Surface-dependent Majorana vortex phases in topological crystalline insulators

TL;DR

This paper investigates Majorana vortex phases in topological crystalline insulator SnTe, revealing how surface symmetries and Dirac fermion properties lead to diverse vortex phase transitions and protected Majorana modes.

Contribution

It provides a comprehensive topological classification and analysis of Majorana vortex phases on different surfaces of SnTe, highlighting symmetry protection and vortex transition mechanisms.

Findings

Majorana vortex end modes are protected by magnetic and rotational symmetries.

Vortex phase transitions depend on surface type and chemical potential.

Number of Majorana modes can change from 0 to 2 with increasing chemical potential.

Abstract

The topological crystalline insulator SnTe exhibits two types of surface Dirac cones: one located at non-time-reversal-invariant momenta on the (001) and (110) surfaces, and the other at time-reversal-invariant momenta on the (111) surface. Motivated by the recent experimental evidence of Majorana vortex end modes (MVEMs) and their hybridization on the (001) surface [Nature 633, 71 (2024)], we present a comprehensive investigation of Majorana vortex phases in SnTe, including topological classification, surface-state Hamiltonians analysis, and lattice model calculations. By utilizing rotational and magnetic mirror symmetries, we present two equivalent methods to reveal the topology of Majorana phases on different surfaces. We find that the MVEMs on the (001) and (110) surfaces are protected by both magnetic group and rotational symmetries. In contrast, the MVEMs on the (111) surface are protected by magnetic group or particle-hole symmetry. Due to the different properties of Dirac fermions in the $\bar{\Gamma}$ and $\bar{M}$ valleys on the (111) surfaces, including Fermi velocities and energy levels, we find that abundant vortex phase transitions can occur for the [111]-direction vortex. As the chemical potential increases, the number of robust MVEMs can change from $0\rightarrow 1\rightarrow 2$. These vortex transitions are characterized by both $Z$ winding number and $Z_2$ pfaffian topological invariants.