Intrinsic Second-Order Topological Superconductors with Tunable Majorana Zero Modes

TL;DR

The work identifies intrinsic second-order topological superconductivity in a 2D nonsymmorphic Dirac semimetal with two Dirac pockets, driven by an even-parity $d_{x^{2}-y^{2}}$-wave pairing that changes sign between pockets. Through self-consistent mean-field theory and symmetry analysis, it shows a gapped superconducting phase hosting Majorana zero modes protected by a bulk quadrupole moment $q_{xy}=1/2$ in the mirror sectors, and demonstrates tunability of MZMs via boundary sublattice terminations. The authors develop a bulk-edge-corner framework, including an edge-state theory with domain-wall physics that explains how edge termination controls MZM positions. This mechanism offers a practical path to intrinsic topological superconductivity with controllable Majorana modes, potentially extendable to 3D Dirac semimetals and cold-atom platforms.

Abstract

Dirac semimetals, with their protected Dirac points, present an ideal platform for realizing intrinsic topological superconductivity. In this work, we investigate superconductivity in a two-dimensional, square-lattice nonsymmorphic Dirac semimetal. In the normal state near half-filling, the Fermi surface consists of two distinct pockets, each enclosing a Dirac point at a time-reversal invariant momentum ($\textbf{X}=(π,0)$ and $\textbf{Y}=(0,π)$). Considering an on-site repulsive and nearest-neighbor attractive interaction, we use self-consistent mean-field theory to determine the ground-state pairing symmetry. We find that an even-parity, spin-singlet $d_{x^{2}-y^{2}}$-wave pairing is favored as it gives rise to a fully gapped superconducting state. Since the pairing amplitude has opposite signs on the two Dirac Fermi pockets, the superconducting state is identified as a second-order topological superconductor. The hallmark of this topological phase is the emergence of Majorana zero modes at the system's boundaries. Notably, the positions of these Majorana modes are highly controllable and can be manipulated simply by tailoring the boundary sublattice terminations. Our results highlight the promise of nonsymmorphic Dirac semimetals for realizing and manipulating Majorana modes.

Figures (8)

• Figure 1: Illustration of how the Fermi surface (solid lines) dictates superconducting properties for a $d$-wave pairing state. The dashed lines denote the pairing nodes. (a) Nodal phase. (b) Fully gapped phase with trivial topology. (c) Fully gapped phase with nontrivial topology, where the Fermi pockets enclose a Dirac point (red dots).
• Figure 2: (a) Sketch of a top-down view of the bilayer lattice, where the two layers are shifted by a vector $(1/2,1/2)a$. The hopping and spin-orbit coupling coefficients are shown. (b) The band structure of the DSM, where the Dirac points appear at $\textbf{X}= (\pi,0)$, $\textbf{Y}= (0,\pi)$. Parameters are $\{t, \lambda_{\rm so}, \eta_1, \eta_2\} = \{0.5, 0.4, 0.8, 0.8\}$.
• Figure 3: BdG spectra corresponding to the pairing functions in the four different IRs. All spectra are calculated with the common parameter set: $\{t, \lambda_{\rm so}, \eta_1, \eta_2, \mu, \Delta_{1x},\Delta_{1y}\} = \{0.5, 0.4, 0.8, 0.8, 0.6, 0.3,-0.3\}$. For $A_{g}$ and $A_{u}$, $\Delta_{0}$ is set to zero. For $B_{g}$ and $B_{u}$, we choose $M_{\Delta}=\tau_{y}\sigma_{z}s_{x}$ and $M_{\Delta}^{\prime}=\tau_{y}s_{x}$, respectively.
• Figure 4: (a) Pairing amplitudes for the gap functions in the four IRs. (b) Free-energy difference $\delta F$ between the normal and superconducting states at zero temperature. Parameters are $\{t, \lambda_{\rm so}, \eta_1, \eta_2, \mu, U\} = \{0.5, 0.4, 0.8, 0.8, 0.6, 0.1\}$.
• Figure 5: Tunability of MZMs via sublattice termination. (a) With complete unit cells at all boundaries, the system hosts one Majorana Kramers pair at each corner. (b) Changing the sublattice termination on the upper ($y$-normal) and right ($x$-normal) edges halves the number of Majorana Kramers pairs. (c) Further adjusting the termination of the upper edge creates a domain wall and relocates a Majorana Kramers pair from the corner to a generic boundary site. The MZM count is quantified by the number of zero-energy states shown in the insets. Common parameters: $\{t, \lambda_{\rm so}, \eta_1, \eta_2, \mu, \Delta\} = \{0.5, 0.4, 0.8, 0.8, 0.6, 0.3\}$