Projective representation of D6 group in twisted bilayer graphene*

Noah F.Q.Yuan

Department of Physics,Massachusetts Institute of Technology,Cambridge,Massachusetts 02139,USA

Keywords: twisted bilayer graphene,tight-binding model,projective representation

1. Introduction

The recent experimental discovery of superconducting and strongly correlated insulating phases in the twisted bilayer graphene (TBG)[1-3]attracts tremendous attention and triggered massive theoretical studies.[4-22]TBG is a bilayer system in which two graphene layers are stacked to each other with rotational misalignment,[26-31]and the electronic band structure sensitively depends on the twist angleθ. Theoretically a smallθcan produce a long-period moir´e pattern, causing a severe folding of original bands and hence creating new low-energy bands in the moir´e Brilluoin zone(MBZ),[32-44]especially whenθis near the so-called the magic angles.[39,41,43]Experimentally in TBG samples atθnear a magic angle of 1.05°, the insulating phase and the superconducting phase are observed around the filling of two electrons or holes per supercell.[1-3]

To understand the insulating and superconducting phases in TBG,one needs to construct Wannier orbitals and hence an effective tight-binding model for the lowest bands which are separated from the rest by energy gaps. We thus focus on the low-energy physics of TBG and neglect other details, which is captured by a long wave length model known as the continuum model of TBG.[42-44]In this work, we discuss the band structure of TBG within the framework of continuum model.

The low-energy band structure of TBG within the continuum model can be computed and turns out to depend on the twist angle. Near the magic angle, the band width is close to a local minimum,and the band symmetry has been studied intensively. Based on these band structure properties, some theoretical studies claim that Wannier orbitals of the lowest bands cannot be constructed to respect essential symmetries in TBG,[5,7,10]due to the so-called fragile topology.[22-24]While in some works of TBG,Wannier orbitals can be and have been constructed explicitly.[6-8]

In this work, we try to address on the Wannier orbital issue of TBG within continuum model, especially the appropriate representation of the symmetry group in the basis of these Wannier orbitals. We show that the Wannier orbitals constructed from the continuum model in Ref.[7]can furnish a projective representation of the point group, and the corresponding tight-binding model can reproduce the band structure of the lowest bands without fine tuning. To further test our construction, we apply external fields to TBG and show that the Wannier orbitals can also correctly describe the response to electromagnetic fields.

2. Projective representation of emergent D6 group

We consider TBG on the long-range (superlattice constantLM～10 nm) and low-energy (bandwidth of the lowest bands～10 meV)scales. Three special spots are found on the long range,namely,AA,AB,and BA spots,which are all centers for three-fold rotations, and AA is also the six-fold rotation center. The low-energy physics can be described by the continuum model,where both the emergent valley U(1)symmetry andD6point group symmetry are respected.

After successive symmetry operationsg2,g1∈D6,the final operation isg1g2up to an U(1)phaseφ

whereUdenotes the representation ofD6andτis the generator of valley U(1)symmetry. This type of representation ofD6is known to be projective.

We then apply the projective representation ofD6to Wannier orbitals of TBG within continuum model. We first consider TBG with atomic point groupD3(Fig. 1(a)). From symmetry arguments, the Wannier orbitals of the lowest four bands reside on the honeycomb lattice formed by AB and BA spots and have (px,py) on-site symmetry underD3,[6]which have been constructed from atomic orbitals explicitly without U(1) symmetry.[8]Within the continuum model, the Wannier orbitals from individual valleys have also been constructed explicitly,[7]which we may call continuum model Wannier orbitals(CMWO).Two generators ofD6are found in TBG:The in-plane six-fold rotationC6zand out-of-plane two-fold rotationC2x. When acting on CMWO, besides real-space operations,in momentum spaceC6zwill flip the valley index whileC2xwill not,hence

where|r,τ〉denotes CMWO at siterwith valleyτ=±.From Ref.[7],we find that underC3z,the CMWO with valleyτhas angular momentum-τ,namely,

AsC3z=C26z, to reconcile Eqs. (2) and (4), we must haveφ(C6z,C6z)=2π/3. Other projective phasesφ(g1,g2)can be worked out similarly. With nontrivial projective phases listed in Table.1,we find that CMWO can furnish the projective representation ofD6satisfying Eqs. (1)-(3), which incorporates the atomic representation ofD3. Under projective representation ofD6,CMWO can have s and f≡fx(x2-3y2)on-site symmetries,and|τ〉=|s〉+iτ|f〉.

Fig.1. Twisted bilayer graphene at θ =1.05° with atomic point groups(a)D3 and(b)D6. Black and red grids denote layers 1 and 2,A and B sublattices are also shown.

Table 1.The projective U(1)phase φ(g1,g2)of the projective representation in D3 structure TBG,where g1,g2 ∈D6 are listed in rows and coloumns.

However, when we consider TBG with atomic point groupD6(Fig. 1(b)), and adopt the atomic representation ofD6, the Wannier orbitals of the lowest four bands cannot be constructed from individual valleys.[5,7]This means that the atomic representation ofD6is different from the projective representation ofD6furnished by CMWO.In fact,the atomic representation ofC2xis different from its projective representation, hence if the atomic point group includesC2x, then its atomic representation will not be compatible with the projective representation ofD6.

In terms of CMWO, we can write down the following tight-binding(TB)model of TBG:

whereciτis the annihilation operator of CMWO at siteiwith valleyτ,rijis the vector from siteitoj, andt(r),φ(r) are the amplitude and phase of hopping integral alongr, respectively. The form of the TB model above manifests the timereversal symmetryT, and point groupD6further restrictsφ.For example, when we applyC6zandT, we findφ(r)=0 whenr‖Cn6zˆy,n=0,1,...,5. Details of this TB model can be found in Ref.[7].

Due to projective representation, the emergent symmetry group of TB model Eq.(5)should be written as a semidirect productD6U(1). To be complete, we can also include spin SU(2)and time reversal symmetryTand write the whole symmetry group in terms of both semidirect and direct productsD6U(1)×T×SU(2),which leads to an eight-fold Dirac point (including spin) at eachKpoint of the moir´e Brillouin zone.

When we break some symmetries and lower the symmetry group to its subgroup by external fields and other means,the band structure can be altered and the Dirac points at±Kpoints can be split. Such altered band structure can also be captured by the TB model in the basis of CMWO, which we will discuss in the following.

3. Responses to electromagnetic fields

As applications of CMWO and the effective TB model,we consider the band structure of TBG under electromagnetic fields.

An out-of-plane electric field will introduce an interlayer potential differenceV.From the continuum model we can find that at eachKpoint,Vwill split the eight-fold Dirac point into two four-fold Dirac points (spin included). AsVbreaks outof-plabe rotationC2xbut preserves in-plane rotationC2z, the Dirac points from individual valleys are protected byC2zT.The numerical results are shown in Fig.2.

In the basis of CMWO,VfurnishesA2representation ofD6and is invariant under U(1)×SU(2)×T,which we may call as a pseudo scalar. The linear response of the system toVcan be described by the HamiltonianHe=Vℋe, and the invariance requiresℋeto have the same symmetry representation asV,namely,a pseudo scalar furnishingA2representation ofD6and invariant under U(1)×SU(2)×T. One then searches for such pseudo scalars order by order in tight-binding (TB)framework. We find that the on-site and nearest-neighbor terms cannot be a pseudo scalar. Hence the leading order TB contribution ofVis the next-nearest neighbor term

whereLn=LMCn3zˆy(n=0,1,2)are three bonding vectors of the next-nearest neighbors with superlattice constantLM, and(-)j=±forj=AB and BA spots,respectively. The dimensionless coupling constantgedepends on twist angleθ. We can add higher order TB terms(i.e.,hopping at longer range)to improve our TB description of the electric field response.

Fig.2. Band structure of TBG θ =1.05° with interlayer potential difference V =5 meV based on continuum model, where blue and red lines are from valleys τ =±, respectively. The moir´e Brillouin zone and high symmetry lines (thick and black) for band structure are also shown at the left upper corner.

Another more subtle way to introduce an electric field is by substrates such as transition metal dichalcogenides or hexagonal boron nitrides,and the induced electric field will be highly nonuniform on atomic scale. In this case, the electric field can be captured by both interlayer potential differenceVand sublattice potential differenceV′.

Similarly, we find thatV′furnishesB1representation ofD6and is invariant under U(1)×SU(2)×T. Hence the leading order TB contribution ofV′is

Fig. 3. Energy contours of the twisted bilayer graphene at θ =1.05°with in-plane magnetic field B=1 T along ˆx direction,based on continuum model. The blue and red lines denote bands from valley τ =±,and e,h denote electron and hole sides,respectively.

In the basis of CMWO,Bis a pseudo vector [E1representation ofD6, odd underTand invariant under U(1)×SU(2)]. Similarly, we search for TB terms which are pseudo vectors and find that unlike the spin effect,the on-site terms cannot be pseudo vectors. The leading order TB contribution of orbital magnetic fieldBis actually the nearestneighbor term

where ˆeijis the unit vector from siteitoj, andgmis the dimensionless coupling constant analogous to sping-factor.Similar toge(θ), the dimensionlessgmis also a function of twist angleθ.Higher order TB terms at longer hopping ranges can be further included to improve our description of magnetic field response in TBG.

Within each valley, the TB term Eq. (8) describes the distortion of hopping integral along magnetic field direction,while the distrotions are opposite for opposite valleys. Thus we may regard the orbital fieldBas a valley-contrasting strain field, which distorts energy contours as shown in the continuum model and Fig.3.

4. Conclusion

We show that Wannier orbitals can be constructed from each valley to furnish the projective representation of the point groupD6in TBG within the framework of continuum model.Such Wannier orbitals and the corresponding tight-binding model can correctly describe the lowest bands without and with external magnetic fields.

Acknowledgement

We thank Liang Fu for important discussions.