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Topological Triviality of Flat Hamiltonians

Pratik Sathe, Rahul Roy

TL;DR

Topological Triviality of Flat Hamiltonians studies 2D tight-binding Hamiltonians with all bands exactly flat and shows that, if the band projectors are strictly local, each flat band has zero Chern number, independent of lattice translational invariance. The analysis uses flux insertion on a torus and shows that the spectrum $H(\phi_x,\phi_y)$ remains unchanged under twists when $L_x, L_y \ge 3q$ and $L_x, L_y \ge 2q+2$, respectively. It constructs a smooth family of many-body wavefunctions $\Psi(\phi_x,\phi_y)$ via a monomer-dimer SL basis and a gauge choice so that $\Psi(0,0)$ continuously maps to $\Psi(\phi_x,\phi_y)$. The many-body Chern number is defined as $C = \frac{1}{2\pi} \iint i\langle \Psi| \nabla_{\boldsymbol{\phi}} \Psi \rangle \cdot d\boldsymbol{\phi}$, and boundary terms cancel to give $C=0$, constituting a no-go theorem for topological flat bands under strict locality, valid beyond translational invariance and requiring a finite-system size bound $L_{x,y} \ge 3nR$.

Abstract

Landau levels play a key role in theoretical models of the quantum Hall effect. Each Landau level is degenerate, flat and topologically non-trivial. Motivated by Landau levels, we study tight-binding Hamiltonians whose energy levels are all flat. We demonstrate that in two dimensions, for such Hamiltonians, the flat bands must be topologically trivial. To that end, we show that the projector onto each flat band is necessarily strictly local. Our conclusions do not need the assumption of lattice translational invariance.

Topological Triviality of Flat Hamiltonians

TL;DR

Topological Triviality of Flat Hamiltonians studies 2D tight-binding Hamiltonians with all bands exactly flat and shows that, if the band projectors are strictly local, each flat band has zero Chern number, independent of lattice translational invariance. The analysis uses flux insertion on a torus and shows that the spectrum remains unchanged under twists when and , respectively. It constructs a smooth family of many-body wavefunctions via a monomer-dimer SL basis and a gauge choice so that continuously maps to . The many-body Chern number is defined as , and boundary terms cancel to give , constituting a no-go theorem for topological flat bands under strict locality, valid beyond translational invariance and requiring a finite-system size bound .

Abstract

Landau levels play a key role in theoretical models of the quantum Hall effect. Each Landau level is degenerate, flat and topologically non-trivial. Motivated by Landau levels, we study tight-binding Hamiltonians whose energy levels are all flat. We demonstrate that in two dimensions, for such Hamiltonians, the flat bands must be topologically trivial. To that end, we show that the projector onto each flat band is necessarily strictly local. Our conclusions do not need the assumption of lattice translational invariance.
Paper Structure (1 section, 1 theorem, 12 equations, 2 figures)

This paper contains 1 section, 1 theorem, 12 equations, 2 figures.

Table of Contents

  1. Proof of Spectrum not changing after Flux Insertion

Key Result

Theorem 1

Consider a SL tight-binding Hamiltonian defined on a 2d system of size $L_{x}\times L_y$ cells with periodic boundary conditions. Let $n$ denote the number of distinct energies and $R$ denote the maximum hopping range of the Hamiltonian. If $L_{x,y} \geq 3nR$, then the Chern number associated with e

Figures (2)

  • Figure 1: Periodic boundary conditions are imposed on the system, so that it lies on a torus. Two magnetic fluxes $\phi_{x}$ and $\phi_y$ are inserted through the two holes of the torus. The directions $\hat{x}$ and $\hat{y}$ are shown for reference.
  • Figure 2: The $(\phi_x,\phi_y)$ parameter space with $\phi_x,\phi_y \in [0,\Phi_0]$. For any pair of values $(\phi_x,\phi_y)$, wavefunction $\ket{\Psi(\phi_x,\phi_y)}$ is obtained in two steps. First, we obtain a family of wavefunctions along the blue horizontal line. Next, for any value of $\phi_x$, we obtain a family of wavefunction as a function of $\phi_y$.

Theorems & Definitions (1)

  • Theorem