Quantized Quadrupole Superconductors

TL;DR

The paper addresses the need for intrinsic invariants in higher-order topological superconductors by defining a half-quantized bulk quadrupole moment $q_{xy}$ protected by particle-hole symmetry, which ensures Majorana corner modes via bulk-corner correspondence. It develops a real-space framework and applies it to two realistic realizations: bilayer Rashba 2DEGs proximitized by $d_{x^{2}-y^{2}}\\pm id'$ pairing and by $d_{x^{2}-y^{2}}\\pm is$ pairing, demonstrating that $q_{xy}=\tfrac{1}{2}$ drives zero-energy Majorana corner modes under open boundaries. The work provides phase diagrams and spectral evidence for robust corner states, showing resilience to phase fluctuations and moderate disorder, and identifies practical platforms such as twisted bilayer cuprates and cuprate–junctions for experimental exploration of Majorana physics in higher-order superconductors. These findings broaden the topological classification of superconductors and offer feasible routes toward Majorana-based quantum information applications.

Abstract

We introduce a class of superconductors termed "quantized quadrupole superconductors" that support Majorana corner modes according to the bulk-corner correspondence, distinct from previous works on the second-order topological superconductors. An intrinsic physical quantity for superconductors, i.e., the quadrupole moment serves as the topological invariant, which is always half-quantized due to the particle-hole symmetry. As examples, two types of mixed pairings, $d_{x^{2}-y^{2}}\pm id_{xy}$ and $d_{x^{2}-y^{2}}\pm is$, induced in the bilayer two-dimensional electron gases with Rashba spin-orbit coupling give the quadrupole phase. Extended discussions indicate that the nontrivial phase is robust against relative phase fluctuations in the mixed pairings and the disorders. Our schemes provide realistic platforms to implement Majorana zero modes, paving the way for studying the Majorana physics.

Figures (3)

• Figure 1: (a) The illustration for the setup to achieve the first model for quadrupole superconductor that supports Majorana corner modes. The bilayer two-dimensional electron gases with Rashba spin-orbit coupling are in proximity to twisted bilayer copper oxides. The second model is by replacing the twisted bilayer copper oxides with Josephson junctions displaying $d\pm is$ pairings. (b) The bands for the bilayer 2DEGs near $\Gamma$ point. The interlayer coupling $t_{z}$ splits the two Rashba bands into bonding (lower) and antibonding states with a hybridization gap.
• Figure 2: (a) The quasiparticle dispersions for a ribbon along the $x$ direction when the Fermi energy $\mu=0$. The ribbon width is adopted $N_{y}=40$. (b) The quadrupole moment $q_{xy}$ with respect to $\mu$. The sample size is chosen as $40\times40$ for the calculations. (c)-(d) Density distribution for the corner modes with the two chemical potential $\mu=0$ and $\mu=4.0$, respectively. The insets show the energy levels around the zero energy. The sample size is $40\times40$. In all the panels, $t_{0}=1.0$, $\alpha=1.0$, $t_{z}=2.0$, $\Delta=0.5$, $\Delta^{\prime}=0.4$.
• Figure 3: (a) The phase diagram in the $t_{z}-\mu$ plane. The sample size is adopted as $20\times20$ for the numerical calculations of $q_{xy}$. The dashed lines are the phase boundary obtained analytically from the band gap closing condition. (b) Density distribution of the MCMs with $\mu=0$ and $t_{z}=2.0$. The inset shows the energy levels around the zero energy. The sample size is adopted as $40\times40$. (c) The averaged $q_{xy}$ with respect to the disorder strength $D_{0}$ for the first model. $E_{g}^{e}$ denotes the edge gap while $E_{g}^{b}$ denotes the bulk gap of quasiparticles. (d) The spatial distributions of the MCMs at disorder strength $D_{0}-0.9$ for the first model.