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Polynomial indicator of flat bands

Ivan Damnjanović, Milan Damnjanović, Ivanka Milošević, Dragan Stevanović

Abstract

We present a universal and straightforward algebraic procedure for flat bands construction, polynomial indicator method. Using only Bloch Hamiltonian eigendeterminant functional to identify conditions that guarantee existence of nondispersive eigenvalues, the polynomial indicator method is applicable to all lattice types, enabling predictions of (topological) flat bands in electronic band structure (across the materials), as well as all possible designs of novel artificial flat band lattices. The method is in detail illustrated on several examples - kagome and dice lattice included.

Polynomial indicator of flat bands

Abstract

We present a universal and straightforward algebraic procedure for flat bands construction, polynomial indicator method. Using only Bloch Hamiltonian eigendeterminant functional to identify conditions that guarantee existence of nondispersive eigenvalues, the polynomial indicator method is applicable to all lattice types, enabling predictions of (topological) flat bands in electronic band structure (across the materials), as well as all possible designs of novel artificial flat band lattices. The method is in detail illustrated on several examples - kagome and dice lattice included.

Paper Structure

This paper contains 4 sections, 2 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Indicating flat band through the characteristic polynomial
  3. Flat bands through link weight engineering
  4. Conclusions

Key Result

Theorem 1

Let us write the characteristic polynomial of the Bloch Hamiltonian of the lattice $L$ as where $z_j=e^{ik_j}$ and $c_{\mathbf{t}}(\lambda)$ is a univariate polynomial in $\lambda$ for each $\mathbf{t}\in T$. Then $L$ has a flat band at $\lambda_0$ if and only if $c_{\mathbf{t}}(\lambda_0)=0$ for each $\mathbf{t}\in T$.

Figures (5)

  • Figure 1: Two possible ways of dividing kagome lattice into unit cells. Unit cells are shown with bold lines, and intercell hopping with dashed lines. Variables $a_1$ and $a_2$ denote the primitive lattice vectors.
  • Figure 2: Left: Lattice with octahedron as the unit cell, with intercell hopping represented with the dashed line. Right: Electron band structure of this lattice.
  • Figure 3: Dice lattice with an extra link in the unit cell. Intercell hopping is shown with dashed lines.
  • Figure 4: Top left: General parameterization of the link weights in the kagome lattice. The lattice has a flat band for arbitrary $a,b,c$ at any $\lambda<-\max\{\frac{abc}{a^2},\frac{abc}{b^2},\frac{abc}{c^2}\}$, provided $d,e,f$ are set as in Eqs. (\ref{['eq-d2b']})--(\ref{['eq-f2b']}). Top right: Setting $a=d=0$ with non-zero $b,c,e,f$ yields 2D Lieb lattice with a flat band at 0. Bottom: Setting $a=d=0$ and additionally $f=0$ yields an infinite union of 1D stub lattices, each with a flat band at 0.
  • Figure 5: General parameterization of link weights in dice lattice. It has a flat band at $\lambda=0$ for each value of the weights $a,b,c,d,e,f$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5