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We investigate correlated insulating states in magic-angle twisted bilayer graphene (TBG) by the exact diagonalization method applied to the extended Hubbard model with interaction parameters recently evaluated in the realistic effective model.
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Direct Coulomb interaction between two point charges is assumed to be proportional to 1 / r 1/r where r r is the distance between the point charges. When two point charges exist in the same place, the direct Coulomb energy between them is evaluated as a finite constant. For detail, see Ref. [ 23 ]
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A sum of the V 0 , V 1 , V 2 , V 3 V_{0},V_{1},V_{2},V_{3} contributions per hexagon is V 0 ( N 2 − N ) / 6 V_{0}(N^{2}-N)/6 where N = ∑ i = 1 6 n i N=\sum_{i=1}^{6}n_{i} provided that V 0 : V 1 : V 2 : V 3 = 3 : 2 : 1 : 1 V_{0}:V_{1}:V_{2}:V_{3}=3:2:1:1 . For example, the V 0 V_{0} contribution per hexagon equals ( V 0 / 3 ) ∑ i = 1 6 n i ( n i − 1 ) / 2 (V_{0}/3)\sum_{i=1}^{6}n_{i}(n_{i}-1)/2 , where the factor 1 / 3 1/3 comes from the fact that each vertex is shared with three neighboring hexagons. The expression V 0 ( N 2 − N ) / 6 V_{0}(N^{2}-N)/6 corresponds to a product of the number of the pairs of fractional charge − e / 3 -e/3 in this hexagon and the on-site interaction per fractional-charge pair, V 0 / 3 V_{0}/3 [see Fig. 1
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