Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$\mathbb{Z}_2$ topological invariants from the Green's function diagonal zeros

Florian Simon, Corentin Morice

Abstract

We investigate the relationship between the analytical properties of the Green's function and $\mathbb{Z}_2$ topological insulators, focusing on three-dimensional inversion-symmetric systems. We show that the diagonal zeros of the Green's function in the orbital basis provide a direct and visual way to calculate the strong and weak $\mathbb{Z}_2$ topological invariants. We introduce the surface of crossings of diagonal zeros in the Brillouin zone, and show that it separates time-reversal invariant momenta (TRIMs) of opposite parity in two-band models, enabling the visual computation of the $\mathbb{Z}_2$ invariants by counting the relevant TRIMs on either side. In three-band systems, a similar property holds in every case except when a trivial band is added in the band gap of a non-trivial two-band system, reminiscent of the band topology of fragile topological insulators.

$\mathbb{Z}_2$ topological invariants from the Green's function diagonal zeros

Abstract

We investigate the relationship between the analytical properties of the Green's function and topological insulators, focusing on three-dimensional inversion-symmetric systems. We show that the diagonal zeros of the Green's function in the orbital basis provide a direct and visual way to calculate the strong and weak topological invariants. We introduce the surface of crossings of diagonal zeros in the Brillouin zone, and show that it separates time-reversal invariant momenta (TRIMs) of opposite parity in two-band models, enabling the visual computation of the invariants by counting the relevant TRIMs on either side. In three-band systems, a similar property holds in every case except when a trivial band is added in the band gap of a non-trivial two-band system, reminiscent of the band topology of fragile topological insulators.

Paper Structure

This paper contains 23 sections, 2 theorems, 25 equations, 5 figures.

Table of Contents

  1. Diagonal zeros and topology
  2. Notation
  3. Formalism
  4. Zero-pole representation of $G$
  5. Eigenvector-eigenvalue relation
  6. Cauchy interlacing inequality
  7. Inversion and time-reversal symmetric systems
  8. $\mathbb{Z}_2$ topological invariants
  9. Two-band models
  10. Proposition
  11. Proof
  12. Non-triviality
  13. Example: Wilson-Dirac model
  14. Model
  15. Hamiltonian and Green's function
  16. ...and 8 more sections

Key Result

Proposition 1

Two TRIMs $\Gamma_1$ and $\Gamma_2$ are of opposite parities $\xi(\Gamma_1)$ and $\xi(\Gamma_2)$ iff the non-trivial zeros surface $\mathscr{S}_0$ separates $\Gamma_1$ and $\Gamma_2$

Figures (5)

  • Figure 1: (a) Visualization of the subsets of TRIMs $\text{T}_j$ corresponding to the three $\mathbb{Z}_2$ topological invariants $\nu_j$, in the cubic Brillouin zone. The set $\text{T}_0$, corresponding to the strong invariant $\nu_0$, comprises the eight TRIM points. (b) Graphical representation of a band inversion between two TRIM in a two-band system. The inset represents the orbital character of the bands, red being purely $\ket{\phi_1}$ and blue purely $\ket{\phi_2}$. (c) Any path (dashed blue lines) between non-trivially inverted TRIMs $\Gamma_1$ and $\Gamma_2$ crosses the zeros surface $\mathscr{S}_0$ (in red), thus separating the two TRIMs. For illustration, a two-dimensional cut of the cubic Brillouin zone is shown.
  • Figure 2: Surface of zero-zero crossings within the cubic Brillouin Zone, for three different topological phases of the Wilson-Dirac model. (a) Zeros surface for the trivial $(0;000)$ phase, with $b_z/b_{\parallel}=-3$ and $m/b_{\parallel}=0$. (b) Zeros surface for the weak $(0;111)$ phase, with $b_z/b_{\parallel}=1$ and $m/b_{\parallel}=0$. (c) Zeros surface for the strong $(1;000)$ phase, with $b_z/b_{\parallel}=1$ and $m/b_{\parallel}=2$.
  • Figure 3: (a) Configuration of band parities for which the Proposition \ref{['Proposition-N-bandes']} does not hold. (b) Schematic behavior of the diagonal zeros with the configuration of Fig. \ref{['fig:Zéros-3band-Gamma1']}a at $\Gamma_1$ and of Fig. \ref{['fig:Zéros-3band-Gamma2']}c at $\Gamma_2$.
  • Figure 4: The three possible configurations of diagonal zeros at $\Gamma_1$. Figs. \ref{['fig:Zéros-3band-Gamma1']}a, \ref{['fig:Zéros-3band-Gamma1']}b and \ref{['fig:Zéros-3band-Gamma1']}c correspond to the first, second and third solutions in Eq. (\ref{['eq:3band-3sol-Gamma1']}), respectively.
  • Figure 5: The three possible configurations of diagonal zeros at $\Gamma_2$. Figs. \ref{['fig:Zéros-3band-Gamma2']}a, \ref{['fig:Zéros-3band-Gamma2']}b and \ref{['fig:Zéros-3band-Gamma2']}c correspond to the first, second and third solutions in Eq. (\ref{['eq:3band-3sol-Gamma2']}), respectively.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2