Classification of two-dimensional topological crystalline superconductors and Majorana bound states at disclinations

We classify discrete-rotation symmetric topological crystalline superconductors (TCS) in two dimensions and provide the criteria for a zero-energy Majorana bound state (MBS) to be present at composite defects made from magnetic flux, dislocations, and disclinations. In addition to the Chern number that encodes chirality, discrete rotation symmetry further divides TCS into distinct stable topological classes according to the rotation eigenspectrum of Bogoliubov-de Gennes quasiparticles. Conical crystalline defects are shown to be able to accommodate robust MBS when a certain combination of these bulk topological invariants is nontrivial as dictated by the index theorems proved within. The number parity of MBS is counted by a ${\mathbb{Z}}_2$-valued index that solely depends on the disclination and the topological class of the TCS. We also discuss the implications for corner-bound Majorana modes on the boundary of topological crystalline superconductors.

Figure 1

Brillouin zone of a two-dimensional fourfold symmetric system. The first-descendant weak invariants ${\nu }_{1,2}$ are defined as 1D ${\mathbb{Z}}_2$ indices along the two perpendicular colored lines marked with arrows. The second-descendant weak invariants $\mu \left({\Gamma }_i\right)$ are defined at the four momenta $\Gamma =(0,0)$, $X=(\pi ,0)$, $X^{\prime }=(0,\pi )$, and $M=(\pi ,\pi )$.

Figure 2

Brillouin zones for systems with (a) fourfold, (b) twofold, (c) sixfold, and (d) threefold rotation symmetries and their rotation fixed points. Shaded regions indicate the fundamental domain that generates the entire Brillouin zone upon rotation around the fixed point at the center of the Brillouin zones $\Gamma =(0,0)$.

Figure 3

Rotation eigenvalues at the fixed-point momenta (a) $\Gamma$, (b) $X$, and (c) $M$ in the Brillouin zone of $C_4$-symmetric crystals.

Figure 4

Restrictions on the fourfold rotation invariants due to PH symmetry. Horizontal lines represent bands that have been sorted out according to their corresponding rotation eigenvalue.

Figure 5

Topological phases of model $H_{u_1,u_2}^{\left(4\right)}$ in Eq. (47). At $u_2=u_1$, the gap closes at the fourfold fixed point $M$, at $u_2=-u_1$, the gap closes at $\Gamma$. At $u_2=0$, the gap closes at $X$ and $X^{\prime }$. Chern numbers and weak invariants are shown for each phase. For rotation invariants, see Table 1. Primitive generators $H_1\left(4\right)$ and $H_2\left(4\right)$ we simulated with parameters as shown by the cross and asterisk, respectively.

Figure 6

Energy bands for primitive Hamiltonians (a) $H_1^{\left(4\right)}$ and (b) $H_2^{\left(4\right)}$ for a strip geometry with periodic boundary conditions in the ${\mathbf{a}}_1$ direction and open boundary conditions in the ${\mathbf{a}}_2$ direction. The dashed blue/ dotted red lines correspond to states localized at the upper/lower edges. The parameters are $u_1/\Delta =1,u_2/\Delta =0.5$ for (a), and $u_1/\Delta =-1,u_2/\Delta =0.5$ for (b). Both models have $Ch=1$.

Figure 7

Tight-binding representations of primitive generators that take the form of 2D $p$-wave wires for various rotation symmetries. (a) $H_3^{\left(4\right)}$, (b) $H_4^{\left(4\right)}$, (c) $H_4^{\left(2\right)}$, and (d) $H_3^{\left(6\right)}$. Black dots indicate Majorana fermions. $H_3^{\left(4\right)}$ and $H_4^{\left(4\right)}$ are fourfold symmetric superconductors with four Majorana fermions per site and first- and second-nearest-neighbor connections, respectively. $H_4^{\left(2\right)}$ has the same atomic arrangement as in (a) and (b), but contains only two Majorana fermions per site and is trivial along ${\mathbf{a}}_2=(0,1)$. Gray vertical lines in (c) serve only as a guide and do not represent terms in the Hamiltonian. $H_3^{\left(6\right)}$ is a sixfold symmetric superconductor with six Majorana fermions per site.

Figure 8

Topological phases of model $H^{\left(6\right)}$ in Eq. (66). For rotation invariants see Table 3. Primitive generators $H_1^{\left(6\right)}$ and $H_2^{\left(6\right)}$ were simulated with parameters marked by the cross and asterisk, respectively.

Figure 9

Energy bands for primitive Hamiltonians (a) $H_1^{\left(6\right)}$ and (b) $H_2^{\left(6\right)}$ for a strip geometry with periodic boundary conditions in the ${\mathbf{a}}_1$ direction and open boundary conditions in the $(0,1)$ direction. The dashed blue/ dotted red lines correspond to states localized at the upper/lower edges. The parameters are $u_1/\Delta =1$, $u_2=0$ for (a) and $u_1=0$, $u_2/\Delta =1$ for (b). The Chern invariants are 1 and 3, respectively.

Figure 10

Energy bands for primitive generator with Hamiltonian $H_3^{\left(3\right)}$ for a strip geometry with periodic boundary conditions in the ${\mathbf{a}}_1$ direction and open boundary conditions in the $(0,1)$ direction. The dashed blue/ dotted red lines correspond to states localized at the upper/lower edges. The parameters are $u_1/\Delta =0.5$, and $\mu /\Delta =0.5$. This model has Chern invariant $-$1.

Figure 11

Holonomy of a disclination around a loop (red path) with a fixed starting point (blue dot).

Figure 12

Lattice disclinations and dislocations. (a) and (b) Dislocations in the form of disclination dipoles. (c), (d) $\pm {120}^{\circ }$ disclinations with opposite Frank angles and different translation types.

Figure 13

Fractional vortices bounded at disclinations in a $p$-wave SC. ${\varphi }_0=hc/2e$ is the flux quantum in a SC.

Figure 14

(a) and (b) Lattice cells of $C_4$-symmetric configurations having $-\pi /2$ and $+\pi$ disclinations. Periodic boundary conditions are imposed, by identifying edges on the unit cell with red, blue and black lines. (c)–(f) Flattened cores of $\Omega =-\pi /2$ [(c) and (e)] and $\Omega =+\pi$ [(d) and (f)] disclinations centered at points $O,K,O^{\prime }$, and $K^{\prime }$ in the unit cells. The disclination types are type (1,0) in (c) and (f), type (1,1) in (d), and type (0,0) in (e).

Figure 15

Simulation of primitive model $H_2^{\left(4\right)}$ with the lattice configuration depicted in Fig. 14. (a) Density of states. The zoomed-in centered region of the insulating gap shows two zero-energy states with corresponding probability density functions exponentially localized around the disclination cores $O_1$ (b), and $K$ (c). The lattice cell has $n=20$ sites per side. The parameters used were $2u_2/\Delta =-u_1/\Delta =1$.

Figure 16

Tight-binding model $H_3^{\left(4\right)}$ with (a) type-(0,0) and (b) type-(1,0) disclinations, and model $H_4^{\left(4\right)}$ with (c) type-(0,0) and (d) type-(1,0) disclinations. Thick red dots in disclination cores are unpaired Majorana bound states.

Figure 17

Tight-binding models $H_3^{\left(2\right)}$ (a)–(d) and $H_4^{\left(2\right)}$ (e)–(h) with $\Omega =+\pi$ disclinations. Disclinations are of type-(0,0) in [(a) and (e)], type-(1,0) in [(b) and (f)], type-(0,1) in [(c) and (g)], and type-(1,1) in [(d) and (h)]. For $H_4^{\left(2\right)}$, gray lines serve only as a guide, and are oriented along the trivial $(0,1)$ direction in a system with no disclinations, as in Fig. 7. Thick red dots in disclination cores are unpaired Majorana bound states.

Figure 18

(a) Lattice cell of a $C_6$-symmetric lattice configuration having $\pm \pi /3$ disclinations. Periodic boundary conditions are imposed, by identifying edges on the unit cell marked with the same color of dashed lines. $O_{1,2}$ indicate centers of $-\pi /3$ disclinations. $K_{1,2}$ indicates centers of $+\pi /3$ disclinations. We also show examples of a (b) $-\pi /3$ disclination and a (c) $+\pi /3$ disclination.

Figure 19

Simulation of primitive model $H_2^{\left(6\right)}$ with the lattice configuration depicted in Fig. 18. (a) Density of states. The zoomed-in centered region of the insulating gap shows four zero-energy states with corresponding probability density functions centered at negative disclinations $O_1$, $O_2$ [(b) and (c)], and positive disclinations $K_1$, $K_2$ [(d) and (e)]. The unit cell has $n=24$ sites per side. The parameters used were $u_1/\Delta =0,u_2/\Delta =1$. The splitting of the states near zero energy is due to the finite size of the lattice. We show in Appendix pp5 that the energies exponentially approach zero as the system size is increased.

Figure 20

Tight-binding model $H_3^{\left(6\right)}$ with a $\Omega =-\pi /3$ disclination. The thick red dot in the disclination core represents an unpaired Majorana fermion.

Figure 21

$C_6$-symmetric 2D $p$-wave wires. (a) Second-nearest-neighbor triangular $p$-wave wire. (b) Kagome $p$-wave wire. For easy of visualization, only lattice sites are shown, and not Majorana fermions. Red dots represent sites that harbor unpaired MBS, as they have an odd number of connections.

Figure 22

(a) Lattice cell of a $C_3$-symmetric lattice configuration having $\Omega =\pm 2\pi /3$ disclinations. Periodic boundary conditions are imposed by identifying edges on the lattice cell marked with the same type of dashed lines. $O$ indicates the center of the $\Omega =-2\pi /3$ disclination, and $K$ indicates the center of the $\Omega =+2\pi /3$ disclination. We show examples of an (b) $\Omega =-2\pi /3$ disclination and an (c) $\Omega =+2\pi /3$ disclination.

Figure 23

Simulation of primitive model $H_1^{\left(3\right)}$ with the lattice configuration depicted in Fig. 22. (a) Density of states showing two zero-energy states with corresponding probability density functions centered at the $\Omega =-2\pi /3$ disclination core $O$ (b), and at the $\Omega =+2\pi /3$ disclination core $K$ (c). Superconducting fluxes of $\pm 8\pi /3$ bind the disclinations. The unit cell has $n=18$ sites per side. The Hamiltonian parameters were set to $u_1/\Delta =1$ and $u_2/\Delta =0$.

Figure 24

(a) Schematics of the (unhybridized) Fermi surfaces of the normal metallic phase of Sr${}_2$RuO${}_4$. In the Raghu-Kapitulnik-Kivelson state the $d_{xz}$ and $d_{yz}$ bands (horizontal and vertical red lines) are responsible for superconductivity while the $d_{xy}$ one (dashed blue circle) is ignored [81]. (b) Tight-binding limit of the superconducting $d_{xz}$ and $d_{yz}$ bands. Dashed lines on the edges represent allowed perturbations that will gap the edge Majorana modes and leave an unpaired MBS (red dot) at each corner.

Figure 25

(a) Fermi surface of graphene at filling $\nu \simeq 3/8$ or $5/8$. (b) BdG excitation spectrum of superconducting graphene. $\Delta =1$ eV for solid lines and $\Delta =0$ for shaded ones.

Figure 26

Boundary states of superconducting graphene in a slab geometry terminating along zigzag edges with (a) $s$-wave pairing or (b) $d\pm id$ pairing.

Figure 27

Three-dimensional $p$-wave wire with corner states. (a) Unit cell showing the connections of its Majorana fermions (black dots). (b) Brillouin zone, showing the rotation fixed points and the axes of rotation.

Figure 28

Rotation eigenvalues at the fixed-point momenta (a) $\Gamma$, (b) $X$, (c) $Y$, and (d) $M$ in the Brillouin zone of $C_2$-symmetric crystals.

Figure 29

Rotation eigenvalues at the fixed-point momenta (a) $\Gamma$, (b) $M$, and (c) $K$ in the Brillouin zone of $C_6$-symmetric crystals.

Figure 30

Rotation eigenvalues at the fixed-point momenta (a) $\Gamma$, (b) $K$, and (c) $K^{\prime }$ in the Brillouin zone of $C_3$-symmetric crystals.

Figure 31

Brillouin zones for systems with (a) fourfold, (b) twofold, (c) sixfold, and (d) threefold rotation symmetries and their rotation fixed points. Shaded regions indicate the fundamental domain that generates the entire Brillouin zone upon rotation around the fixed point at the center of the Brillouin zones $\Gamma =(0,0)$. Arrows indicate direction of integration in the calculation of the Chern invariant.

Figure 32

(Color online) Brillouin zone of a $C_2$-symmetric superconductor showing the lines of integration (red lines) for the calculation of the weak index ${\nu }_1$.

Figure 33

Deformation of Hamiltonian $H_0$ into $H_1$ over a fundamental domain, blue for $H_0$ and red for $H_1$.

Figure 34

Construction of a $\Omega =-\pi /2$ disclination by the Voltera process. Green lines mark the quadrants' frame axes.

Figure 35

Quadrants that make the lattice cells for the simulation of $p_x+i{p}_y$ models $H_1^{\left(4\right)}$, $H_2^{\left(4\right)}$ (a), $H_1^{\left(6\right)}$, $H_2^{\left(6\right)}$ (b), and $H_1^{\left(3\right)}$, $H_2^{\left(3\right)}$, $H_3^{\left(3\right)}$ (c). To each quadrant a superconducting phase winding as in Fig. 36 is assigned.

Figure 36

Phase winding function $R\left({\mathbf{r}}^q\right)$ for the winding of the superconducting order parameter in the quadrants that make the lattice cells used in simulation.

Figure 37

Absolute value of MBS energies as function of system size $n$ for all primitive generators with $C_4$ symmetry.

Figure 38

Absolute value of the lowest 40 energies as function of system size $n$ for the $C_6$-symmetric primitive generators $H_1^{\left(6\right)}$ and $H_2^{\left(6\right)}$, as well as for a $C_6$ model with third-nearest-neighbor hopping $H_{3\mathrm{NN}}^{\left(6\right)}$. The first (second) row corresponds to a phase winding of $\pm \pi /3$ ($\pm 7\pi /3$) at disclinations. The parameters used were $(u_1,u_2,u_3)=(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ for $H_1^{\left(6\right)}$, $H_2^{\left(6\right)}$, and $H_3^{\left(6\right)}$, respectively.

Figure 39

Absolute value of the lowest 40 energies as a function of system size $n$ for primitive generators $H_1^{\left(3\right)}$, $H_2^{\left(3\right)}$, and $H_3^{\left(3\right)}$. The first (second) row corresponds to a phase winding of $\pm \pi /3$ ($\pm 7\pi /3$) at disclinations.

Figure 40

Topological phases of model in Eq. (66) with $C_6$ symmetry when third-nearest-neighbor hopping terms are added, with strength $u_3$.