Fermion bound states at the center of the core of a vortex in a two-dimensional superconductor are investigated by analytically solving the Bogoliubov–de Gennes equations in a magnetic field. The metallic surface states of a strong topological insulator become superconducting via proximity effect with an s-wave superconductor. Due to the magnetic field, the states undergo Landau quantization. A zero-energy Majorana state arises for the Landau level \(n=0\) together with an equally spaced (\(\Delta ^2_{\infty }/E_F\)) sequence of fermion excitations. The spin-momentum locking due to the Rashba spin–orbit coupling is key to the formation of the Majorana state. Extending previous results in zero magnetic field, we present analytical expressions for the energy spectrum and the wave functions of the bound states in a finite magnetic field. The solutions consist of harmonic oscillator wave functions (associated Laguerre polynomials) times a function that falls off exponentially with distance \(\rho \) from the core of the vortex as \(\exp [-\int _0^{\rho } d\rho ' \Delta (\rho ')/v_F]\). An analytic expression for the local density of states (LDOS) for the bound states is obtained. It depends on two length scales, \(1/k_F\) and the magnetic length, \(l_H\), and the angular momentum index \(\mu \). The particle-hole symmetry is broken in the LDOS as a consequence of the spin–orbit coupling and the chirality of the vortex. We also discuss the spin polarization of the bound states.

通过在磁场中解析求解 Bogoliubov-de Gennes 方程，研究了二维超导体中涡旋中心的费米子束缚态。强拓扑绝缘体的金属表面态通过与s波超导体的邻近效应变为超导。由于磁场，状态经历朗道量子化。朗道能级\(n=0\)和等距 ( \(\Delta ^2_{\infty }/E_F\)) 费米子激发序列。由于 Rashba 自旋轨道耦合引起的自旋动量锁定是马约拉纳态形成的关键。扩展先前在零磁场中的结果，我们提出了有限磁场中束缚态的能谱和波函数的解析表达式。解决方案包括谐振子波函数（相关的拉盖尔多项式）乘以一个函数，该函数随着距涡旋核心的距离\(\rho \)呈指数衰减为\(\exp [-\int _0^{\rho } d \rho ' \Delta (\rho ')/v_F]\)。获得了束缚态的局部态密度 (LDOS) 的解析表达式。它取决于两个长度尺度\(1/k_F\)和磁长度\(l_H\)，以及角动量指数\(\mu \)。由于自旋轨道耦合和涡旋的手性，LDOS 中的粒子孔对称性被破坏。我们还讨论了束缚态的自旋极化。