topologically nontrivial (trivial) character. For weak-pairing superconductors the gaps $\Delta^{\pm}_{\bf k}$ are non-zero only in a small neighborhood of the Fermi surface and hence equation (24) can be simplified to [22]

$$\begin{equation} N_{\mathcal{L}} = {\rm sgn} \left( \Delta^+_{\mathcal{L}} \right) {\rm sgn} \left( \Delta^-_{\mathcal{L}} \right) , \end{equation} \tag{ 25 }$$

where ${\rm sgn} \left( \Delta^{\pm}_{\mathcal{L}} \right)$ represents the sign of the gap at the intersection of $\mathcal{L}$ with the positive/negative helicity Fermi surface. By the bulk-boundary correspondence [65], $N_{\mathcal{L}} = -1$ for a one-dimensional contour $\mathcal{L}$ indicates the appearance of a helical Majorana cone at a surface of the NCS. (See figure 1(a)). Similarly, for a two-dimensional contour $\mathcal{L}$, $N_{\mathcal{L}} = -1$ gives rise to an arc surface state. (See figure 1(b).

In the previous subsections we have discussed the topological classification of nodal superconductors in terms of internal symmetries, i.e. global symmetries that act locally in position space (see table 1). Recently it became apparent that besides global symmetries, crystal symmetries which operate non-locally in position space can also lead to the protection of superconducting nodes [60, 61, 72–75]. Indeed, a complete topological classification of reflection-symmetry-protected nodal superconductors was recently obtained in [61]. Moreover, it has been argued that the heavy fermion superconductor UPt₃ can be viewed as a crystalline topological nodal superconductor, since its topological surface states are protected by mirror symmetry [76].

We consider the surface of a topological superconductor, with normal defining the axis r_⊥. The superconductor is assumed to occupy the half-space r_⊥ > 0, and translational invariance parallel to the surface ensures that the transverse momentum k_∥ is a good quantum number. Our aim is to solve the BdG equation

$$\begin{equation} \tilde{H}\Psi({\bf k}_\parallel; r_\perp) = E\Psi({\bf k}_\parallel;r_\perp) \end{equation} \tag{ 26 }$$

for states with energy comparable to the superconducting gap. The Hamiltonian $\tilde{H}({\bf k}_{\parallel})$ appearing here is related to the bulk Hamiltonian H(k) by

$$\begin{equation} \tilde{H}({\bf k}_\parallel) = \int \frac{{\rm d}k_\perp}{2\pi} H({\bf k}) {\rm e}^{{\rm i}k_\perp r_\perp} . \end{equation} \tag{ 27 }$$

The BdG equations can be solved numerically, e.g. by exact diagonalization of a lattice regularization of the Hamiltonian. Superconductivity is a low-energy phenomenon, however, where the characteristic energy scale of the gap Δ is typically much smaller than the Fermi energy E_F, and it is at such low energy scales that the surface states exist. Not only is it computationally advantageous to restrict our solution of equation (26) to the low-energy sector, but it also permits at least semi-analytic solutions which provide a more transparent understanding of the physics. This is the motivation for the quasiclassical theory of superconductivity. A full development of this theory, in particular the construction of the quasiclassical Green's function and the self-consistent determination of the pairing potential, is rather involved and beyond the scope of this review. We therefore sketch the theory below, and refer interested readers to the extensive literature on the subject, e.g. [77–81].

The first step in constructing an effective theory is to bring the noninteracting Hamiltonian h (k) into a diagonal form. The low-energy quasiparticle excitations in the superconductor occur within a narrow momentum shell about the normal-state Fermi surface of width ∼Δ/ℏ|v_F|≪|k_F| where v_F is the Fermi velocity. The Fermi velocity is regarded as constant across this momentum shell, which allows us to linearize the low-energy Hamiltonian [h (k)]_{α, α} ≈ ℏv_F,α · (k − k_F,α) where k_F,α lies on the Fermi surface of the α band. We now examine the pairing matrix $\hat{\Delta}({\bf k})$. The superconducting gap is taken to be constant along the direction of the Fermi velocity at each point on the Fermi surface, i.e. $\hat{\Delta}({\bf k}) \approx \hat{\Delta}({\bf k}_{{\rm F},\alpha})$. The solution of the low-energy effective theory is therefore only sensitive to the value of the superconducting gap in the immediate vicinity of the Fermi surface. Pairing between states on nondegenerate Fermi surfaces is not considered.

An appropriate ansatz for an eigenstate of the low-energy Hamiltonian is

$$\begin{equation} \Psi({\bf k}_\parallel;{\bf r}) = \sum_{j}a_j \left(\begin{array}{@{}c@{}} {\bf u}_j \\ {\bf v}_j \end{array} \right){\rm e}^{{\rm i}({\bf k}_{\parallel} \cdot{\bf r}_\parallel + [k_{j} + \delta k_{j}] r_\perp)}, \end{equation} \tag{ 28 }$$

where the sum j is over all scattering channels, a_j are $\mathbb{C}$-number coefficients, the vector k_j = (k_∥, k_j) lies on the Fermi surface, |δk_j|≪|k_j| is a small energy-dependent deviation from the Fermi wavevector k_j, and the multi-component BCS coherence factors u_j and v_j are functions only of k_j and the energy. The coefficients a_j in equation (28) are determined by imposing boundary conditions on the wavefunction at the surface, which can in principle be derived from the underlying microscopic theory. This presents the greatest challenge faced by the low-energy effective theory, as there is no general prescription of how this should be done, especially for multiband systems or those with strong spin-orbital mixing.

In order to avoid this complication, in the following we restrict our attention to a commonly-used microscopic model of the normal state where boundary conditions can be explicitly and easily derived [1, 14, 15, 22, 23, 31, 34, 45, 78, 80, 82–88]. Specifically, we adopt the noninteracting Hamiltonian

$$\begin{equation} h({\bf k}) = \left(\frac{\hbar^2}{2m}|{\bf k}|^2 - \mu\right)\hat{\sigma}_{0} + \lambda {\bf l}_{\bf k}\cdot\hat{\pmb{\sigma}} + V_{{\rm work}}\Theta(r_\perp) \end{equation} \tag{ 29 }$$

This describes a free electron gas, including antisymmetric SOC of strength λ so as to model both centrosymmetric (λ = 0) and noncentrosymmetric (λ ≠ 0) systems. In the latter case, the spin-orbit vector l_k has the following continuum-limit representations for the different point groups

$$\begin{eqnarray} \fl {\bf l}_{\bf k} = \left\{\begin{array}{@{}ll@{}} (a_1 k_x + a_2 k_y){\bf x} + ( a_3 k_x + a_4 k_y){\bf y} + a_5 k_z{\bf z}, & C_2 \\ k_y {\bf x} - k_x{\bf y}, & C_{4v} \\ k_x(1 + g_2[k_y^2+k_z^2]){\bf x}\\ + k_y(1 + g_2[k_x^2+k_z^2]){\bf y} + k_z(1 + g_2[k_x^2+k_y^2]){\bf z}, & {\mathcal O} \\ k_x(k_y^2-k_z^2){\bf x} + k_y(k_z^2-k_x^2){\bf y} + k_z(k_x^2-k_y^2){\bf z}, & T_{d} \end{array}\right.\nonumber\\ \end{eqnarray} \tag{ 30 }$$

The coefficients a_j=1,...,5 and g₂ appearing in the expressions for the C₂ and ${\mathcal O}$ point groups are arbitrary. Our Hamiltonian also includes a confining potential V_work for the surface which we identify with the work function. For our low-energy theory it is reasonable to take the limit V_work → ∞, and so we implicitly include this potential as the boundary condition that the wavefunctions vanish at the surface.

As an illustration of the quasiclassical method, we consider the subgap bound states appearing at the (1 0 0) surface of the Weyl superconductor models introduced in section 2.1. Take a point k_∥ = (k_y, k_z) in the two-dimensional surface Brillouin zone that lies within the projection of the Fermi surface, i.e. |k_∥| < k_F. This surface wavevector corresponds to two points on the Fermi surface ${\bf p}^{}=(k^{}_{x},{\bf k}_\parallel)$ and ${\bf p}^\prime=(-{k}^{}_{x},{\bf k}_\parallel)$, where $k^{}_{x} = \sqrt{k_{\rm F}^2 - |{\bf k}_\parallel|^2}$. The bound state wavefunction is then a superposition of evanescent states at these momenta

$$\begin{eqnarray} &&\fl \Psi({\bf k}_\parallel;{\bf r}) = \left\{a_{{\bf p}} \left(\begin{array}{@{}c@{}} 1\\ \gamma_{{\bf p}} \end{array}\right){\rm e}^{({\rm i}k_x - \kappa_{{\bf p}})x} + a_{{\bf p}^{\prime}} \left(\begin{array}{@{}c@{}} 1\\ \gamma_{{\bf p}^{\prime}} \end{array}\right){\rm e}^{-({\rm i}k_x + \kappa_{{\bf p}^{\prime}})x}\right\}{\rm e}^{{\rm i}{\bf k}_\parallel\cdot{\bf r}} \nonumber\\ \end{eqnarray} \tag{ 31 }$$

where a_k are $\mathbb{C}$-number coefficients, and the coherence factors γ_k and inverse decay length κ_k are defined

$$\begin{equation} \gamma_{{\bf k}} = \frac{1}{\Delta_{\bf k}}\left[E - {\rm i} \mathop{{\rm sgn}}(k_x)\sqrt{|\Delta_{\bf k}|^2 - E^2}\right]\,, \end{equation} \tag{ 32 }$$

$$\begin{equation} \kappa_{{\bf k}} = \frac{m\sqrt{|\Delta_{{\bf k}}|^2 - E^2}}{\hbar^2|k_{x}|}. \end{equation} \tag{ 33 }$$

The ansatz equation (31) satisfies the criteria that the wavefunction vanishes in the bulk. The requirement that the wavefunction also vanishes at the surface leads to the condition on the coherence factors

$$\begin{equation}\gamma_{{\bf p}} - \gamma_{{\bf p}^{\prime}} = 0. \end{equation} \tag{ 34 }$$

This can be analytically solved for the surface bound state energies of the p-wave and d-wave Weyl superconductor:

$$\begin{eqnarray} \fl E({\bf k}_\parallel) = \left\{\begin{array}{@{}ll@{}} \Delta_0\left(1 - \tilde{k}_z^2\right)^{1/2}\tilde{k}_y & p_x+{\rm i}p_y \\ \Delta_0\left(1 - \tilde{k}_z^2\right)\left(1- |\tilde{\bf k}_\parallel|^2 - \tilde{k}_y^2\right)\\ \times{\rm sgn}(\tilde{k}_y) & d_{x^2-y^2}+{\rm i}d_{xy} \end{array}\right. \end{eqnarray} \tag{ 35 }$$

where $\tilde{k}_\mu = k_\mu/k_{\rm F}$. The bound state spectrum is plotted in figure 4. In both cases, we observe zero-energy arc surface states running between the nodes at the north and south poles of the Fermi surface. Whereas there is only a single such arc state in the p-wave Weyl superconductor, there are two arc states in the d-wave Weyl superconductor, consistent with the Chern numbers equation (15) of $N_{k_z} = 1$ and $N_{k_z}=2$ for |k_z| < k_F, respectively.

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We now turn our attention to the surface bound states of a NCS. We restrict ourselves to the limit E_F ≫ λmax{|l_k|}≫|Δ|, so that we can ignore the spin-splitting of the Fermi surfaces in the low-energy theory, but the triplet d-vector is still constrained to be parallel to the SOC vector, i.e. d_k = Δ_t l_k, and hence the physical description outlined in section 2.2 is still applicable. This is a common approximation [22, 23, 80, 88–93] as accounting for the spin-orbit splitting of the Fermi surfaces complicates the analysis but generally only quantitatively modifies the surface state spectra.

A similar (but more involved) argument to that for the Weyl superconductors above gives the implicit equation [22, 23, 80] for the bound states energies

$$\begin{eqnarray} \fl 0 & =& (\gamma^{+}_{{\bf p}^\prime} - \gamma^{-}_{{\bf p}})(\gamma^{-}_{{\bf p}^\prime} - \gamma^{+}_{{\bf p}})(|{\bf l}_{\bf p}||{\bf l}_{{\bf p}^{\prime}}| - {\bf l}_{\bf p}\cdot{\bf l}_{{\bf p}^{\prime}}) \nonumber\\ \fl && + (\gamma^{+}_{{\bf p}^\prime} - \gamma^{+}_{{\bf p}})(\gamma^{-}_{{\bf p}^\prime} - \gamma^{-}_{{\bf p}})(|{\bf l}_{\bf p}||{\bf l}_{{\bf p}^{\prime}}| + {\bf l}_{\bf p}\cdot{\bf l}_{{\bf p}^{\prime}}) . \end{eqnarray} \tag{ 36 }$$

where

$$\begin{equation} \gamma^{\pm}_{\bf k} = \frac{1}{\Delta^\pm_{\bf k}}\left[E - {\rm i} \mathop{{\rm sgn}}(v_{{\rm F},\perp}({\bf k}))\sqrt{|\Delta^\pm_{\bf k}|^2 - E^2}\right] \end{equation} \tag{ 37 }$$

where v_F,⊥(k) is the component of the Fermi velocity in the direction normal to the boundary. Bound-state energies must lie within the minimum gap, i.e. $|E|<\min\{|\Delta^{\pm}_{{\bf p}}|,|\Delta^{\pm}_{{\bf p}^{\prime}}|\}$. Although the general solution must be obtained numerically, analytic conditions can be derived for the existence of zero-energy states. Substituting the zero-energy form of $\gamma^{\pm}_{{\bf k}} = -{\rm i} \mathop{{\rm sgn}}(v_{{\rm F},\perp}({\bf k})\Delta^{\pm}_{\bf k})$ into equation (36) we obtain five conditions for zero-energy states in terms of the SOC vector and the gap signs [22]:

• (a)  
  The spin-orbit vectors l_p and ${\bf l}_{{\bf p}^{\prime}}$ are parallel, and the gap signs satisfy $\mathop{{\rm sgn}}(\Delta^{+}_{\bf p}) = \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}^\prime}) \neq \mathop{{\rm sgn}}(\Delta^{+}_{{\bf p}^\prime}) = \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}})$.
• (b)  
  The spin-orbit vectors l_p and ${\bf l}_{{\bf p}^{\prime}}$ are antiparallel, and the gap signs satisfy $\mathop{{\rm sgn}}(\Delta^{+}_{\bf p}) = \mathop{{\rm sgn}}(\Delta^{+}_{{\bf p}^\prime}) \neq \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}^\prime}) = \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}})$.
• (c)  
  The gap signs satisfy $\mathop{{\rm sgn}}(\Delta^{+}_{\bf p}) = \mathop{{\rm sgn}}(\Delta^{+}_{{\bf p}^\prime})$ and $\mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}^\prime}) \neq \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}})$.
• (d)  
  The gap signs satisfy $\mathop{{\rm sgn}}(\Delta^{+}_{\bf p}) \neq \mathop{{\rm sgn}}(\Delta^{+}_{{\bf p}^\prime})$ and $\mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}^\prime}) = \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}})$.
• (e)  
  The gap signs satisfy $\mathop{{\rm sgn}}(\Delta^{+}_{\bf p}) \neq \mathop{{\rm sgn}}(\Delta^{+}_{{\bf p}^\prime})$, $\mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}^\prime}) \neq \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}})$, $\mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}^\prime}) \neq \mathop{{\rm sgn}}(\Delta^{+}_{{\bf p}})$, and $\mathop{{\rm sgn}}(\Delta^{+}_{{\bf p}^\prime}) \neq \mathop{{\rm sgn}}(\Delta^{-}_{{\bf p}})$.

These five conditions for a zero-energy state include the topological criteria defined in section 3.3.3. Only a state satisfying the first condition is not topological: It cannot be a Kramers-degenerate Majorana mode as the antisymmetric spin-orbit vectors at p and p' are not parallel if k_∥ lies at a time-reversal invariant point in the surface Brillouin zone; It also cannot be an arc surface state protected by a $\mathbb{Z}_2$ number, since the condition on the gap signs requires that there be a node in the plane defined by p and p'. On the other hand, since condition (b) requires different sign of the positive and negative helicity gaps, it is clearly consistent with the definition of the topological number equation (25) for both these topological states. It also describes nontopological states, however, as these conditions are not sufficiently restrictive. The conditions (c)–(e) exclusively describe flat band states with nontrivial winding number, and they are equivalent to the topological criteria equation (23). Specifically, conditions (c) and (d), which correspond to a situation where only one of the helical gaps has a sign change between the two sides of the Fermi surface, describe topologically protected nondegenerate zero-energy states with winding number W_(lmn) = ±1. In contrast, (e) requires sign changes of both helical gaps and gives doubly degenerate states of winding number W_(lmn) = ±2.

4.3.1. Helical Majorana.

Helical Majorana states appearing at the centre of the surface Brillouin zone (k_∥ = 0) are relatively common for majority-triplet NCS at high-symmetry surfaces. In each of the examples shown in figure 5 the zero-energy state at the zone centre has this topological protection. The other zero-energy states that are present elsewhere in the surface Brillouin zone in each of these examples, on the other hand, do not satisfy the necessary criterion.

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4.3.2. Arc states.

4.3.3. Flat-band states.

Flat-band states are the most generic topological surface state of nodal NCSs, appearing at any surface where the projections of the nodal rings enclose a finite area and do not exactly lie on top of another. We illustrate this with an example from each point group in figure 6. In the top row we plot the spectrum in the surface Brillouin zone, while in the bottom row we show the corresponding variation of the winding number W_(lmn)(k_∥). The regions where the topological number evaluates to ±1 are bounded by the projected line nodes of the gap, and exactly matches the location of the zero-energy flat bands in the surface spectrum. By the bulk-boundary correspondence this ensures the topological protection of the zero-energy flat bands and implies that these states are nondegenerate.

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4.3.4. Spin polarisation of surface states.

The surface bound states of NCSs typically display a strong spin polarization, which is a consequence of the broken inversion symmetry. The μ-component of the spin polarization of a bound state at transverse momentum k_∥ and energy E is defined [85] as the expectation value

$$\begin{eqnarray} \fl \rho^{\mu}( E, {\bf k}_{\parallel}) = \int_{0}^{\infty} {\rm d}r_\perp \Psi^{\dagger}({\bf k}_\parallel;r_\perp) \left(\begin{array}{@{}cc@{}} \hat{\sigma}^{\mu} & 0 \\ 0 & - \left[ \hat{\sigma}^{\mu} \right]^{\ast} \\ \end{array}\right) \Psi({\bf k}_\parallel;r_\perp) \nonumber\\ \end{eqnarray} \tag{ 38 }$$

where Ψ(k_∥; r_⊥) is the wavefunction. Due to the symmetries of the Hamiltonian (time-reversal, particle-hole, and chiral), the polarization of the edge states is even in energy and odd in the surface momentum, i.e. ρ^μ(E, k_∥) = ρ^μ(−E, k_∥) = −ρ^μ(E, −k_∥), and there is no spin accumulation at the surface. The spin polarization is sometimes defined only with reference to the electron-like components of the wavefunction [86, 89, 90], but the definition equation (38) is more reasonable as this quantity couples to an applied Zeeman field.

The spin polarization of the edge states at the (1 0 1) surfaces of a C_4v NCS is shown in figure 7. Both the dispersing and the flat-band edge states display a pronounced spin texture. The spin polarization does not follow from the topology, however, but rather arises from the unequal contribution of positive- and negative-helicity states to the wavefunction. The spin polarization of the surface states must match that of the continuum states where the two intersect, in particular at the bounding nodes of the flat band states. Despite its nontopological origin, the spin polarization of the topological flat-band states is exploited in several proposed experimental tests of their existence.

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Conductance spectroscopy experiments represent the dominant approach to probing the surface states of unconventional superconductors [78, 98, 99]. Since the conductance is sensitive to the surface density of states of the superconductor, it can provide direct evidence of surface states. A number of authors have studied the conductance signatures of topologically protected surface states of NCS [22–24, 31, 45, 80, 84, 85, 91, 100–103].

The simplest model of a conductance experiment consists of a junction between a topological superconductor and a normal metal with an insulating barrier at their interface. The conductance σ_S(E) at zero temperature is given by a generalization of the familiar Blonder–Tinkham–Klapwijk formula [101, 104],

$$\begin{equation} \sigma_{S}(E) = \sum_{{\bf k}_\parallel}\left\{1 + \frac{1}{2}\sum_{\sigma,\sigma^{\prime}}\left(|a^{\sigma,\sigma^{\prime}}_{{\bf k}_\parallel}|^2 - |b^{\sigma,\sigma^{\prime}}_{{\bf k}_\parallel}|^2\right)\right\}, \end{equation} \tag{ 39 }$$

where $a^{\sigma,\sigma^{\prime}}_{{\bf k}_\parallel}$ and $b^{\sigma,\sigma^{\prime}}_{{\bf k}_\parallel}$ are the Andreev and normal reflection coefficients, respectively, for spin-σ electrons injected into the superconductor at interface momentum k_∥. In the presence of SOC, both spin-preserving (σ' = σ) and spin-flip (σ' = −σ) Andreev and normal reflection coefficients are included. The scattering coefficients are determined by solution of the BdG equation for an electron at bias energy E = eV injected into the superconductor from the normal lead. The insulating tunneling barrier is commonly modeled as a δ-function with strength U = ZE_F/k_F [78]. The value of the dimensionless barrier parameter Z defines different experimental regimes: small values of Z < 1 are least sensitive to surface states, and are typically used to model point contact Andreev spectroscopy experiments [105]; larger values Z ≳ 1 give clear signatures of the surface states and are characteristic of tunnel junctions and scanning tunneling microscopy. We will concentrate upon the latter regime.

Tunneling conductance spectra at the (1 0 0) and (1 0 1) interfaces of a C_4v NCS are shown in figure 8. While the majority-singlet cases (r > 0.5) show a clear gap feature, majority-triplet pairing (r < 0.5) is characterized by substantial subgap conductance due to resonant tunneling through the surface states. At the (1 0 0) interface, the dispersing arc states give a broad feature centred at zero bias which completely fills the gap. This is also present at the (1 0 1) interface, but here the most striking aspect is the sharp zero-bias conductance peak due to tunneling into the zero-energy flat-band states [78]. The sharp zero-bias peak is a robust indicator of the flat-bands, as it arises from their singular contribution to the surface density of states. On the other hand, fine structures in the dispersing states give rise to a much richer variation in their contribution to the conductance, and there is no universal signature [23, 80]. In particular, there is no way to distinguish the dispersing surface states of a NCS from those of a Weyl superconductor using simple tunneling conductance measurements.

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Many variations on the basic junction geometry are possible, e.g. replacing the normal lead by a metallic ferromagnet [93, 103, 106], or assuming a spin-active barrier [85]. This allows conductance measurements to probe the spin structure and, indirectly, the degeneracy of the surface states. Applying a Zeeman field to the sample is predicted to shift the spin-polarized flat bands away from zero energy, thus splitting the zero-bias peak [24, 57, 107]. The absence of splitting for certain field orientations would strongly suggest nondegenerate bands, as doubly-degenerate bands are expected to be spin degenerate and split for any orientation of the applied field [108]. On the other hand, the stability of nodal NCSs in such Zeeman fields is uncertain [109, 110]. A less invasive variation on this approach is to replace the insulating tunneling barrier with a ferromagnetic insulator [85, 108].

The topological classification of nodal superconductors is only approximately valid in the presence of disorder. Even for a pristine bulk system, however, disorder at the surface or at the interface in a tunneling junction is inevitable. So long as the disorder is not too strong the conductance features discussed above remain detectable, although they are broadened by the finite lifetime of the quasiparticles [31, 111–114]. Extremely disordered surfaces produce 'dead' insulating layers, which redefine the edge of a superconductor [113–115]. This is illustrated in figures 9(a) and 9(b), which show the layer-resolved density of states of a (d_xy+p)-wave NCS [45] on a square lattice in the presence and absence of strong edge disorder, respectively. For very strong edge impurities, the states in the outermost layer are almost fully localized, while in the second and third inward layers new weakly disordered states reemerge.

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A key signature of the chiral surface states of Weyl superconductors is that they show anomalous spin and thermal Hall conductance in the direction of their propagation [49, 116]. Although the zero-temperature Hall conductances are directly related to the Chern number, they also depend upon the number of chiral edge modes in the system, and so they are proportional to the separation of the Weyl points in momentum space. This is in contrast to the universal quantization in a fully-gapped time-reversal-symmetry breaking topological superconductor [117–119]. Furthermore, the spin Hall conductance in the triplet Weyl superconductor depends on the orientation of the magnetic field with respect to the d-vector, as not all spin directions are conserved [119]. While the measurement of the anomalous spin Hall conductance is likely to be very difficult [117], the anomalous thermal Hall conductance is predicted to be experimentally accessible for realistic materials [44, 120].

The chiral surface states may also carry a surface charge current, but it is not topologically protected since charge is not conserved in a superconductor. In particular, its detection is extremely difficult as it will be compensated for by Meissner screening currents [121]. While the current carried by the bound states is localized within a few coherence lengths ξ of the surface, however, the screening currents build up on the scale of the penetration depth λ, and so in extreme type-II superconductors a finite surface current density is expected. Precision magnetometry has so far failed to detect surface charge currents in Sr₂RuO₄ [122], despite otherwise compelling evidence that this is a chiral p-wave superconductor [123, 124]. The charge current is not universally related to the Chern number, however, and even ignoring the Meissner effect, it depends sensitively upon microscopic details of the superconductor [125, 126].

Surface currents might also arise in time-reversal invariant NCS. Ignoring SOC, the simplest model of a majority-triplet C_4v NCS is a superconducting analogue of the quantum spin Hall insulator [20], and the helical edge states therefore carry a surface spin current [86, 90, 127]. On the other hand, a surface spin current carried by states above the gap is present in the nontopological phase [89], and the spin current is strongly suppressed by the SOC [90]. Charge currents are also predicted to arise at the interface between NCSs and ferromagnets [83, 128, 129]. In particular, at surfaces with spin-polarized flat bands, the coupling to the exchange field of the ferromagnet generates a chiral electronic structure at the interface. The resulting charge current is very pronounced at low temperatures and shows nonanalytic dependence on the exchange field strength [83, 128].

There are a number of proposals for evidencing chiral or helical edge states based upon interference effects in the magnetoconductance of superconductor loops [130, 131], or nonlocal conductance at domain walls in Weyl superconductors [132]. Although such experiments could give unambiguous demonstration of the edge states, these proposals assume a fully-gapped pairing state, so it is unclear if they would apply to a nodal system.

The electromagnetic response of surface states may differ significantly from the bulk, which can be exploited as an experimental test of their existence. In particular, the flat bands at the (1 1 0) surface of the cuprate superconductors show a paramagnetic response [133] to an applied magnetic field, which has been detected in tunneling [134] and penetration depth [135, 136] measurements. Although there is also a bulk contribution to this so-called nonlinear Meissner effect from the nodal quasiparticles [137], this has a very different temperature dependence compared to the surface state contribution [138], and recent experiments have been able to distinguish the two [139]. The nonlinear Meissner effect may therefore be a useful probe of the surface physics of other nodal topological superconductors.

The surface states of a topological superconductor could be more directly demonstrated using quasiparticle interference spectroscopy [66, 140], which has already been successfully used to evidence topological insulator surface states [141]. This can probe the spin character of the surface states, as scattering between states with opposite spin polarization (i.e. backscattering processes) are suppressed, unless the scattering is by magnetic impurities. Predictions [66] for NCS arc states are shown in figure 10: whereas nonmagnetic scattering gives a weak and diffuse signal (panel (a)), magnetic scattering produces a strong and well-defined image of the arc states (panel (b)).

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A nodal spin-triplet pairing state has been established unambiguously in superfluid ³He [52, 54]. That is, the A phase of ³He is known to be described by the Anderson-Brinkman-Morel (ABM) state [53], a three-dimensional superfluid with nodal points at the north and south poles of the Fermi sphere. The ABM state is a member of symmetry class D with codimension p = 3 (see section 3.3.1). Hence, according to the classification of table 1, the point nodes of ³He-A are topologically stable and protected by a two-dimensional Chern number [25–28].

Here, however, we want to focus on nodal topological superconductors rather than superfluids. A very promising group of materials for nodal topological superconductivity are the noncentrosymmetric superconductors (NCSs) [143, 144]. These systems have extremely diverse properties, however, as they are defined by their crystal structure as opposed to chemical composition. In particular, the parity-mixing of the gap depends upon microscopic details. The most interesting case for us is when the spin-triplet and spin-singlet components are comparable in magnitude, which generically leads to a nodal superconducting state as discussed in section 3. Sizable spin-triplet pairing components can be expected to occur in strongly correlated NCSs where the onsite Coulomb repulsion suppresses the spin-singlet s-wave channel, and where there are strong noncentrosymmetric spin fluctuations [145–147]. These conditions are believed to hold in the heavy fermion C_4v point group NCS CePt₃Si [148], leading to an (s + p)-wave state [146]. Indeed, there is significant evidence of line nodes and an unconventional spin pairing state in this material [149–152]. A similar pairing state seems to be realized in CeIrSi₃ [153, 154] and CeRhSi₃ [155], which are also heavy fermion NCS with C_4v point group. Not all NCS with a sizable spin-triplet gap are so strongly correlated: Replacing Pd by Pt in the O point group NCS Li₂(Pd_1−xPt_x)B appears to tune the system from a conventional s-wave superconductor to a nodal triplet superconductor [156–158], but there is no evidence that the correlation strength is dramatically altered by this substitution. The gap structure of LaNiC₂(C_2v point group) is controversial [159–161]. Muon spin rotation experiments evidence time-reversal-symmetry breaking in LaNiC₂ [159] and the related (centrosymmetric) LaNiGa₂ [162], suggesting nonunitary triplet pairing.

A number of Uranium-based heavy fermion compounds show strong evidence of unconventional triplet pairing states with nontrivial topological properties [175–178]. Among them, UPt₃ is probably the most promising candidate for spin-triplet superconductivity, and has received much attention [171]. This material has three different nodal superconducting phases, and there is strong evidence that one of these realizes a time-reversal-symmetry breaking chiral f-wave state, which is predicted to show Weyl nodes [120, 179]. The superconductivity of UBe₁₃ is less well-understood, but evidence of surface states has been reported [180]. Intriguingly, some of these systems, in particular URhGe, UCoGe, and UGe₂, show an apparently cooperative coexistence of ferromagnetism and superconductivity [175, 178]. That is, close to a ferromagnetic quantum critical point ferromagnetic spin fluctuations mediate spin-triplet pairing. The natural pairing state for such systems would involve equal-spin-pairing |↑↑〉 and |↓↓〉 on the majority- and minority-spin Fermi surfaces, respectively. The crystal space group of most ferromagnetic superconductors is of low symmetry, which results in strong anisotropies of the magnetic fluctuations, favourable to the observed nodal pairing states. Topological properties of these systems are largely uninvestigated, although the existence of arc states connected with Weyl nodes has been proposed [140].

The relative rarity of intrinsic spin-triplet superconductors has led to a number of proposals to engineer such states in heterostructures involving spin-singlet superconductors [49, 50, 82, 181, 182]. A key to these proposals is the presence of strong SOC, which, under appropriate conditions, can accomplish a change in parity of the Cooper pairs, resulting in a sizable proximity-induced spin-triplet gap. For example, superlattices of topological insulators and s-wave superconductors are proposed to realize a p-wave Weyl superconductor [49, 50]. Incorporating high-temperature superconductors into such devices has the great advantage that the engineered spin-triplet superconductor could be realized at much higher temperatures than their intrinsic counterparts [82, 181, 182]. Particularly interesting is proximity-induced pairing in the surface states of a topological insulator [13, 181]. Recently, cuprate superconductor-topological insulator devices have been fabricated [183, 184]. However, the proximity-induced gap in these devices appears to be nodeless.

High-temperature cuprate superconductors are probably the most extensively studied unconventional superconductors [142]. While there is no common consensus on the pairing mechanism, it is by now generally accepted that the pairing symmetry in the cuprates is of a $d_{x^2-y^2}$-wave form [35, 36]. That is, the superconducting gap changes sign along the Fermi surface, giving rise to four topologically protected point nodes. The superconducting state of the cuprates belongs to symmetry class CII in table 1, since it preserves both time-reversal and SU(2) spin-rotation symmetry. Hence, it follows from the classification of table 1, that the point nodes are protected by a $2 \mathbb{Z}$ topological number, i.e. by a winding number which takes on the values ±2. Due to the bulk-boundary correspondence, the topological characteristics of these point nodes lead to the appearance of doubly degenerate (i.e. spin-degenerate) flat band edge states [33, 34]. These zero-energy edge modes have been observed in tunneling experiments on the cuprate material YBa₂Cu₃O_6−x [94–96].

A superconducting gap with d-wave symmetry also occurs in the Cerium-based heavy fermion compounds CeCu₂Si₂ [191] and CeXIn₅, where X is a transition metal (Co, Ir, Rh) [37, 38]. In all these materials superconductivity appears close to an antiferromagnetic quantum critical point and is believed to be mediated by spin fluctuations [142, 177, 178]. In contrast to the ferromagnetic superconductors, however, magnetic order in the Cerium compounds competes with superconductivity. NMR measurements indicate spin-singlet pairing and in combination with specific heat and thermal conductivity measurements yield evidence for line nodes in the superconducting gap. Therefore, it seems likely that the order parameter symmetry in these heavy fermion superconductors is of a d-wave form. This has been confirmed for CeCoIn₅ by a careful analysis of the quasiparticle interference patterns observed in scanning tunneling spectroscopy [39, 40].

The table 3 lists two compounds which show signs of unconventional spin-singlet superconductivity with broken time-reversal symmetry. The first one is URu₂Si₂ [192], in which superconductivity emerges out of a somewhat mysterious 'hidden order phase'. Recently, a polar Kerr effect has been observed, which appears with the onset of superconductivity [188]. Together with specific heat [42] and thermal conductivity measurements [41], this suggests that URu₂Si₂ is a chiral d-wave superconductor, which breaks time-reversal symmetry. Such a state can support Weyl nodes [44]. The second candidate for time-reversal symmetry breaking superconductivity is the pnictide material SrPtAs. Muon spin-rotation measurements on SrPtAs have revealed time-reversal-symmetry breaking in the superconducting state [47]. This material is also predicted to realize a chiral d-wave state with Weyl nodes [48].

Artificial superconducting structures also hold much promise in realizing gapless topological singlet superconductors. Recently, superlattices of the $d_{x^2-y^2}$-wave superconductor CeCoIn₅ and the non-superconducting metal YbCoIn₅ have been grown, such that inversion symmetry is broken in the CeCoIn₅ layers [189]. This material is predicted to realize a nodal NCS with nondegenerate flat bands [45, 102]. Another intriguing compound is Cu_x(PbSe)₅(Bi₂Se₃)₆, where the doped topological insulator Cu_xBi₂Se₃ is combined with the topologically trivial insulator PbSe [190]. Cu_xBi₂Se₃ has attracted much attention as a candidate topological superconductor with a full (odd-parity) gap [193–195]. Surprisingly, specific heat measurements appear to indicate that the superconducting gap in Cu_x(PbSe)₅(Bi₂Se₃)₆ has line nodes. Note that we have conservatively classified the pairing as spin singlet, although the upper critical field hints at a more exotic gap symmetry. More speculatively, s-wave singlet superconductivity in doped Weyl semimetals is predicted to realize a Weyl superconductor [196], which may possess exotic crossed arc surface states [197].