Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor

Conventional $n$-dimensional topological superconductors (TSCs) have protected gapless $(n-1)$-dimensional boundary states. In contrast to this, second-order TSCs are characterized by topologically protected gapless $(n-2)$-dimensional states with the usual gapped $(n-1)$ boundaries. Here, we study a second-order TSC with a two-dimensional (2D) magnetic topological insulator proximity coupled to a high-temperature superconductor, where Majorana bound states (MBSs) are localized at the corners of a square sample with gapped edge modes. Due to the mirror symmetry of the hybrid system considered here, there are two MBSs at each corner for both cases: $d$-wave and $s_{\pm }$-wave superconducting pairing. We present the corresponding topological phase diagrams related to the role of the magnetic exchange interaction and the pairing amplitude. A detailed analysis, based on edge theory, reveals the origin of the existence of MBSs at the corners of the 2D sample, which results from the sign change of the Dirac mass emerging at the intersection of any two adjacent edges due to pairing symmetry. Possible experimental realizations are discussed. Our proposal offers a promising platform for realizing MBSs and performing possible non-Abelian braiding in 2D systems.

Figure 1

Schematic view showing a magnetic TI approximated by a high-temperature superconductor. The Majorana bound states are located at the four corners of the green square sample. I, II, III, and IV label the four edges of the lattice.

Figure 2

Energy-band structure of a 2D magnetic TI nanoribbon along the (a) $x$ and (b) $y$ directions. The red curves in (a) and (b) denote two degenerate gapless chiral edge states. The zero-energy edge states exist at the $k_x=0$ and $k_y=0$ points, respectively. The BdG spectrum with a $d_{x^2-y^2}$-wave pairing along the (c) $x$ and (d) $y$ directions. In the presence of $\mathit{d}$-wave pairing, the edge states are gapped out (red curves). (e) The probability density distributions of the BdG wave functions with zero energies for a sample size with $30\times 30$. There are two MBSs localized at each corner due to the mirror symmetry of the BdG Hamiltonian. The inset shows the eigenenergies of the same sample with energies around zero. Note that there are eight zero-energy modes in the gap shown as eight red dots. The parameters are chosen to be $m_0=-0.8$, ${\lambda }_z=0.4$, ${\lambda }_{\mathrm{so}}=1$, $m_x=m_y=1$, $\mu =0$, and ${\mathrm{\Delta }}_1=0.5$.

Figure 3

BdG energy spectrum with open boundary conditions in the $x$ and $y$ directions as a function of ${\lambda }_z$. The sample size is $20\times 20$. The red line corresponds to eightfold-degenerate MCSs, which exist only for finite amplitudes of the exchange field. The parameters are chosen to be $m_0=-0.8$, ${\lambda }_{\mathrm{so}}=1$, $m_x=m_y=1$, $\mu =0$, and ${\mathrm{\Delta }}_1=0.5$.

Figure 4

Topological phase diagram of the magnetic $\mathrm{TI}/d_{x^2-y^2}$ high-temperature superconductor hybrid system in the $(m_0,{\lambda }_z)$ plane, which reveals three distinct phases: (i) second-order topological superconductor with MCSs, (ii) chiral MZMs characterized by a finite Chern number $\mathcal{N}$, and (iii) topologically trivial states $\mathcal{N}=0$ with zero chiral MZMs and MCSs. The parameters are chosen to be ${\lambda }_{\mathrm{so}}=1$, $m_x=m_y=1$, $\mu =0$, and ${\mathrm{\Delta }}_1=0.5$.

Figure 5

Energy-band structure of a 2D magnetic TI nanoribbon along the (a) $x$ and (b) $y$ directions. The red curves in (a) and (b) denote two degenerate gapless chiral edge states. The zero-energy edge states exist at the $k_x=0$ and $k_y=\pi$ points, respectively. The BdG spectrum with an $s_{\pm }$-wave pairing along the (c) $x$ and (d) $y$ directions. In the presence of $s_{\pm }$-wave pairing, the edge states are gapped out (red curves). (e) The probability density distributions of BdG wave functions with zero energies for a sample of size $30\times 30$. There are two MBSs localized at each corner due to the mirror symmetry of the BdG Hamiltonian. The inset shows eigenenergies of the same sample with energies around zero. Note that there are eight zero-energy modes in the gap shown as eight red dots. The parameters are chosen to be $m_0=-0.8$, ${\lambda }_z=0.4$, ${\lambda }_{\mathrm{so}}=1$, $m_x=-m_y=1$, $\mu =0$, ${\mathrm{\Delta }}_0=0$, and ${\mathrm{\Delta }}_2=0.4$.

Figure 6

The probability density distributions (with the scale shown in the right vertical bar) of the BdG wave functions with zero energies for a sample size of $60\times 60$, with different chemical potentials: (a) $\mu =0$ and (b) $\mu =0.1$. There are two MBSs localized at each corner. A finite chemical potential $\mu$ will not destroy the MCSs. The inset shows the eigenenergies of the same sample with the energies around zero. Note that there are eight zero-energy modes in the gap shown as eight red dots. Other parameters are chosen to be $m_0=-0.8$, ${\lambda }_z=0.1$, ${\lambda }_{\mathrm{so}}=1$, $m_x=-m_y=1$, ${\mathrm{\Delta }}_0/{\mathrm{\Delta }}_2=0.6$, and ${\mathrm{\Delta }}_2=0.4$.

Figure 7

Topological phase diagram of a magnetic $\mathrm{TI}/s_{\pm }$ high-temperature superconductor hybrid system in the $({\lambda }_z,{\mathrm{\Delta }}_0/{\mathrm{\Delta }}_2)$ plane, which reveals three distinct phases: (i) second-order topological superconductor with MCSs, (ii) chiral MZMs characterized by a finite Chern number $\mathcal{N}$, and (iii) topologically trivial states $\mathcal{N}=0$ with zero chiral MZMs and MCSs. Recall that ${\mathrm{\Delta }}_x={\mathrm{\Delta }}_y={\mathrm{\Delta }}_2$, as shown in Eq. (30). The parameters are chosen to be $m_0=-1$, ${\lambda }_{\mathrm{so}}=1$, $m_x=-m_y=1$, $\mu =0$, and ${\mathrm{\Delta }}_2=0.4$.

Figure 8

(a) BdG energy spectra of a $d_{x^2-y^2}$-wave pairing superconductor with open boundaries along the $x$ and $y$ directions as a function of ${\lambda }_x$. The red dots denote eightfold-degenerate MCSs, and the black dots represent the fourfold-degenerate MCSs. The probability density distributions of mid-gap states for (b) ${\lambda }_x=0.3$ and (c) ${\lambda }_x=0.5$. The inset shows the eigenenergies with energies around zero. The parameters are chosen to be $m_0=-0.8$, ${\lambda }_z=0.4$, ${\lambda }_{\mathrm{so}}=1$, $m_x=m_y=1$, $\mu =0$, and ${\mathrm{\Delta }}_1=0.5$.

Figure 9

(a) BdG energy spectra of an $s_{\pm }$-wave pairing superconductor with open boundaries along the $x$ and $y$ directions as a function of ${\lambda }_x$. The red dots denote eightfold-degenerate MCSs, and the black dots represent the fourfold-degenerate MCSs. The probability density distributions of mid-gap states for (b) ${\lambda }_x=0.2$ and (c) ${\lambda }_x=0.5$. The inset shows the eigenenergies with energies around zero. The parameters are chosen to be $m_0=-0.8$, ${\lambda }_z=0.4$, ${\lambda }_{\mathrm{so}}=1$, $m_x=-m_y=1$, $\mu =0$, ${\mathrm{\Delta }}_0=0$, and ${\mathrm{\Delta }}_2=0.4$.