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Twisted TMDs in the small-angle limit: exponentially flat and trivial bands

Simon Becker, Mengxuan Yang

Abstract

Recent experiments discovered fractional Chern insulator states at zero magnetic field in twisted bilayer MoTe$_2$ [C23,Z23] and WSe$_2$ [MD23]. In this article, we study the MacDonald Hamiltonian for twisted transition metal dichalcogenides (TMDs) and analyze the low-lying spectrum in TMDs in the limit of small twisting angles. Unlike in twisted bilayer graphene Hamiltonians, we show that TMDs do not exhibit flat bands. The flatness in TMDs for small twisting angles is due to spatial confinement by a matrix-valued potential. We show that by generalizing semiclassical techniques developed by Simon [Si83] and Helffer-Sjöstrand [HS84] to matrix-valued potentials, there exists a wide range of model parameters such that the low-lying bands are of exponentially small width in the twisting angle, topologically trivial, and obey a harmonic oscillator-type spacing with explicit parameters.

Twisted TMDs in the small-angle limit: exponentially flat and trivial bands

Abstract

Recent experiments discovered fractional Chern insulator states at zero magnetic field in twisted bilayer MoTe [C23,Z23] and WSe [MD23]. In this article, we study the MacDonald Hamiltonian for twisted transition metal dichalcogenides (TMDs) and analyze the low-lying spectrum in TMDs in the limit of small twisting angles. Unlike in twisted bilayer graphene Hamiltonians, we show that TMDs do not exhibit flat bands. The flatness in TMDs for small twisting angles is due to spatial confinement by a matrix-valued potential. We show that by generalizing semiclassical techniques developed by Simon [Si83] and Helffer-Sjöstrand [HS84] to matrix-valued potentials, there exists a wide range of model parameters such that the low-lying bands are of exponentially small width in the twisting angle, topologically trivial, and obey a harmonic oscillator-type spacing with explicit parameters.
Paper Structure (12 sections, 10 theorems, 126 equations, 4 figures)

This paper contains 12 sections, 10 theorems, 126 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Agmon estimates, tight-binding limits & related work
  3. The continuum Hamiltonian for twisted transition metal dichalcogenides
  4. Symmetries & Absence of flat bands
  5. Bloch-Floquet theory
  6. Absence of flat bands
  7. Diagonalization of the TMD Hamiltonian
  8. Harmonic approximation
  9. Harmonic approximation with remainder terms
  10. Topologically trivial bands
  11. Agmon estimate for matrix-valued potentials
  12. Exponentially narrow bands

Key Result

Theorem 1

Let $\Gamma$ be a lattice in $\mathbb R^n$. The Hamiltonian $H_V=-\Delta + V$, where $V \in L^{\infty}(\mathbb R^n/\Gamma;\mathbb C^{n \times n})$ is Hermitian, does not have any flat bands, i.e., there does not exist a $\lambda$ such that $\lambda \in \mathop{\mathrm{Spec}}\nolimits_{L^2(\mathbb R^

Figures (4)

  • Figure 1: Contour plot of $x \mapsto \lambda_-(V(x))$ with $(U,\beta)=(2,0),(2,5),(0,5),(0,0)$ from top left (clock-wise) with $\phi=4\pi/3$.
  • Figure 2: Contour plot of $x \mapsto \lambda_-(V(x))$ with $U=0,\beta=1$ and $\phi=1.32,1.94,2.31$ from left to right exhibiting the features of a honeycomb, Kagome, and triangular lattice.
  • Figure 3: Different $h$, $\alpha=\beta=1$. All lowest bands for $h=9$ and lowest band for different parameters of $h$ showing exponential flattening.
  • Figure 4: Lowest two bands for $h=1/9$, $\alpha=1$ and $\beta=1/9$ with $\phi=0$ (left) almost exhibiting Dirac points and $\phi = 2\pi/3$ (right).

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1: Honeycomb-lattice potentials
  • Remark 2: Coordinates for numerics
  • proof : Proof of Theo. \ref{['theo:no_flat']}
  • Lemma 3.1
  • proof
  • ...and 11 more