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The theory of topological-topological flat bands

Rui-Heng Liu, Jiangping Hu, Chen Fang

Abstract

Electronic flat bands have localized Wannier-like orbitals as zero modes. In the Lieb or the kagome models, the localized orbitals satisfy a topological condition that entails two non-contractible loop eigenstates along $x/y$-axis in real space, and one topological band touching point with other bands in momentum space. In these topological-flat bands, the Bloch state at the touching point is ill-defined, and so is any topological invariant for the entire band. We propose a new topological condition that the loop states in different directions be linearly dependent. Its satisfaction removes the singularity at the band touching point, and enforces nontrivial, well-defined topological invariants. Enforcing the new condition, we obtain topological-topological (top$^2$)-flat bands in 2D and 3D that have nontrivial invariants including the Chern numbers, the $\mathbb{Z}_2$ invariants, and the topological-crystalline invariants. Under small, generic interactions, top$^2$-flat bands flow to correlated topological insulators with a dynamically generated, symmetric mass term; and specially designed interacting models can have top$^2$-flat bands as exact zero modes.

The theory of topological-topological flat bands

Abstract

Electronic flat bands have localized Wannier-like orbitals as zero modes. In the Lieb or the kagome models, the localized orbitals satisfy a topological condition that entails two non-contractible loop eigenstates along -axis in real space, and one topological band touching point with other bands in momentum space. In these topological-flat bands, the Bloch state at the touching point is ill-defined, and so is any topological invariant for the entire band. We propose a new topological condition that the loop states in different directions be linearly dependent. Its satisfaction removes the singularity at the band touching point, and enforces nontrivial, well-defined topological invariants. Enforcing the new condition, we obtain topological-topological (top)-flat bands in 2D and 3D that have nontrivial invariants including the Chern numbers, the invariants, and the topological-crystalline invariants. Under small, generic interactions, top-flat bands flow to correlated topological insulators with a dynamically generated, symmetric mass term; and specially designed interacting models can have top-flat bands as exact zero modes.

Paper Structure

This paper contains 13 equations, 3 figures, 1 table.

Figures (3)

  • Figure 1: An explicit CLS satisfying the first topological condition. (a) the summation of $b_\mathbf{R}$ in the grey rectangle equals the sum of $a_{x/y}$ along the red loop. (b) If the length spans the whole lattice, the boundaries are two separate loops at $y_{1,2}$. (c) If the height spans the whole lattice, the boundaries are two separate loops at $x_{1,2}$. (d) Graphical expressions of $b_\mathbf{R}$ and $a_{x,y}$ defined on the bonds along $x/y$-direction.
  • Figure 2: The second topological condition. (a) As $\mathbf{k}$ moves towards the origin along $x/y$-axis, the Bloch state converges to $\Theta_{x/y}$. (b) The configuration of explicit CLSs in 2D satisfying the second topological condition while having the fourfold or the sixfold rotation. (c) An explicit 3D CLS satisfying the second topological condition and the cubic symmetry. The spinor wavefunctions $O,R,A$ are given in Table \ref{['table']}, and $C_4=\exp(-i\sigma_z\pi/4),C_3=\exp(-i\sigma_{111}\pi/3),I=\sigma_{0}\tau_{z}$ where $\tau_{z}=1(-1)$ for $A(B)$ orbitals.
  • Figure 3: A many-body gap opens upon adding Hubbard interaction on top$^2$-flat Chern/$\mathbb{Z}_2$ bands. Colors of the band represents spin or orbital polarization. In the middle, zoom-in band structures near $\mathbf{k}=0$ are shown.