Abstract
The honeycomb lattice possesses a novel energy band structure, which is characterized by two distinct Dirac points in the Brillouin zone, dominating most of the physical properties of the honeycomb structure materials. However, up till now, the origin of the Dirac points is unclear yet. Here, we discover a hidden symmetry on the honeycomb lattice and prove that the existence of Dirac points is exactly protected by such hidden symmetry. Furthermore, the moving and merging of the Dirac points and a quantum phase transition, which have been theoretically predicted and experimentally observed on the honeycomb lattice, can also be perfectly explained by the parameter dependent evolution of the hidden symmetry.
Highlights
For the general honeycomb lattice with β = 1 (β is defined as the hopping amplitude ratio β = t 2/t1), the corresponding Dirac points locate at (±2π/3 cos θ,0) in the Brillouin zone
We have proved above that the Dirac points on the honeycomb lattice are protected by the hidden symmetry Λθ,β
We have found a hidden symmetry on the honeycomb lattice and proved that the hidden symmetry protects the Dirac points on the honeycomb lattice
Summary
For the general honeycomb lattice with β = 1 (β is defined as the hopping amplitude ratio β = t 2/t1), the corresponding Dirac points locate at (±2π/3 cos θ,0) in the Brillouin zone. The mapping ω1,θ has the effect on the Bloch functions and the wave vectors as ω1,θΨθ,k (r) = Ψb,p(r) and ω1,θ: (kx, ky ) (px , py ) = (cos θkx ,(1 + sin θ)ky ).
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