Winding Vector: How to Annihilate Two Dirac Points with the Same Charge

Area of Science:

• Condensed matter physics
• Topological materials
• Solid-state physics

Background:

• Dirac points in electronic band structures are crucial for understanding topological phenomena.
• The classification of Dirac point merging/emergence relies on winding numbers, typically categorized as ++ (identical) or +- (opposite) scenarios.
• Flat bands in lattice models can host unique topological properties and influence Dirac point dynamics.

Purpose of the Study:

• To elucidate the apparent contradiction in the evolution of winding numbers during Dirac point merging and emergence.
• To demonstrate the role of Bloch sphere vector rotation in resolving the winding number classification.
• To investigate this phenomenon in a minimal two-band lattice model featuring a flat band.

Main Methods:

• Analysis of a two-band lattice model (Mielke model) exhibiting a flat band.
• Classification of Dirac point dynamics based on winding numbers and their evolution.
• Investigation of the rotation of the unit vector on the Bloch sphere during Dirac point motion.

Main Results:

• Two Dirac points with identical winding numbers emerge from a band touching point under distortion, following the ++ scenario.
• These Dirac points subsequently merge into a pair with opposite winding numbers (the +- scenario) under further distortion.
• The winding number's definition relative to a rotating Bloch sphere vector reconciles the transition between the ++ and +- scenarios.

Conclusions:

• The observed evolution between ++ and +- winding number scenarios for Dirac points is explained by the rotation of the Bloch vector.
• The simplest Mielke lattice model with a flat band serves as a clear platform to demonstrate this general principle.
• The findings suggest that this evolution is a general feature in various lattice systems exhibiting Dirac points.