Correlations and electronic order in a two-orbital honeycomb lattice model for twisted bilayer graphene

Jörn W. F. Venderbos and Rafael M. Fernandes

Phys. Rev. B 98, 245103 – Published 3 December 2018

Abstract

The recent observation of superconductivity in proximity to an insulating phase in twisted bilayer graphene (TBG) at small “magic” twist angles has been linked to the existence of nearly flat bands, which make TBG a fresh playground to investigate the interplay between correlations and superconductivity. The low-energy narrow bands were shown to be well described by an effective tight-binding model on the honeycomb lattice (the dual of the triangular Moiré superlattice) with a local orbital degree of freedom. In this paper, we perform a strong-coupling analysis of the proposed $(p_{x}, p_{y})$ two-orbital extended Hubbard model on the honeycomb lattice. By decomposing the interacting terms in the particle-particle and particle-hole channels, we classify the different possible superconducting, magnetic, and charge instabilities of the system. In the pairing case, we pay particular attention to the two-component ( $d − wave$ ) pairing channels, which admit vestigial phases with nematic or chiral orders, and study their phenomenology. Furthermore, we explore the strong-coupling regime by obtaining a simplified spin-orbital exchange model which may describe a putative Mott-type insulating state at quarter-filling. Our mean-field solution reveals a rich intertwinement between ferromagnetic and antiferromagnetic orders with different types of nematic and magnetic orbital orders. Overall, our work provides a solid framework for further investigations of the phase diagram of the two-orbital extended Hubbard model in both strong- and weak-coupling regimes.

Received 11 September 2018
Revised 6 November 2018

DOI:https://doi.org/10.1103/PhysRevB.98.245103

Physics Subject Headings (PhySH)

Superconducting order parameter Superconductivity

Honeycomb lattice Mott insulators

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jörn W. F. Venderbos^1,2 and Rafael M. Fernandes³

¹Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
²Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
³School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

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Vol. 98, Iss. 24 — 15 December 2018

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Images

Figure 1
Orbital and spin ordering. Schematic picture of the intertwined spin and orbital orderings appearing in the Mott insulating state at quarter-filling, as discussed in Sec. 5. Solid and dashed orbitals refer to the different $p_{y}$ and $p_{x}$ orbitals. (a) Antiferromagnetic ferro-orbital order; (b) ferromagnetic antiferro-orbital order; (c) antiferromagnetic ferro-orbital-magnetic order with complex orbitals.
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Figure 2
Twisted bilayer graphene. Figure of two twisted graphene sheets, shown as black and red honeycomb nets, with commensurate Moiré superlattice periodicity. In this commensurate realization of twisted bilayer graphene, the twist angle is $θ = 6 . 01^{\circ}$ and the twist center is a pair of registered carbon atoms which defines the origin. The triangular superlattice vectors connecting regions of $A A$ stacking are shown by dashed arrows. The black and red dots indicate the sites of the dual honeycomb (super)lattice and correspond to regions of $A B$ and $B A$ stacking, respectively.
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Figure 3
Symmetry of TBG. (Left panel) Example of TBG structure with $D_{3}$ point-group symmetry. The twist rotation axis is coincident with a pair of registered carbon atoms. The structure has a twofold rotational symmetry $C_{2 y}$ about the $y$ axis, and $C_{3 z}$ threefold rotational symmetry about the $z$ axis. (Right) For comparison, we show a TBG structure where the twist rotation axis is coincident with the center of a hexagon, resulting in $D_{6}$ point-group symmetry. This implies an additional twofold rotational symmetry $C_{2 x}$ , and a $C_{6 z}$ sixfold rotational symmetry about the $z$ axis. Both structures, left and right, have the same twist angle (and Moiré period), which was chosen large for illustrative purposes. Importantly, the twist center is also the center for the $C_{3 z}$ rotations, both for the $D_{3}$ and $D_{6}$ structures.
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Figure 4
Honeycomb superlattice model. (Left) Sketch of the effective honeycomb lattice extracted from twisted bilayer graphene with commensurate twist angle (see Fig. 2). The triangular Moiré superlattice, defined by the regions of $A A$ stacking, is shown by solid lines. Red and black solid dots represent the sites of the honeycomb lattice (indicated by dashed lines), with different colors corresponding to the triangular sublattices of the honeycomb lattice. The sublattice sites coincide with regions of $A B$ and $B A$ stacking. (Right) Definition of lattice vectors. Here, $a_{1, 2, 3}$ are lattice vectors of the (triangular) Moiré superlattice and ${\hat{e}}_{1, 2, 3}$ are unit vectors corresponding to the directions of nearest-neighbor bonds.
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Figure 5
Phase diagram of the classical mean-field solution of the spin-orbital exchange model in the isotropic case ( $t_{σ} = t_{π}$ ), obtained by minimizing the bond energy $E_{bond}$ (here plotted in units of $Δ = t^{2} / U$ ) as function of the ratio $J / U$ . AFM refers to antiferromagnetic order, FM to ferromagnetic order, FO to ferro-orbital order, and AFO to antiferro-orbital order. For $J < 0$ , the orbital order lowers the point-group symmetry of the honeycomb lattice, and is thus nematic [Fig. 1]. For $0 < J < U / 3$ , there is an enlarged $SU (2)$ symmetry in the orbital degrees of freedom, and the orbital order can be either nematic or magnetic [Fig. 1 illustrates the nematic case]. For $J > U / 3$ , the system has orbital-magnetic order [Fig. 1].
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