Multi-orbital Dirac superconductors and their realization of higher-order topology

TL;DR

The paper addresses the realization of 3D Dirac superconductors with Dirac nodes at general Brillouin-zone positions and demonstrates higher-order topology via symmetry-breaking perturbations. It develops an effective theory for multi-orbital SCs showing that an effective pairing potential $ ilde{m{ ilde{oldsymbol{ abla}}}}(m{k})$ preserves the same $ ext{T}$ and $ ext{P}$ constraints as the original pairing, enabling a reduced $4 imes4$ BdG description. Two constructive schemes are provided to realize Dirac SCs with general-position DPs, and by introducing a $ ext{T}$-breaking even-parity pairing one obtains HO topology with Majorana hinge modes, while a separate $ ext{T}$-breaking Dirac SC yields surface Majorana arcs; the work also analyzes stability under symmetry breaking and discusses potential experimental relevance. The results extend the landscape of HO topological superconductivity in multi-orbital systems and offer concrete models with Majorana bound states at hinges, corners, or surfaces, depending on the perturbations and geometry. Overall, the study provides a rigorous framework and tangible constructions for realizing and tuning higher-order topological Dirac SCs through symmetry-respecting and symmetry-breaking perturbations, with implications for Majorana-based platforms.

Abstract

Topological nodal superconductors (SCs) have attracted considerable interest due to their gapless bulk excitations and exotic surface states. In this paper, by establishing a general framework of the effective theory for multi-orbital SCs, we realize a class of three-dimensional (3D) time-reversal (T )-invariant Dirac SCs, with their topologically protected gapless Dirac nodes being located at general positions in the Brillouin zone. By introducing T -breaking pairing perturbations, we demonstrate the existence of Majorana hinge modes in these Dirac SCs as evidence of their realization of higher-order topology. We also propose a new kind of T -breaking Dirac SCs, whose Dirac nodes possess nonzero even chiralities and so are characterized by surface Majorana arcs.

Figures (5)

• Figure 1: Odd-parity Dirac SCs with DPs at general positions or on high-symmetry lines. The upper (lower) panels correspond to the 1st (2nd) scheme. (a), (f) FSs (blue and purple surfaces) and the nodal lines (green) of the effective $\bm{d}$-vector $\tilde{\bm{d}}(\bm{k})$ located on the surface of $g=0$ (light yellow), their intersections giving the DPs (denoted by red solid dots). (b), (g) Horizontal plane at $k_z=k_0$ containing general-position DPs in the upper half of BZ, where the black hexagon is the 2D BZ boundary and the closed loops (blue and purple) represent the 1D FS cross-sections. (c), (h) [(d), (i)] Energy spectra of $k_z=k_0$ subsystem with boundary along $\bm{a}_1$ ($\bm{a}_2$) direction. (e), (j) Energy spectra of $k_2=0$ subsystem with boundary along $\bm{a}_z$ direction. Parameters: $(t_z,t_2,t_z^{'},\gamma_{so},\lambda_{so},k_0,\mu,\Delta_0)=(0,0,1.5,0.4,0.4,\pi/2,-3.5,0.1)$ and $(t_z,t_2,t_{1g},t_{2g},\gamma_{so},\lambda_{so},\mu,\Delta_0)=(0.8,0,1,0.2,0.4,0.4,-2.5,0.1)$ for the upper and lower panels, respectively.
• Figure 2: Higher-order topology of the $\mathcal{T}$-breaking Dirac SCs with mixed parity, where the odd-parity Dirac SCs with DPs in the 1st (left panels) and 2nd scheme (right panels) in the last section, are perturbed by a $\mathcal{T}$-breaking even-parity pairing term. (a), (e) [(b), (f)] Energy spectra for $k_z=0$ ($k_z=\pi$) planes with boundary along $\bm{a}_2$ direction. The blue solid (red dashed) lines are the edge states without (with) the $\mathcal{T}$-breaking term. (c), (g) Energy spectra for an infinitely long hexagonal prism with side lengths of 30 and 40, respectively, where the red segments denote the Majorana hinge modes. (d), (h) Real-space distribution of the Majorana corner states at $k_z=0$ and $k_z=\pi$ in (c) and (g), respectively, where the insets show energy eigenvalues near zero. Parameters: $(\Delta_0,\Delta_1)=(0.1,0.1)$ and $(0.1,0.25)$ for (a)-(f) and (g)-(h), respectively.
• Figure 3: $\mathcal{T}$-breaking Dirac SCs with DPs on high-symmetry lines, where the left (right) panels correspond to $\mu=-6$ (7). (a), (d) FSs centered at $\bm{k}=(0,0,\pi)$ and $(\pm 2\pi/3, 2\pi/\sqrt{3}, \pi)$, respectively, with DPs (red solid dots) located on high symmetry lines. (b), (e) Surface Majorana arcs at ($010$) surface BZ, where the numbers represent the chiralities of DPs. (c), (f) Energy spectra of $k_z=\pi$ plane with open boundary along $\bm{a}_1$ direction, where green (red) solid lines correspond to the Majorana edge bands localized at the right (left) boundary. Parameters: $(t_z,t_2,t_z^{'},\gamma_{so},\lambda_{so},k_0,\Delta_2)=(0,-0.2,1,0.2,0.2,0,0.1)$.
• Figure 4: Schematic diagram of a single layer of the layered model adopted in the main text, where the black solid dots represent the primary atoms, possessing two orbitals ($1$ and $2$) per site and forming a triangular lattice, while the red (blue) hollow dots denote the decorated apical atoms with orbital-$1$ ($2$)-like orbitals.
• Figure 5: Odd-parity $s$-wave Dirac SC and its realization of higher-order topology. (a) FS (blue surface) and the effective DPs (red solid dots). (b) [(c)] Energy spectra of the $k_1=0$ ($k_z=\pi$) subsystem with boundary along $\bm{a}_z$ ($\bm{a}_1$) direction. The solid blue (dashed red) lines in (c) represent the edge states of $k_z=\pi$ before (after) the inclusion of the additional pairing potential $\Delta_e(\bm{k})$. (d) Energy spectra for an infinitely long hexagonal prism with side lengths of 40, where the red segments denote the Majorana hinge modes. (e) Real-space distribution of the Majorana corner states at $k_z = \pi$ in (d), where the insets show energy eigenvalues near zero. Parameters: $(t_z,t_2,t_z^{'},\gamma_{so},\lambda_{so},k_0,\mu,\Delta_3)=(0,0,1,0.5,0.5,0,-7,0.2)$, $\Delta_4=0.05$ and $0.15$ for (c) and (d)-(e), respectively.