Band model of the graphene bilayer

23and because this is really just a 2 × 2matrixinthelow-energysubspacewecanwriteitas:H eff =⎛ ⎞{ v2 F p2 [2t⊥t 2 ⊥ − v 4 +∆(1+v 2∆2 4 )] ⎝ 1 0 ⎠0 1⎛− [ t ⊥ (1 + v4)+2v 2 4 ∆ ] ⎝ 0The corresponding eigenvalues are:E eff,± ≈v2 F p2 [t 2 ⊥ − 2t⊥ v 4 +∆(1+v4) 2 ]∆2√{ v2± (v 3 v F p) 2 + Fp 2[ t ⊥ (1 + v4 2)+2v 4∆ ]e−i2φe i2φ 0⎞ ⎛ ⎞}⎠ + v 3 v F p ⎝ 0 eiφ⎠ . (3.8)e −iφ 0} 2 2v 3 v −F 3 p3[ t ⊥ (1 + v4 2)+2v 4∆ ]t 2 ⊥ − ∆2 t 2 ⊥ − ∆2cos(3φ).(3.9)This expression shows that v 4 and ∆ weakly breaks the particle-hole symmetry of thesystem and that v 3 is responsible for breaking the cylindrical symmetry and the so-called“trigonal warping” of the energy bands.A simplified model that takes only the termsinvolving t ⊥ and the trigonal warping v 3 into account isH eff = − v2 F p2t ⊥⎛⎝ 0e−i2φe i2φ 0⎞ ⎛ ⎞⎠ + v 3 v F p ⎝ 0 eiφ⎠ . (3.10)e −iφ 0An even simpler model which neglects both the electron-hole asymmetry and the trigonalwarping is:H eff = − v2 F p2t ⊥⎛⎝ 0e−i2φe i2φ 0⎞⎠ . (3.11)This form is interesting since it gives rise to massive “chiral” quasi-particles (McCann andFal’ko, 2006). Here “chirality” means that there exist an operator Ĉ defined by⎛Ĉ≡−⎝ 0e i2φ 0e−i2φ⎞⎠ , (3.12)