Engineering second order topological superconductor hosting tunable Majorana corner modes in magnet/$d$-wave superconductor hybrid platform

TL;DR

The paper addresses realizing a tunable second-order topological superconductor hosting Majorana corner modes in a 2D heterostructure of a $d$-wave superconductor, a quantum spin Hall insulator, and a noncollinear spin texture. It combines a real-space lattice model with an effective low-energy continuum theory and an edge theory, using the quadrupolar winding number $N_{xy}$ to classify phases. The authors show that the system can host 4 or 8 Majorana corner modes, tunable by the exchange coupling $J$ and pitch vector $\mathbf{g}$, with $N_{xy}=1$ for 4 MCMs and $N_{xy}=2$ for 8 MCMs, and provide a microscopic pairing analysis revealing emergent $s$- and $p$-wave components. They also outline experimental paths and discuss the role of Rashba SOC, highlighting the potential realization of tunable MCMs in magnetic adatom–superconductor–QSHI platforms.

Abstract

We theoretically study the noncollinear magnetic texture effect on second-order topological superconductor (SOTSC) phase generated in unconventional $d$-wave superconductors and two-dimensional (2D) quantum spin Hall insulators (QSHI). While the interplay of the $d$-wave superconductor and QSHI has been studied as a platform to realize Majorana corner modes (MCMs), we show that the addition of the spin texture enables the tunability of these MCMs. Each corner of this hybrid system can host one or two Majorana modes depending on the system parameters, in particular, exchange strength and pitch vector of the spin texture. To characterize the higher order bulk topology, we compute the quadrupolar winding number, which directly corresponds to the number of MCMs acquiring a value of one for four corner modes and two for eight corner modes. We investigate and show the close resemblance in the topological phase diagrams obtained from the low energy effective Hamiltonian that reveals an emergent in-plane Zeeman field and spin-orbit coupling induced by the spin texture, and the real space tight binding lattice model. The microscopic pairing mechanism responsible for the appearance of SOTSC phase is investigated via an effective bulk pairing analysis, while a low-energy edge theory captures the mechanism behind tunability of MCMs. Our result paves the way for realizing SOTC with multiple MCMs which can be tuned via system parameters.

Figures (8)

• Figure 1: Schematic diagram of our 2D heterostructure consisting of an unconventional $d$-wave SC shown in orange at the bottom, with $d$-wave pairing lies in the $x$-$y$ plane. A noncollinear spin texture, represented by blue arrows, is deposited on the surface of the $d$-wave SC. On top of this spin texture, a layer of quantum spin Hall insulator (QSHI), depicted in green, is deposited resulting in a sandwich-like configuration. The four edges of the 2D heterostructure are labeled as I, II, III, and IV. The system hosts localized MCMs, indicated by four black spheres at the corners of the heterostructure.
• Figure 2: Panel (a) depicts the energy eigenvalue spectrum, obtained via OBC, as a function of the exchange interaction $J$, with fixed model parameters $g_x = g_y = g = 0.1$, $\Delta_0 = 0.5t$, and $\lambda_x = \lambda_y = \lambda = \epsilon_0=t$. The red dotted lines indicates the two critical values of $J$ represented as $J_{c_1}$ and $J_{c_2}$. For the former case topological phase transition takes place from one SOTSC phase (8 MCMs) to second SOTSC phase (4 MCMs) and the latter showcase the transition from topological to trivial phase. Region I corresponds to the presence of 8 MCMs, as illustrated in the inset of panel (c) choosing $J = 0.8t$, while region II denotes the presence of 4 MCMs, as shown in the inset of panel (d) for $J = 1.5t$. In panels (c) and (d) LDOS spectra of the MCMs are depicted in the $L_x-L_y$ plane. The insets of (c) and (d) represent the eigenvalue spectrum which we have zoomed around zero energy. In panel (b), we present the phase diagram for number of MCMs in the $g$-$J$ plane, where the yellow region corresponds to 8 MCMs and the pink region to 4 MCMs. All other model parameters remains same as mentioned above. Here we choose a finite size system with $30\times 30$ lattice sites.
• Figure 3: We depict the edge state spectrum obtained under OBC along $y$ and PBC along $x$ direction. Panel (a) and (c) showcase the gapped edge states in SOTSC phases with 8 and 4 MCMs for $J = 0.8t$ and $J = 1.8t$, respectively. Panels (b) and (d) display the gapless edge states at the topological transition points obtained for $J_{c_1} = 1.3t$ and $J_{c_2} = 2.1t$. The other model parameters are chosen as $\Delta_0 = 0.5t$, $g = 0.05$, $t = 1$, $\lambda = t$, and $\epsilon_0 = t$. Here, we consider finite size of $100$ latice sites along $y$-direction.
• Figure 4: Topological characterization of SOTSC phase via $N_{xy}$ is illustrated for both the exact and effective lattice regularized models. In panels (a) and (c), we depict $N_{xy}$ for the effective model in the $g$-$J$ plane choosing $\lambda = t$ and in the $g$-$\lambda$ plane at $J = 2t$, respectively. On the other hand, in panels (b) and (d), we display the corresponding results for the exact lattice model in the same parameter space. In the phase diagrams, the yellow and pink regions correspond to $N_{xy} = 2$ and $N_{xy} = 1$ respectively. The phase diagram in panel (b) corroborates with the phase diagram of number of MCMs presented in Fig. \ref{['Fig2']}(b). The calculations are performed with chosen model parameters $\Delta_{0} = 0.5t$, $\epsilon_0 = t$, and on a finite size system of $30 \times 30$ lattice sites.
• Figure 5: (a) We show the gapped edge spectrum of the lattice-regularized model employing OBC along the $x$-direction and PBC along the $y$-direction. The model parameters used are $\lambda_{x}=\lambda_{y}=\lambda = 2t$, $g = 0.8$, $J = 1.2t$, $\epsilon_0 = t$, $\Delta_0 = 0$, and $t = 1$. Panel (b) depicts the emergence of eight MCMs, with two modes localized at each corner, in a system comprising of a $d$-wave SC and a QSHI. Parameters are set as $\lambda = t$, $\epsilon_0 = t$, $J = g = 0$, and $\Delta_0 = 0.5t$. The inset displays the eight MCMs in the zoomed-in eigenvalue spectrum and the corresponding LDOS distribution displays their corner localization.