Topological invariants for spin-orbit coupled superconductor nanowires

Area of Science:

• Condensed Matter Physics
• Quantum Materials
• Topological Superconductivity

Background:

• Semiconductor nanowires are promising platforms for realizing topological states.
• Spin-orbit coupling and superconductivity are key ingredients for topological superconductivity.
• Chiral symmetry plays a crucial role in the topological classification of Hamiltonians.

Purpose of the Study:

• To investigate the chiral symmetry of a multiband spin-orbit coupled semiconductor nanowire with Zeeman splitting and s-wave superconductivity.
• To determine the conditions under which chiral symmetry becomes exact.
• To explore the implications for Majorana fermion modes in realistic scenarios.

Main Methods:

• Analytical investigation of the Hamiltonian for a multiband semiconductor nanowire.
• Analysis of the symmetry properties under different conditions of Zeeman splitting.
• Application of the topological classification of band Hamiltonians (BDI symmetry class).

Main Results:

• The Hamiltonian exhibits approximate chiral symmetry in multiband nanowires.
• Exact chiral symmetry is achieved in an idealized case with a single confinement band and parallel Zeeman splitting.
• This leads to an integer number of zero-energy Majorana fermion modes.
• In realistic multiband wires, approximate chiral symmetry results in multiple near-zero-energy end states.
• Increasing Zeeman splitting causes the minigap to vanish, which can be restored by breaking chiral symmetry with a second Zeeman field.

Conclusions:

• Multiband semiconductor nanowires with spin-orbit coupling and superconductivity host rich topological phenomena.
• Approximate chiral symmetry leads to exotic low-energy states with potential applications.
• External magnetic fields, specifically Zeeman splitting, play a critical role in tuning topological properties and can be used to control Majorana modes.