[Paper Review] Chern bands of twisted bilayer graphene: fractional Chern insulators and spin phase transition

Cécile Repellin, T. Senthil|arXiv (Cornell University)|Dec 24, 2019

Quantum and electron transport phenomena7 citations

TL;DR

This paper investigates fractional quantum anomalous Hall (FQAH) states in twisted bilayer graphene/hBN (TBG/hBN) with nearly flat Chern bands. Using exact diagonalization, it identifies fractional Chern insulator (FCI) ground states at fillings ν_T = 10/3 and 17/5, with a quantized Hall conductivity and a charge gap of ~0.2 meV (~2 K), indicating experimental feasibility. The ground state is spin-polarized for realistic parameters but can transition to a spin singlet under band parameter tuning, highlighting a sensitive competition between spin phases.

ABSTRACT

When one of the graphene layers of Magic Angle Twisted Bilayer Graphene is nearly aligned with its hexagonal boron nitride substrate (a configuration dubbed TBG/hBN), the active electronic bands are nearly flat, and have a Chern number $C=\pm1$. Recent experiments demonstrated a quantum anomalous Hall effect and spontaneous valley polarization at integer filling $ν_T=3$ of the conduction band in this system. Motivated by this discovery, we ask whether fractional quantum anomalous Hall states (FQAH) could also emerge in TBG/hBN. We focus on the range of filling fractions where valley ferromagnetism was observed experimentally. Using exact diagonalization, we find that the ground states at $ν_T = \frac{10}{3}$ and $ν_T=\frac{17}{5}$ are fractional Chern insulator states in the flat band limit (in the hole picture, these are the fractional quantum Hall fractions $\frac{2}{3}$ and $\frac{3}{5}$). The ground state is either spin polarized or a spin singlet depending sensitively on band parameters. For nominally realistic band parameters, spin polarization is favored. Flattening the Berry curvature by changing a band parameter gives way to the spin singlet phase. Our estimation of the charge gap in the flat band limit shows that the FQAH state may be seen at accessible temperatures in experiments. We also study the effect of a non-zero bandwidth and show that there is a reasonable range of parameters in which the FQAH state is the ground state.

Motivation & Objective

• To determine whether fractional Chern insulator (FCI) states can emerge in TBG/hBN at fractional fillings, motivated by observed integer filling ferromagnetism.
• To assess the stability of FQAH states in the presence of realistic band parameters and non-zero bandwidth.
• To investigate the competition between spin-polarized and spin-singlet FCI ground states in multicomponent Chern bands.
• To estimate the experimentally accessible energy gap for FQAH states in the flat band limit.

Proposed method

• Exact diagonalization of a many-body Hamiltonian for interacting electrons in the nearly flat Chern bands of TBG/hBN.
• Use of a model Hamiltonian with tunable band parameters, including the ratio w₀/w₁, to probe the effect on spin and Chern band structure.
• Analysis of the ground state wavefunction, charge gap, and Hall conductivity to identify FCI and FQAH states.
• Extrapolation of energy gaps and critical bandwidths (W_c) from finite-size systems to estimate stability in the thermodynamic limit.
• Comparison of spin-polarized and spin-singlet states by varying w₀/w₁ to explore phase competition.
• Estimation of the activation gap in the flat band limit and its robustness under finite bandwidth perturbations.

Experimental results

Research questions

• RQ1 Can fractional Chern insulator states emerge in TBG/hBN at fractional fillings ν_T = 10/3 and 17/5, despite the absence of an external magnetic field?
• RQ2 What is the role of spin polarization in the ground state of FCI states in multicomponent Chern bands of TBG/hBN?
• RQ3 How does the competition between spin-polarized and spin-singlet FCI phases depend on band structure parameters such as w₀/w₁?
• RQ4 What is the magnitude of the charge gap in the flat band limit, and is it accessible at experimentally relevant temperatures?
• RQ5 How robust are FQAH states to finite bandwidth effects in realistic TBG/hBN systems?

Key findings

• The ground state at ν_T = 10/3 and ν_T = 17/5 is identified as a fractional Chern insulator with a quantized Hall conductivity σ_xy = (4 - ν_T)e²/h.
• In the flat band limit, the charge gap is estimated at approximately 0.01U ≈ 0.2 meV ≈ 2 K, indicating potential experimental observability.
• For nominally realistic band parameters (w₀/w₁ ≈ 0.8), the ground state is spin-polarized, favoring a FQAH state.
• A transition to a spin-singlet FCI ground state occurs when the band parameter w₀/w₁ is tuned toward zero, indicating strong competition between spin phases.
• The FQAH state remains stable over a reasonable range of non-zero bandwidth, with the critical bandwidth W_c^FCI ≈ 0.1U for FCI formation.
• The critical bandwidth for spin polarization (W_c^FM) is of the same order of magnitude (~0.14U for ν_T=3), suggesting that both FCI and ferromagnetic metal phases are competing in the same parameter regime.