Interacting topological bound states and Majorana fermions in strained nodal superconductors

Landau levels (LL) have been predicted to emerge in systems with Dirac nodal points under applied nonuniform strain. We consider two-dimensional (2D), $d_{xy}$ singlet (2D-S), and three-dimensional (3D) $p_x\pm i{p}_y$ equal-spin triplet (3D-T) superconductors. We demonstrate the topological bound state and spinful Majorana nature of the bulk gapless zeroth LLs in the singlet and triplet cases, respectively. Strain along certain directions can induce two topologically distinct phases in the bulk, with zeroth LLs localized at the interface. These modes are unstable toward ferromagnetism for 2D-S cases. True Majorana fermions in 3D-T allow for more exotic possibilities.

Figure 1

Typical Fermi surfaces (FSs) and nodal Bogoliubov–de Gennes (BdG) spectra for (a) 2D $d_{xy}$ spin-singlet and (b) 3D $(p_x\pm i{p}_y)$ equal spin-triplet pairing in the absence of any applied strain. Note the labeling of the valleys. The dashed arrows indicate two directions of applied uniaxial strain along different axes of symmetry of the SC sample. In the case of $d_{xy}$ pairing, the green and red arrows represent strain along the (10) and [11] axes, respectively. In the case of 3D $(p_x\pm i{p}_y)$ pairing, the green arrow indicates strain along the (100) axis. (c) Emergence of the zeroth LLs at the boundary between strain-induced topologically distinct phases in the bulk. The top panel represents an unstrained, trivially gapped SC with open boundary conditions along the $y$ directions (see Sec. 2). The bottom panel shows the same system in the presence of nonuniform strain applied along $x$. The pink peak on the right is a schematic of a topological zero-energy edge mode due to the induced topological SC phase in the bulk of the strained sample. The emergence of the illustrated zero modes with strain is the subject of Secs. 3 and 4a.

Figure 2

(a) Spectrum for the 2D-S case about valley (1) as a function of conserved momentum and strain along the $x$ direction for the spin-up sector. See text for the parameters of the calculation. (b) Evolution of the spectral weight with momentum for the two degenerate states at zero energy. Note the formation of a broad LL state in the bulk accompanied by an edge state.

Figure 3

2D-S case with strain along the (10) axis: (a) suppression of the bulk gap $E_0$ for $k_y=1.84$ as a function of increasing strain parameter $\epsilon$. Beyond $\epsilon \approx 1.2\times {10}^{-4}$ the gap effectively vanishes. (b) Evolution of the spatially resolved spectral weight with increasing strain indicated in (a). Although the vanishing strain cannot close the gap, spectral weight accumulates at the edge. (c) Beyond a threshold strain $\epsilon =1.2\times {10}^{-4}$, the gap suppression indicated in (a) is accompanied by the emergence of the bulk zeroth LL and edge states. In-between the two localized states, the chain is in a topologically nontrivial phase, as illustrated in Fig. 1. The parameters of the calculation are the same as in Fig. 2.

Figure 4

2D-S case with strain along the (11) axis. (a) Spectrum as a function of conserved momentum. The degenerate states at zero energy are both in the bulk. See text for the parameters of the calculation. (b) Real BdG coefficients as functions of position along the 1D chain for the two states indicated in (a). The two sets of coefficients are related via $u_{II}=-v_I,v_{II}=u_I$ due to the mirror symmetry as discussed in the main text.

Figure 5

3D-T case (a) spectrum of spin-up sector under uniaxial strain along $x$ for $k_z=K_F$ corresponding to valley (1) in Fig. 1, as a function of $k_y$. The parameters of the calculation are discussed in the main text. (b) Spectral weights of the two counterpropagating modes indicated in (a) as functions of position along $x$. Only the right-moving modes are bulk zeroth LL states. Panels (c) and (d) are the same as (a) and (b) for the spin-down sector. Here, the left-moving mode is in the bulk.

Figure 6

Lattice model for 2D-S with strain along the diagonal of the BZ. The $x,y$ directions are rotated with respect to the axes of the BZ. In order to preserve translational symmetry in the $y$ direction along the edges, we use a two-site unit cell, illustrated by the dashed line. The strain is along the $x$ direction.