Anisotropic magnetic responses of topological crystalline superconductors

Majorana Kramers pairs emerged on surfaces of time-reversal-invariant topological crystalline superconductors show the Ising anisotropy to an applied magnetic field. We clarify that crystalline symmetry uniquely determines the direction of the Majorana Ising spin for given irreducible representations of pair potential, deriving constraints to topological invariants. Besides, necessary conditions for nontrivial topological invariants protected by the n-fold rotational symmetry are shown.


I. INTRODUCTION
Topological superconductors are gapped systems hosting gapless states on their surfaces 1-4 as Andreev bound states [5][6][7][8] . The gapless surface states behave as Majorana fermions, which are self-conjugate particles and protected by the topological invariant associated to (broken) symmetries. Due to the stability and the so-called non-Abelian statistics derived from the self-conjugate property, one would expect that topological superconductors can be a platform of the fault-tolerant topological quantum computation 9 .
Shiozaki and Sato 23 have unveiled that the underlying mechanism of Majorana Ising spin is the protection by crystalline symmetry, i.e., an extension from topological crystalline superconductivity [24][25][26][27] . In this paper, we develop the theory and find that the direction of Majorana Ising spin is uniquely determined for a given irreducible representation 28 of the pair potential. The obtained result can be applied to all the space groups hence we believe that it is useful for studies on topological-superconductor materials and experiments.
The paper is organized as follows. We start with a brief review on Majorana Ising spin in Sec. II and clearly summarize issues to be addressed. We derive conditions for nontrivial topological invariant in Sec. III in systems with time-reversal and crystalline symmetries. From the obtained conditions, one finds the direction of Majorana Ising spin and summarizes it in tables (Appendix C). Besides Majorana Ising spin, in Sec. IV, we also derive the winding number corresponding to the surface Majorana fermions protected by n-fold rotational symmetry, in a manner similar to that in Sec. III. An example of the application of our general theory is shown in Sec. V. We finally summarize the paper in Sec. VI.

II. PRELIMINARY
Before going into the main discussion, we first review zero-energy states and the associated topological invariants in superconductors. A BdG Hamiltonian H(k) has the form in the basis of (c ↑ (k), c ↓ (k), c † ↓ (−k), −c † ↑ (−k)), where ↑ and ↓ denote the spin up and down, respectively, and the spin indices in h(k) and ∆(k) are implicit. Note that one can choose ∆(k) = ∆(k) † for time-reversal-invariant superconductors. The Hamiltonian preserves time-reversal T symmetry and particle-hole C symmetry arXiv:1611.09642v2 [cond-mat.mes-hall] 20 Feb 2017 Combining these symmetries, chiral symmetry holds; Next, we introduce the topological invariant corresponding to the number of zero-energy states on the surface, which are located on x ⊥ = 0 [ Fig. 1(a)]. The time-reversal-invariant momentum at which the zero-energy states appear is set to k = Γ [ Fig. 1(b)]. The one-dimensional topological invariant W 29 is given by where H(k ⊥ ) = H(k ⊥ + 2π/a ⊥ ). This invariant is equal to the number of the zero-energy surface states (see Appendix A). In time-reversal-invariant spinful systems, however, the above topological invariant always vanishes owing to timereversal symmetry [10][11][12]30 , which requires {T , Γ} = 0 and using T H(k)T −1 = H(−k), hence W = 0. The topological invariant can take a finite value with the help of an order-2 symmetry operation U that involves the spin and respects the surface: [U, H(k ⊥ )] = 0. Now we introduce a modified chiral operator Γ U as where the phase φ U is chosen to satisfy Γ 2 U = 1. The modified topological invariant W [U ] is given by replacing Γ with Γ U ; which is free from the condition of Eq. (6) when the following condition is satisfied; Normally, order-2 symmetry operations stem from crystalline point/space-group symmetries such as two-fold rotations and reflections with respect to the x ⊥ axis. This means that systems with W [U ] = 0 are interpreted as a onedimensional topological crystalline superconductor. In the last part of this section, we review that W [U ] naturally explains the Ising-anisotropic response to a magnetic field 21,23 . The symmetry operation U is taken to be a two-fold rotation or a reflection. The symmetry operation U flips or keeps the direction of applied magnetic field, i.e., {U, H mag } = 0 or [U, H mag ] = 0, respectively. Here H mag denotes the Hamiltonian of magnetic field including the Zeeman and vector potential terms. These operations are summarized in Table I  I. Symmetry of magnetic field B applied along the x ⊥ , x 1 , and x 2 directions, which are depicted in Fig. 1. C2(x ⊥ ) is the two-fold rotation along the x ⊥ axis. σ(xixj) is the mirror reflection with respect to the xixj plane. These are symmetry operations of the semi-infinite system with the surface of x ⊥ = 0. − (+) indicates that the magnetic field is (not) flipped by the symmetry operation. S denotes the direction of Majorana Ising spin protected by the topological invariant W [U ] for U = C2(x ⊥ ), σ(x ⊥ x 1 ), and σ(x ⊥ x 2 ). the absence of magnetic field as long as H mag is small enough, while not in the latter case ([U, H mag ] = 0). From Table  I, the latter case is realized only in the case that magnetic field is applied for a specific direction in each symmetry operation. Therefore, the zero-energy surface states protected by W [U ] are annihilated only by the magnetic field along the specific direction. Namely, Majorana fermions on the surface acts as an Ising spin under a magnetic field.
In the following sections, developing the theory, we show only one winding number among , and W [σ(x ⊥ x 2 )] is possible to take a finite value for a given surface and an irreducible representation of pair potential, i.e., the anisotropy of magnetic response is uniquely determined, irrespective of the details of the system.

III. TOPOLOGICAL INVARIANTS FOR IRREDUCIBLE REPRESENTATIONS
We show that only one among three possible topological invariants can become finite in a given superconducting pair potential. This is the decisive evidence of Majorana Ising spin.

A. Symmetry of crystalline systems including a surface
In crystalline systems, all the symmetry operations other than time reversal and particle-hole transformation are elements of a space group. Here we focus on the momentum line including the time-reversal-invariant momentum k = Γ along which the one-dimensional topological invariant is defined. Symmetry operations respecting the k = Γ point are (screw) rotation and (glide) reflection, which are classified into those preserving (type-I) and inverting (type-II) the surface of x ⊥ = 0. The type-I symmetry operations are two-fold (screw) rotation C 2 (x ⊥ ) along the x ⊥ axis and mirror (glide) reflections σ(x ⊥ x 1 ) with respect to x ⊥ x 1 plane and σ(x ⊥ x 2 ) with respect to x ⊥ x 2 plane [ Fig. 2(a)]. The type-II symmetry operations, on the other hand, are two (screw) rotations [C 2 (x 1 ) and C 2 (x 2 )] and one mirror (glide) reflection [σ(x 1 x 2 )], as shown in Fig. 2(b). Afterwards, we denote a type-I operation by U i then we have The spatial inversion I is represented in terms of U i as for i = 1, 2, 3, where P i is a type-II symmetry operation, i.e.,

B. Symmetry operations in superconducting states
Now define symmetry operations in a superconductor. A superconductor keeps a crystal symmetry S (U i or P i ) where the pair potential ∆(k) is a one-dimensional representation of S; where χ(S) is the character of the one-dimensional representation and k is the momentum transformed by S. Then, the symmetry operationS in the superconducting state is defined as In these cases, one obtains the following relationS † ΓS = χ(S)Γ, for Γ = τ y [see Eq. (4)].

C. Topological invariant
In the following, we derive necessary conditions for finite-valued topological invariants, which is defined by x ⊥ indicates the direction normal to the surface, i.e., W [Ũ i , x ⊥ ] is the number of the Majorana zero modes on the surface perpendicular to x ⊥ . Note that the type-II symmetries P i may define an topological invariant but it does not correspond to the zero-energy surface states since the surface is not invariant against P i . This is why only the type-I symmetries U i are considered here. Glide reflection along the direction parallel to the surface, e.g., a-glide with respect to the ac plane for the ab surface, is one of the possible type-I U i symmetries for the winding number. Screw rotation, however, is not used for the winding number because the surface is not invariant by the operation. Glide reflection that translates a system along the direction normal to the surface and screw rotation may define a bulk invariant although the bulk-edge correspondence does not hold, as type-II P i symmetry. Henceforth, for the rotational symmetries, we suppress the suffix , due to the uniqueness of the directions of the integrals, i.e., x ⊥ must be along the rotational axis. Now we derive the constraint to W [Ũ i , x ⊥ ] by the symmetries. One gets These equations are derived by applying unitary transformations byŨ l and byP l . Here we introduce p(A, B) as Note thatŨ l includes the n-fold rotation (if exist) in addition to the two-fold rotations. In consequence, the conditions of and of [T , ΓŨ i ] = 0 [Eq. (9)] are necessary for W [Ũ i , x ⊥ ] = 0. From the above condition, χ(U i ) = 1 is derived because of p(U i , U i ) = 1.
Next, we prove that the two-fold symmorphic symmetry operations, rotations and reflections, satisfy the condition of Eq. (9) while the nonsymmorphic ones, glide reflections, do not on the Brillouin zone boundary. Symmetry operations are represented by the direct product of real-space part O i and spin part Σ i , U i = O i Σ i . For two-fold rotations and mirror reflections, the real-space part O i is an orthogonal matrix with O 2 i = 1 and [O i , T ] = 0. The spin part is given by Pauli matrices then Σ † i = Σ i , Σ 2 i = 1, and {Σ i , T } = 0. As a result, the chiral operator is given by ΓŨ i =Ũ i τ y so that the condition of Eq. (9) holds. For glide reflections, on the contrary, the orbital part O i on the Brillouin zone boundary is purely-imaginary matrix hence the condition Eq. (9) is not satisfied, i.e., withŨ i being a glide reflection, where τ is the translation vector of the glide reflection (for details, see Appendix B). For symmorphic space groups, the necessary condition for W [Ũ i , x ⊥ ] = 0 is easily obtained as follows. The commutation relations of the representations for symmetry operations in a point group are uniquely determined to be in spinful systems. With the help of the above relation, the condition Eq. (20) reduces to Here, χ(O) is the character of O hence the possible topological invariant is determined only from the representation theory of point group, irrespective of details of the system, as summarized in the tables in Appendix C. An example for a nonsymmorphic space group is also shown in Appendix C. The condition of χ(U i )χ(P i ) = χ(U i P i ) = −1 is extracted from the above equations. This means that the character of the spatial inversion I = U i P i must be −1 for the existence of topological superconductivity. That is consistent with the absence of time-reversal-invariant Majorana fermion in even-parity superconductors 31 .
is the mirror or glide reflection with respect to the x ⊥ x j plane. The statement is immediately seen from Eq. (23) for symmorphic space groups: χ(U i ) = 1 and χ(U j ) = 1 are not simultaneously satisfied. This is also true at k = 0 for nonsymmorphic space groups since the commutation relations of symmetry operations are the same as those for the symmorphic space group. When σ(x ⊥ x 1 ) is the x 1 -glide reflection, the commutation relation changes from the symmorphic one at the boundary k 1 = π/a 1 . W [σ(x ⊥ x 1)], however, vanishes from Eq. (21). In consequence, it is impossible that two of W [C 2 (x ⊥ )], W [σ(x ⊥ x 1 )], and W [σ(x ⊥ x 2 )] simultaneously take nontrivial values.

IV. WINDING NUMBER PROTECTED BY n-FOLD ROTATIONAL SYMMETRY
Besides order-2 symmetries, we clarify the winding number protected by the n-fold (n ≥ 3) rotational C n symmetry, [C n , H(k)] = 0. We derive the necessary condition for nonzero topological invariant associated with C n for spinful systems. The spinless case was discussed in Ref. 32 .
A. Definition C n is represented by C n = e −ijz2π/n , where j z denotes the total angular momentum along the rotational axis. For spinful systems, C n n = −1 and the eigenvalue of C n is obtained to be e −iµ2π/n for µ = 0, · · · , n − 1. A Hamiltonian of C n -symmetric system is block diagonalized to be where Hamiltonian in the C n = e −iµ2π/n eigenspace. g µ is the degeneracy of the eigenvalue of e −iµ2π/n then n−1 µ=0 g µ = dim H(k). In a superconductor with the n-fold rotational symmetry, [C n , H(k)] = 0, chiral symmetry in the eigenspaces is found when [Γ,C n ] = 0 holds: Hereafter we assume that the pair potential is the A representation of C n , i.e., ∆(k) = C † n ∆(k )C n , because [Γ,C n ] = 0 holds only in this case. The winding number in each eigenspace is which corresponds to the number of zero-energy end states of H µ .

B. Time-reversal symmetry
Since the angular momentum is time-reversal odd, one finds and These lead to As a result, one finds The above relation is a natural extension from Eq. (6).

C. Spatial symmetry
The commutation relation of C n and spatial symmetries, U l and P l , is given by [U l , C n ] = 0 for [j z , U l ] = 0 and U † l C n U l = C † n for {j z , U l } = 0. The same equations hold for P l . This gives the transformation of Hamiltonian; H µ (k) = U † l,p(jz,U l )µ H p(jz,U l )µ (k)U l,p(jz,U l )µ , H µ (k) = P † l,p(jz,U l )µ H p(jz,U l )µ (−k)P l,p(jz,U l )µ , where U l,µ = V † µ U l V µ and P l,µ = V † µ P l V µ . The chiral operator is transformed by U l,µ as This is the same for P l,µ . The winding numbers satisfy the following relations; Combining these and Eq. (30), one finds a necessary condition for W µ = 0. In symmorphic space groups, from the above conditions, W µ takes a finite value only for the A 1u (or its compatible) representation. The (anti)commutation relations of j z and the symmetry operations, U 1 = C n , U 2 = σ(x ⊥ x 1 ), U 3 = σ(x ⊥ x 2 ), P 1 = σ(x 1 x 2 ), P 2 = C 2 (x 1 ), and P 3 = C 2 (x 2 ), are given by From this and Eq. (34), the necessary condition is given by This holds for the A 1u representation of the pair potential.  As an example, we show the magnetic response of Majorana Ising spin in the bilayer Rashba superconductor 33 , which are depicted in Fig. 3. The Hamiltonian in the normal state reads where s and σ denote the Pauli matrices representing the spin and layer degrees of freedom, respectively. Timereversal-invariant Bogoliubov-de Gennes (BdG) Hamiltonian has the form in the basis of (c k↑ , c k↓ , c † −k↓ , −c † −k↑ ), where the arrows ↑ and ↓ denote the up and down spins, respectively. When the Fermi level µ is located within the hybridization gap, as shown in Fig. 3(b), the Z 2 topological invariant takes the nontrivial value 33 . The above Hamiltonian is regularized on the square lattice as We consider the six types of odd-parity pair potentials, which are summarized in Table II. The corresponding finitesized Hamiltonian defined in (1 ≤ x ≤ N x ) along the x direction is given by with and∆ We calculate the energy spectrum in the presence of a Zeeman field along x, y, and z directions, which are expressed by the Hamiltonian For topological superconducting states, the Majorana zero modes still remain gapless in the presence of a magnetic field perpendicular to the Majorana Ising spin. The direction of the Majorana Ising spin for each pair potential is derived by the general theory studied in the previous section and shown in Table II. This is verified by the numerical results, which are shown in Fig. 4. For the A 1u pairing, Majorana zero modes exist at k y = 0 for the cases of B y and of B z, while they vanish and a gap is generated for the case of B x. For the A 2u pairing, Majorana zero modes vanish only for the case of B y. Note that the A 2u -pairing state under a magnetic field along the z direction is the same as the pair-density-wave (PDW) state studied in Ref 34,35 . There is no Majorana zero mode for the B 1u pairing because the bulk superconducting gap closes at k y = 0. For the B 2u pairing, on the other hand, the bulk superconducting gap closes at k y = 0 and remains finite at k y = 0. Hence Majorana zero modes are emerged at k y = 0 and killed by a magnetic filed along the y direction. The E u pairings are similar to the B 1u and B 2u pairings. The bulk gap vanishes at k y = 0 for the E u (x) pairing but survives for the E u (y) paring. The emerged Majorana zero modes are gapped only when a magnetic field is applied along the z direction. Namely, A 1u , A 2u , B 2u , and E u (y) pair potentials, Majorana zero modes vanish for a specific direction of magnetic field, i.e., the Majorana zero modes respond to the field as a Ising spin. This results on the Majorana Ising spins totally coincide with those in Table II.

VI. CONCLUSION
We have derived possible nonzero topological invariants and the direction of Majorana Ising spin for each irreducible representation of pair potential for time-reversal-invariant superconductors. The obtained result is the detailed classification in the class-DIII superconductors in one spatial dimension. Another point of view is a topological extension to the classification of superconducting pair potential. Our result is general and does not depend on the detail of systems therefore it is useful for all researchers on superconducting materials. The anisotropy can be detected by the tunneling spectroscopy under a magnetic field or with a ferromagnetic junction because a zero-bias peak appears in the presence of the Majorana zero modes 36 . Several examples have been shown for the bilayer Rashba superconductor, D 4h , C 4v , C 2v point groups, and the P mma space group. In the other point groups, D 6h , D 3d , and D 3h , another type of anisotropic response can arise. This will be demonstrated in a separate paper. As for the topological invariants, we focused only on the Z topological invariant in the paper. To complete the classification, we also need to clarify the Z 2 topological invariant and the related Majorana Ising spins. This issue will be also addressed in a future paper.
Multiplying τ 3 to the above equation, one finds that the zero-energy states are chirality eigenstates, i.e., φ is given by φ = φ τ χ τ for τ 2 χ τ = τ χ τ with τ = ±1 chirality. φ τ is obtain by solving q τ (λ)φ τ = 0 with For a nontrivial solution of φ τ , the secular equation det q τ (λ) = 0 holds and has [r dim H(k)] solutions. As a result, the fundamental solutions for the zero-energy end states are obtained to be The number of the independent solutions for the definite chirality τ is |Q τ |.
So as to obtain the physical solutions with a definite chirality τ , a boundary condition is imposed on the system end n = 0, e.g., the fixed boundary condition ψ τ,q = 0 for q ≥ −r + 1. The boundary condition gives [r dim H(k)/2] conditions. Consequently, one obtains the number of the zero-energy end states with the chirality of τ to be One obtains only the trivial solution, ψ τ,n = 0, for |Q τ | = r dim H(k)/2. det q τ (λ) = 0 is equivalent to det q τ (λ) * = 0, which is explicitly shown as This means that if solution is given by λ for τ = +1 then the solution for τ = −1 is given by λ * −1 . Namely, one finds From the above condition, the possible (N + , N − ) are classified into three cases: (A10)

Bulk-edge correspondence
Next, we calculate the winding number of the translational-invariant system, which is described by The winding number is given by In this way, the basis in which the chiral operator is diagonalized makes it easy to calculate the winding number. Then we first off-diagonalize the Hamiltonian as This reduces the winding number to be where the integral runs over the unit circle in the complex λ plane along the counter-clockwise direction. The last line of the above equation is derived with the use of the argument principle because det q τ (λ) has |Q τ | zeros within the unit circle and is asymptotically given by which has the order-[r dim H(k)]/2 pole. Thus W = −N − for |Q − | > [r dim H(k)]/2. Because W is an integer, the above relation is rewritten as The winding number equals W = N + for |Q + | > [r dim H(k)]/2. We finally arrive at the bulk-edge correspondence: Ising spin protected by W [Ũ i , x ⊥ ] is obtained to be parallel to the rotational axis forŨ i =C 2 or normal to the mirror plane forŨ i = σ, respectively, as explained in Sec. II.