Symmetry-based indicators for topological Bogoliubov–de Gennes Hamiltonians

We develop a systematic approach for constructing symmetry-based indicators of a topological classification for superconducting systems. The topological invariants constructed in this work form a complete set of symmetry-based indicators that can be computed from knowledge of the Bogoliubov–de Gennes Hamiltonian on high-symmetry points in the Brillouin zone. After excluding topological invariants corresponding to the phases without boundary signatures, we arrive at a natural generalization of symmetry-based indicators [H. C. Po, A. Vishwanath, and H. Watanabe, Nat. Commun. 8, 50 (2017)] to Hamiltonians of Bogoliubov–de Gennes type.

Figure 1

Topological band labels associated with an atomic-limit superconductor are calculated as the difference of band labels of arrays with the 0D generator Hamiltonian $H_{\mathrm{gen}}$ and with the 0D reference Hamiltonian $H_{\mathrm{ref}}$ at Wyckoff position $x$. The figure shows this schematically for the Wyckoff position $x=\frac{1}{2}$.

Figure 2

Topological phases of a two-dimensional superconductor in tenfold-way class D and with additional $C_4$ symmetry with one-dimensional representation $\mathrm{\Theta }=A,B$. For each boundary signature, the boundary subgroup sequence (top row) and the nonzero symmetry-based indicators for a generator of that phase are given (middle row).

Figure 3

The only topological phase with nontrivial boundary signature for a two-dimensional superconductor in tenfold-way class D and with additional $C_4$ symmetry with one-dimensional representation $\mathrm{\Theta }=\mathrm{\Theta }={}^{1,2}E$ is a Chern superconductor with an even number of chiral Majorana boundary modes. The boundary subgroup sequence for this phase is given in the top row; the symmetry-based indicator for a generator of the phase is given in the middle row.

Figure 4

Topological phases of a three-dimensional superconductor in tenfold-way class D and with additional $C_i$ symmetry with one-dimensional representation $\mathrm{\Theta }=A_g$. For each boundary signature, the boundary subgroup sequence (top row) and the nonzero symmetry-based indicators for a generator of that phase are given (middle row).

Figure 5

Topological phases of a three-dimensional superconductor in tenfold-way class D, with additional $C_i$ symmetry and one-dimensional representation $\mathrm{\Theta }=A_u$. For each boundary signature, the subgroup sequence and the symmetry-based indicators of the generators of that phase are given. The third-order topological superconductor with $z_3=4$ can be constructed as the direct sum of two second-order topological superconductors with $z_3=2$.

Figure 6

Topological phases of a three-dimensional superconductor in tenfold-way class DIII, with additional $C_i$ symmetry and one-dimensional representation $\mathrm{\Theta }=A_u$. For each boundary signature, the subgroup sequence and the symmetry-based indicators of the generators of that phase are given. The two-dimensional second-order topological superconductor with $z_{2;l}=2$ can be constructed as the direct sum of two two-dimensional first-order topological superconductors with $z_{2;l}=1$, $l=x,y,z$. Similarly, the three-dimensional third-order topological superconductor with $z_3=4$ can be constructed as the direct sum of two three-dimensional second-order topological superconductors with $z_3=2$.

Figure 7

Topological phases of a three-dimensional superconductor in tenfold-way class C, with additional $C_i$ symmetry and one-dimensional representation $\mathrm{\Theta }=A_u$. For each boundary signature, the subgroup sequence and the symmetry-based indicators of the generators of that phase are given.

Figure 8

In tenfold-way class CI with additional $C_i$ symmetry and one-dimensional representation $\mathrm{\Theta }=A_u$, there is a single topological phase with first-order boundary signature consisting of pairs of Majorana Dirac cones. The subgroup sequence and the symmetry-based indicator of the generator of this phase is given.

Figure 9

Topological phases of a three-dimensional superconductor in tenfold-way class D and with translation symmetry only (point group $C_1$). For each boundary signature, the boundary subgroup sequence (top row) and the nonzero symmetry-based indicators for a generator of that phase are given (middle row).

Figure 10

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_s$ and representation $\mathrm{\Theta }=A^{\prime }$.

Figure 11

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_s$ and representation $\mathrm{\Theta }=A^{″}$.

Figure 12

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_2$ and representation $\mathrm{\Theta }=A$.

Figure 13

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_2$ and representation $\mathrm{\Theta }=B$.

Figure 14

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_4$ and representation $\mathrm{\Theta }=A$ or $B$.

Figure 15

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_4$ and representation $\mathrm{\Theta }={}^{1}E$ or ${}^{2}E$.

Figure 16

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_{2v}$ and representation $\mathrm{\Theta }=A_2$. The parity of the gapless states under corresponding mirror symmetry is denoted by $\pm$.

Figure 17

Topological phases of a three-dimensional superconductor in tenfold-way class D with point group $C_{2v}$ and representation $\mathrm{\Theta }=B_2$. The parity of the gapless states under corresponding mirror symmetry is denoted by $\pm$.

Figure 18

(a) A $C_{2v}$-symmetric two-dimensional crystal with two perpendicular mirror symmetries with gapped edges may be deformed into four one-dimensional chains in a crosslike arrangement. Corner states of the two-dimensional crystal are in one-to-one correspondence with end states of the one-dimensional structure. (b) A $C_{2v}$-symmetric crystal may be decorated symmetrically with Kitaev chains, resulting in the appearance of extrinsic pairs of corner states at all four mirror-symmetric corners.

Figure 19

(a) Support of Majorana bound states for the Hamiltonians $H_{2,\pm }^{\prime }$ of Eq. (B12). A symmetry-preserving next-nearest-neighbor term $m_1{\rho }_2{\tau }_{1}sin{k}_{x}sin{k}_y$ with $m_1=0.4$ is added to remove a spurious gapless edge mode. (b) Support of Majorana bound states for $H_{2,+}^{\prime }\oplus H_{2,-}^{\prime }$, after adding an additional weak symmetry-preserving hybridization term $m_2{\rho }_0{\tau }_1{\mu }_2$ with strength $m_2=0.1$. The corner modes may appear at the other pair of corners if a different hybridization term is chosen.

Figure 20

A $C_4$-symmetric two-dimensional crystal may be deformed into four one-dimensional chains in a crosslike arrangement. Corner states of the two-dimensional crystal are in one-to-one correspondence with end states of the one-dimensional structure.

Figure 21

(a) A $C_{2v}$-symmetric three-dimensional crystal with two perpendicular mirror symmetries may be deformed into four two-dimensional planes in a crosslike arrangement. Hinge states of the two-dimensional crystal are in one-to-one correspondence with edge states of the two-dimensional structure. (b) A $C_{2v}$-symmetric crystal may be decorated symmetrically with quantum Hall planes, resulting in the appearance of extrinsic configurations of chiral hinge modes at all four mirror-symmetric hinges.

Figure 22

(a) Decoration consisting of four copies of one-dimensional Kitaev chain. This decoration results in two Majorana fermions with angular momentum $j=\frac{1}{2}$ and $\frac{5}{2}$. (b) Decoration consisting of four copies of two-dimensional $p$-wave superconductors, resulting in single Majorana fermion with angular momentum either $j=\frac{1}{2}$ or $\frac{5}{2}$.