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Topology of contact points in Lieb-kagomé model

G. Abramovici

TL;DR

The paper develops a comprehensive topological framework for the three-band Lieb–kagomé model by introducing a universal classification surface S in a four-dimensional Bloch-parameter space. By enforcing projector constraints on the eigenstate octuplet, it identifies a reduced effective surface E ≅ S2 × R × C × T and defines four independent winding numbers ω2, ω3, ω4, ω5 (with ω1 = ω4/4 in the Lieb limit) that characterize all topological singularities and Dirac-point mergers across 0 < t' ≤ 1. In Lieb’s limit (t' = 0), the topology collapses to a single winding ω1 on S1, whereas kagomé (t' = 1) preserves the four-number classification, with predictable periodicities and explicit mappings of loops to holes on multiple projected surfaces. The results yield a robust, geometrically grounded description of protected states and their topological invariants, illuminating how Dirac points merge and how band topology evolves with t'. Overall, the work provides a unified, multi-surface description of topological singularities in a three-band system, offering concrete winding-number diagnostics and a foundation for analyzing similar lattice models.

Abstract

We analyse Lieb-kagomé model, a three-band model with contact points showing particular examples of the merging of Dirac contact points. We prove that eigenstates can be parametrized in a classification surface, which is a hypersurface of a 4-dimension space. This classification surface is a powerful device giving topological properties of the energy band structure; the analysis of its fundamental group proves that all singularities of the band structure can be characterized by four independent winding (integer) numbers. Lieb case separates: its classification surface differs and there is only one winding number.

Topology of contact points in Lieb-kagomé model

TL;DR

The paper develops a comprehensive topological framework for the three-band Lieb–kagomé model by introducing a universal classification surface S in a four-dimensional Bloch-parameter space. By enforcing projector constraints on the eigenstate octuplet, it identifies a reduced effective surface E ≅ S2 × R × C × T and defines four independent winding numbers ω2, ω3, ω4, ω5 (with ω1 = ω4/4 in the Lieb limit) that characterize all topological singularities and Dirac-point mergers across 0 < t' ≤ 1. In Lieb’s limit (t' = 0), the topology collapses to a single winding ω1 on S1, whereas kagomé (t' = 1) preserves the four-number classification, with predictable periodicities and explicit mappings of loops to holes on multiple projected surfaces. The results yield a robust, geometrically grounded description of protected states and their topological invariants, illuminating how Dirac points merge and how band topology evolves with t'. Overall, the work provides a unified, multi-surface description of topological singularities in a three-band system, offering concrete winding-number diagnostics and a foundation for analyzing similar lattice models.

Abstract

We analyse Lieb-kagomé model, a three-band model with contact points showing particular examples of the merging of Dirac contact points. We prove that eigenstates can be parametrized in a classification surface, which is a hypersurface of a 4-dimension space. This classification surface is a powerful device giving topological properties of the energy band structure; the analysis of its fundamental group proves that all singularities of the band structure can be characterized by four independent winding (integer) numbers. Lieb case separates: its classification surface differs and there is only one winding number.

Paper Structure

This paper contains 86 sections, 36 equations, 24 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Notations
  3. Lieb-kagomé model
  4. Atomic structure
  5. Bloch hamiltonian
  6. Energy spectrum
  7. Eigenstates
  8. Determination of the universal classification surface
  9. One projector conditions
  10. Two projector conditions
  11. Completeness relation
  12. Other relations reduce to a unique equation
  13. Classification of singularities
  14. Description of universal classification surface $\cal S$
  15. Bidimensional surface $\widetilde{\cal S}_1$
  16. ...and 71 more sections

Figures (24)

  • Figure 1: Atomic structure of Lieb-kagomé model. First-near links are represented by solid lines, second-near ones by dashed lines.
  • Figure 2: Sign of $q(k_x,k_y)=1+\cos(k_x)+\cos(k_y)+\cos(k_x+k_y)$. Dark area corresponds to ${-}1$.
  • Figure 3: Energy bands in the Brillouin zone $]{-}\pi,\pi]\times]{-}\pi,\pi]$ for $t'=\frac{1}{2}$. The four contact points are artificially enlarged.
  • Figure 4: Contour lines of $e_{\sm1}-e_1$ and $e_1-e_0$ and positions of all contact points in the reciprocal space zone $]{-}2\pi,2\pi]\times ]{-}2\pi,2\pi]$ for $t'=\frac{1}{2}$. In black, Brillouin zone $]{-}\pi,\pi]\times ]{-}\pi,\pi]$ boundary.
  • Figure 5: Curves of $\alpha_{t'}$ and $\beta_{t'}$ versus $t'$.
  • ...and 19 more figures