Hofstadter butterflies and metal/insulator transitions for moiré heterostructures

Abstract

We consider a tight-binding model recently introduced by Timmel and Mele for strained moiré heterostructures. We consider two honeycomb lattices to which layer antisymmetric shear strain is applied to periodically modulate the tunneling between the lattices in one distinguished direction. This effectively reduces the model to one spatial dimension and makes it amenable to the theory of matrix-valued quasi-periodic operators. We then study the charge transport and spectral properties of this system, explaining the appearance of a Hofstadter-type butterfly and the occurrence of metal/insulator transitions that have recently been experimentally verified for non-interacting moiré systems. For sufficiently incommensurable moiré lengths, described by a diophantine condition, as well as strong coupling between the lattices, which can be tuned by applying physical pressure, this leads to the occurrence of localization phenomena.

Figures (6)

• Figure 1: Superposition of two honeycomb lattices. On the left, one of the lattices has been exposed to uniaxial strain in the horizontal direction. On the right, the lattices have been exposed to anti-symmetric shear strain. Both cases exhibit an effectively one-dimensional moiré pattern in the horizontal direction.
• Figure 2: Time-evolution of Gaussian wavepackets. (Top row) Time-evolved discretized Gaussian state for chiral (left) and anti-chiral (right) model with $L=3$ with weak coupling $w_0=0.1$ after time $T_{\text{fin}}=5000$ (Space-Amplitude plot). Gaussian state for strong coupling $w_1=1.9$, $w_0=0$, and $L=\pi$ on the lower left figure for the chiral model after $T_{\text{fin}}=2 \cdot 10^4$. Localization effects are clearly visible. On the lower right we see the time evolution (Time-Amplitude plot) corresponding to the amplitude for a Gaussian wavepacket started at the upper lattice. Here, $(1)$ and $(2)$ refer to the respective components labeling atoms of type A and B, respectively. The wavepacket oscillates between the different layers.
• Figure 3: Lowest eigenfunction of chiral Hamiltonian restricted to interval $\{-100,-99,\ldots,100\}$. Figures on the left are for rational length scales $L=20,$ whereas on the right we study the strongly irrational (diophantine) $L=1/\text{golden mean}.$ The top figures correspond to $w_0= \frac{3}{2}$, the bottom ones to $w_0=6.$
• Figure 4: Hofstadter butterflies (1/L-Spectrum plots). The left figure shows the spectrum for the anti-chiral ($w_0=1$) and the right figure for the chiral potential ($w_1=1$), both for the case $\theta=0$ for $1/L \in [0,1].$
• Figure 5: Hofstadter butterflies (1/L-Spectrum plots). Spectrum of chiral Hamiltonian with subcritical tunneling $w_1= \frac{2}{5}.$ Similarly to the AMO with coupling constant away from the critical coupling, the spectrum starts to become more dense.

Theorems & Definitions (45)

• Remark 1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Remark 2
• Proposition 3.1
• proof