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Updated: Jan 31, 2026

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

Gilles Montambaux1, Lih-King Lim2,3, Jean-Noël Fuchs1,4

  • 1Laboratoire de Physique des Solides, CNRS, Université Paris-Sud, Université Paris-Saclay, F-91405 Orsay, France.

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|January 5, 2019
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Summary

The emergence and merging of Dirac points are classified by winding numbers. This study explains how Dirac points evolve between identical and opposite winding numbers using a flat band lattice model.

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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.