Hybrid Wannier Chern bands in magic angle twisted bilayer graphene and the quantized anomalous Hall effect

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Hybrid Wannier Chern bands in magic angle twisted bilayer graphene and the quantized anomalous Hall effect

Kasra Hejazi, Xiao Chen, and Leon Balents

Phys. Rev. Research 3, 013242 – Published 15 March 2021

Abstract

We propose a method for studying the strong interaction regimes in twisted bilayer graphene using hybrid Wannier functions that are Wannier-like in one direction and Bloch-like in the other. We focus on the active bands as given by the continuum model proposed by Bistritzer and MacDonald, [Proc. Natl. Acad. Sci. 108, 12233 (2011)] and discuss the properties of corresponding hybrid Wannier functions. We then employ the method for a study of the fillings of $\pm 3$ electrons per moiré cell using the Hartree-Fock method. We discuss at length different regimes under which a quantized anomalous Hall effect is seen in these two fillings.

Received 30 June 2020
Revised 12 February 2021
Accepted 16 February 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.013242

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Chern insulators Electronic structure Quantum anomalous Hall effect

Graphene Two-dimensional electron gas

Wannier function methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kasra Hejazi¹, Xiao Chen^2,3, and Leon Balents ^4,5

¹Department of Physics, University of California Santa Barbara, Santa Barbara, California, 93106, USA
²Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA
³Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
⁴Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA
⁵Canadian Institute for Advanced Research, Toronto, Ontario M5G 1M1, Canada

Article Text

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Issue

Vol. 3, Iss. 1 — March - May 2021

Subject Areas

Condensed Matter Physics

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Images

Figure 1
(a) The moiré lattice in real space and the corresponding BZ. A rectangular BZ is chosen $- \frac{\sqrt{3}}{4} < k_{x} < \frac{\sqrt{3}}{4}, - \frac{3}{2} < k_{y} < \frac{3}{2}$ ; note that this is contrary to the usual hexagonal choice so the top and bottom of the BZ are identified—note that this is crucial for the usual properties of the one-dimensional Wannier transform in the $y$ direction to hold. The equations governing the translational properties of the HWFs are also presented. (b) WCC positions (solid black lines) and single band Berry phases in the original Bloch bases (dashed red lines) of the two active bands. The top plot corresponds to the chiral limit, i.e., $η = 0$ , and the bottom one corresponds to the physical value of $η = 0.8$ . A small sublattice potential is added, $Δ = 0.19 meV$ . The configuration of the dashed lines and the solid lines mean that the two bands carry $+ 1$ and $- 1$ Chern numbers in the original Bloch representation and the parallel transport representation, respectively. This is a robust feature present in a wide range of parameter choices. Note that $k_{x}$ is rescaled and instead of plotting the interval $[- 0.5, 0.5)$ , equivalently [0,1) is drawn.
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Figure 2
The filling $ν = - 3$ , $η = 0$ , and $ℓ_{ξ} = 0.17 a_{M}$ plots within the first study, the angles are chosen around the magic value ( $α_{0} = 0.586$ corresponding to $θ = 1 . 05^{\circ}$ ). (a) Density of electrons (number of electron per unit length of the cylinder) versus $k_{x}$ (momentum across the cylinder) for different flavors where $s, ξ, m$ stand for spin, valley, and band. $α = 0.58$ , $Δ = 0$ and furthermore $g_{int} = 0.05$ have been chosen here, this value for the latter corresponds to being close to the strong interaction limit since the band width is very small. Almost full polarization is seen here. (b) The HF eigenvalues (energies), with the same parameters as described in (a). The Fermi surface is shown with the dashed red line, one can observe a HF gap which will be used as a criterion for determining whether QAHE has been stabilized under HF iterations or not. (c) The same plot as in (b) with $Δ = 1.9 meV$ . One can observe a second gap that is formed above the interacting one, due to the relatively large $Δ$ chosen; it separates the states belonging to the opposite sublattices. (d) The HF gap divided by $g_{int}$ as a function of $g_{int}$ for several parameter choices. It can be inferred that approaching the magic angle and making $Δ$ larger makes the QAHE more HF stable.
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Figure 3
Normalized gaps as functions of $g_{int}$ with longer range interaction and nonzero $η$ in the first study. $α = 0.58$ , and $Δ = 1.9 meV$ have been chosen. (a) Here both of the fillings are considered still in the chiral limit ( $η = 0$ ). The QAHE is more stable at larger screening lengths; interestingly, it is stabilized even for $ν = + 3$ for large interactions, if a large enough screening length is chosen. Note that this feature is lost for larger $η$ values and, in particular, $η_{phys} = 0.8$ , as discussed in the main text. (b) Gaps are drawn for $ν = - 3$ with $ℓ_{ξ} = 1.2 a_{M}$ here, as $η$ is increased. Increasing $η$ makes QAHE less stable until it is not stabilized at all even at large $g_{int}$ at $η \approx 0.9 - 0.95$ .
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Figure 4
The HF results obtained within the projected model, i.e., the model used in our second study. (a) The converged HF energies normalized by $g_{int}$ versus $k_{x}$ . The plots on the left and right columns correspond to $η = 0$ and $η = 0.8$ , respectively, while plots on the first and second rows correspond to $ν = - 3$ and $ν = + 3$ , respectively. In all four subplots, we take $α = 0.58, Δ = 0, ℓ_{ξ} = 0.5 a_{M}$ . The interaction strength is chosen as $g_{int} = 0.05$ and 0.002 for the left and right columns, respectively. Note that plots in each column have the same parameter values except for the filling factor. It can be seen that the HF energies at the two fillings $ν = \pm 3$ can be transformed into each other by the particle-hole symmetry of the CM, which, in particular, needs $k_{x} \to - k_{x}$ . It can be seen that for $η = 0$ , the two sets of HF energies are also related by the chiral symmetry of the CM. (b) The gaps as functions of $g_{int}$ . $α = 0.58, Δ = 0$ are chosen for this plot and $η$ and $ℓ_{ξ}$ are varied.
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Figure 5
Eigenvalues of the single hole potential shown in Eq. (D8). The following set of paramters has been used: $η = 0$ , $α = 0.58$ , $Δ = 1.9 meV$ , $ℓ_{ξ} = 0.1 a_{M}$ , $g_{int} = 0.1$ . Lower energies are available for smaller $k_{x}$ values. A magnified view of smallest energies is shown in panel (b).
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