Abstract
In this work, we determine states of electronic order of small-angle twisted bilayer graphene. Ground states are determined for weak and strong couplings which are representatives for varying distances of the twist-angle from its magic value. In the weak-coupling regime, charge density waves emerge which break translational and C 3-rotational symmetry. In the strong coupling-regime, we find rotational and translational symmetry breaking Mott insulating states for all commensurate moiré band fillings. Depending on the local occupation of superlattice sites hosting up to four electrons, global spin-(ferromagnetic) and valley symmetries are also broken which may give rise to a reduced Landau level degeneracy as observed in experiments for commensurate band fillings. The formation of those particular electron orders is traced back to the important role of characteristic non-local interactions which connect all localized states belonging to one hexagon formed by the AB- and BA-stacked regions of the superlattice.
Highlights
The temperature–gate voltage phase diagram of twisted bilayer graphene (TBG) in the small-angle regime is characterized by correlated insulator states as seen in transport experiments [1,2,3,4,5]
Their regular pattern of occurrences at commensurate fillings of the weakly dispersing moiré bands with bandwidths as small as 10meV [6] indicates an enhanced role of interaction effects, including strong-coupling Mott physics complemented by other complex electron phenomena such as superconductivity [1,2,3,4,5], linear-in-temperature resistivity [5], correlated electron states observed in scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) measurements [7,8,9,10], ferromagnetism and quantum hall physics [11]
By assuming the presence of a spin-rotation and a valley symmetry, which is justified in the limit of vanishing spin–orbit coupling [43] and small twist-angles [15], this observation may be interpreted as follows: it implies for the insulator state at charge neutrality (ν = 0) which is found to be 4-fold degenerate the presence of both the spin and valley symmetry, for half electron- or hole-filling (ν = ±1/2) which are two-fold degenerate that either the spin or the valley symmetry is broken, and for the band fillings ν = ±3/4 which are single degenerate that both the spin and the valley symmetry are absent [2,3,4]
Summary
We first obtain the weakly dispersing moiré bands by considering a continuum model [6, 44] where states near the slightly twisted Dirac cones of the two graphene layers hybridize due to a finite inter-layer coupling. Longer-distance interactions processes are numerically smaller due to the absence of a direct overlap of Wannier functions This trend is further enhanced by screening effects which are present due to the short distance of the TBG sample to the back gate which can be of order of the superlattice unit cell. Up to an overall change of the amplitudes, which can be compensated in a redefinition of β, the quantitative changes of the ratio of the various interaction elements are small for twist-angles in the vicinity of the magic-angle and are found to not affect the results of the subsequent analysis This is traced back to the fact that the predominant contribution to the interaction matrix elements arises from the areas of a direct overlap of Wannier functions where screening is inefficient. Is characteristic for small-angle TBG and will decisively determine the nature of the electronic ground states discussed
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