Topological Nature of Edge States for One-Dimensional Systems without Symmetry Protection

Area of Science:

• Condensed Matter Physics
• Topological Materials
• Quantum Mechanics

Background:

• Topological phases are crucial in condensed matter physics.
• Symmetry-protected topological invariants classify phases but have limitations.
• Existing methods struggle with complex couplings and open boundaries.

Purpose of the Study:

• To develop a robust winding number invariant for one-dimensional, two-band models.
• To predict the number of edge states in systems with complex couplings and open boundaries.
• To establish a broader topological criterion distinct from traditional classifications.

Main Methods:

• Numerical verification and analytical proof of a winding number invariant.
• Utilizing analytical continuation of the wave-vector into the complex plane.
• Identifying bulk eigenvector degeneracies on the Riemann surface band structure.

Main Results:

• The proposed winding number invariant accurately predicts edge states.
• The invariant is valid for complex couplings and open boundaries.
• It remains invariant under unitary or similarity transforms.

Conclusions:

• The novel winding number provides a broader topological invariant for complex systems.
• This invariant correctly identifies non-zero energy edge states.
• It generalizes well-known topological invariants under specific symmetry conditions.