Altermagnetism Induced Bogoliubov Fermi Surfaces Form Topological Superconductivity

Abstract

We propose a novel type of topological superconductivity based on Bogoliubov Fermi surfaces (BFSs) in an altermagnetic topological insulator proximitized by an s-wave superconductor. The 3D altermagnetic topological insulator is characterized by zero-energy surface states in bulk nodal-ring phases and anisotropically shifted surface Dirac cones in topological insulating phases. It is potentially realized in \mathrm{EuIn_{2}As_{2}}. The altermagnetic order in combination with superconductivity gives rise to highly anisotropic superconducting gaps with crystal-facet-dependent BFSs at the physical boundaries. These particular BFSs provide distinct platforms to realize Majorana zero modes (MZMs). We propose a quasi-1D nanowire in which the anisotropic BFSs experience topological phase transitions due to quantum confinement leading to MZMs at its ends. We further consider vortex phase transitions in the superconducting altermagnetic topological insulators. Remarkably, we find that the altermagnetic order allows us to transit between two distinct type of MZMs, one type is located at the vortex line, while the other type is located at the physical boundaries. Our work paves a new avenue utilizing altermagnetism-induced BFSs to engineer topological superconductivity through crystal anisotropy and quantum confinement.

Figures (8)

• Figure 1: (a) Schematic of the anisotropic surface states in an AMTI. (b) A quasi-1D triangular prism nanowire of AMTI/SC heterostructure hosting a pair of MZMs at the two ends. The spectrum illustrates the evolution of BFSs to energy levels (blue lines) driven by AM order strength $t_{AM}$ inside the superconducting gap $\Delta$. (c) AMTI/SC hybrid structure with a vortex line, hosting MZMs at vortex core ($\gamma_{1,2}$) or side surfaces ($\gamma_{3,4}$), depending on the value of $t_{AM}$. The energy spectrum depicts the evolution of BFSs to discrete surface energy levels (red lines) as a function of chemical potential $\mu$ within the bulk band gap.
• Figure 2: (a) Phase diagram of the AMTI in the parameter space of $m$ and $t_{AM}$. The four different phases are weak topological insulators (WTIs), strong topological insulators (STIs), normal insulators (NIs), and topological nodal-ring semimetals (TNRS), respectively. (b) 3D Brillouin zone and band structure for TNRS [purple hexagon in (a)] near the high-symmetry points $(\pi,0,\pi)$, $(0,\pi,\pi)$ and $(\pi,\pi,0)$ (marked by black dots). The plots show the band dispersion for $k_{z}=0$ or $\pi$. Red and blue colors denote the horizontal mirror eigenvalues $\sigma_{h}=+i$ and $-i$, respectively. (c) Band structure of a thin-film topological AM in the $xy$-plane with width $L_{z}=20$, plotted along high-symmetry lines. Parameters: $t_{AM}=0.4$, $\lambda=1$, and $m=0.8$. (d) Schematic of the AM-induced anisotropic shift of surface Dirac cone on (010) and (100) side surfaces in the STI phase.
• Figure 3: (a, b) Local density of states on (010) and (110) surfaces at $k_{z}=\pi$, calculated from the surface Green's function. (c) Topological phase diagram for a triangular prism geometry with surfaces $(110)$, ($1\bar{1}0$), and $(010)$. The color map represents the energy gap obtained from numerical diagonalization of the quasi-1D nanowire at $k_{z}=\pi$, plotted as a function of $L_{x}$ and $t_{AM}$. Black curves indicate phase boundaries obtained from analytical solutions. (d) Energy spectra for quasi-1D nanowire of length $L_{z}=80$. The red dots indicate MZMs. The inset shows the spatial profile of the MZMs. Other Parameters are: $m=1.5$, $\mu=0.1$, $\lambda=1$, and $\Delta=0.1$.
• Figure 4: (a) Topological phase diagram in the parameter space of altermagnetic strength $t_{AM}$ and chemical potential $\mu$ for a vortex line along the $z$-axis. Roman numerals label distinct phases: (I) Majorana vortex phase, (II) trivial vortex phase, (III) bulk nodal phase, and (IV) surface nodal phase. The color scale represents the energy gap near zero energy. (b) Energy spectrum $E(k_{z}=\pi)$ as a function of $t_{AM}$ at a fixed chemical potential $\mu/\Delta=1$ [horizontal dashed line in (a)]. Red and blue lines denote states in $C_{2z}=+i$ and $-i$ eigensectors, with squares and circles marking the corresponding topological winding numbers $(w_{+i},w_{-i})$. (c, d) Energy spectrum $E(k_{z}=\pi)$ as a function of $\mu$ for a fixed altermagnetic strength: In (c) $t_{AM}/\Delta=0.5$ and in (d) $t_{AM}/\Delta=2.3$. The color of each data point indicates the spatial localization of the corresponding state, quantified by its average distance from the vortex core $\langle r\rangle$ (red: outer surface states; blue: vortex core-bound states). (e, f) Energy spectrum with open boundary conditions along $z$-direction ($L_{z}=80$) and spatial profile of the corresponding MZMs (inset) for parameters marked in (a): (e) $t_{AM}/\Delta=1$ (triangle), (f) $t_{AM}/\Delta=2.4$ (hexagon). Parameters for all panels: $L_{x}=L_{y}=20$, $m=2.5$, $\lambda=1$, and $\Delta=0.2$.
• Figure S1: (a) Fermi surface topology of the altermagnetic topological insulators as a function of the mass parameter $m$ and chemical potential $\mu$. (b) Nested Fermi sphere geometry in the normal state (blue and red surfaces) and the resulting 3D BFS (yellow surfaces) centered at ($\pi,\pi,0$) for $\mu=0.7$ [purple hexagon in (a)]. (c) Energy dispersion and Pfaffian sign along $k_{x}$ in the vicinity of ($\pi,\pi,0$). Red and blue dashed curves represent the particle and hole bands in normal state, respectively. Black solid curves indicate the Bogoliubov quasiparticle spectrum. (d, e) Corresponding plots near $(0,\pi,\pi)$ for $\mu=0.1$[purple square in (a)], showing a torus Fermi surface and its associated BFS. Other parameters are $t_{AM}=0.35$, $m=0.7$, $\Delta=0.05$.