Twisted bilayer graphene.VI. An Exact Diagonalization Study of Twisted Bilayer Graphene at Non-Zero Integer Fillings

TL;DR

This study uses exact diagonalization to analyze the electronic properties of twisted bilayer graphene at various fillings, revealing ground state characteristics, phase transitions, and the validity of the Flat Metric Condition across different parameters.

Contribution

It provides the first exact diagonalization analysis of twisted bilayer graphene at non-zero integer fillings, exploring ground states, excitations, and phase transitions in the chiral and nonchiral limits.

Findings

Ground states are well-described by Slater determinants in a Chern basis.

Charge ±1 excitations are the lowest charge excitations in the chiral-flat limit.

Phase transitions occur at large w_0/w_1 ratios, with competing orders identified.

Abstract

Using exact diagonalization, we study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene. Employing the U(4) (U(4)$\times$U(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around $\nu=\pm 3,\pm2,\pm1$ fillings. In the first chiral limit $w_0/w_1=0$ where $w_0$ ($w_1$) is the $AA$ ($AB$) stacking hopping, we find that the ground-states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge $\pm1$ excitations found in [arXiv:2009.14200] are the lowest charge excitations up to system sizes $8\times8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in [arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a series of exact ground-states and excitations holds in a large parameter space. For $\nu=-3$, the ground state is the spin and valley polarized Chern insulator with $\nu_C=\pm1$ at $w_0/w_1\lesssim0.9$ (0.3) with (without) FMC. At $\nu=-2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_0/w_1\gtrsim0.5t$ where $t\in[0,1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [arXiv:2009.13530]. For $\nu=-3$ with/without FMC, when $w_0/w_1$ is large, the finite-size gap $\Delta$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $\nu=-3$ suggests a competition among (nematic) metal, momentum $M_M$ ($\pi$) stripe and $K_M$-CDW orders at large $w_0/w_1$.