  Using first-principles calculations of graphene having high-symmetry
distortion or defects, we investigate band gap opening by chiral symmetry
breaking, or intervalley mixing, in graphene and show an intuitive picture of
understanding the gap opening in terms of local bonding and antibonding
hybridizations. We identify that the gap opening by chiral symmetry breaking in
honeycomb lattices is an ideal two-dimensional (2D) extension of the Peierls
metal-insulator transition in 1D linear lattices. We show that the spontaneous
Kekule distortion, a 2D version of the Peierls distortion, takes place in
biaxially strained graphene, leading to structural failure. We also show that
the gap opening in graphene antidots and armchair nanoribbons, which has been
attributed usually to quantum confinement effects, can be understood with the
chiral symmetry breaking.
