3 FIG. 2: Shifting of the DP | k 0 | normalized to the y coordi- nates of this point ( K y ) as a function of h 0 /τ (solid line). We also shown ˆ v$_{F}$ v$_{F}$ giving the reduction of the Fermi velocity (dotted line). Bloch theorem. They will have the form: A low energy Hamiltonian could be obtained from (3) expanding the Fourier transform of the electron opera- tors around two of the inequivalent points where the dis- persion relation vanish, namely the Dirac points (DP). We choose these points as given by K = 2 π 3 $_{a}$(1 , 1 √$_{3}$ ) and K ′ = 2 π 3 $_{a}$(1 , − 1 √$_{3}$ ). In addition we assume a smooth vari- ation of z ( x ), i.e. τ >> a . The low energy physics is described by the sum of two Dirac Hamiltonians for massless particles in a pseudo-magnetic field. It is given by: B. Dirac equation H = v$_{F}$ ∫ dr $^{2}$( Ψ † $_{1}$( r ) σ ∗ · ( − i ∇ + A ( r )) Ψ $_{1}$( r ) + Ψ † $_{2}$( r ) σ · ( − i ∇ − A ( r )) Ψ $_{2}$( r )) (4) Ψ$_{n,}$ k ( x, y ) = e i k · $^{r}$ψ$_{n,k}$$_{x}$ ( x ) (7) where n is a band index and k$_{x}$ belong to the Brillouin zone of the imposed periodic lattice i.e. − 2 π τ ≤ k$_{x}$ < 2 π τ . Otherwise there is not restrictions on the values of k$_{y}$ . Note that (6) is of the form of (7) with a quasi-momenta of the zero energy eigenstates given by k$_{0}$ = (0 , − k$_{0}$ ). and function ψ$_{0}$$_{,k}$$_{x}$ ( x ) given by: where v$_{F}$ = 3 t$_{0}$a 2 is the Fermi velocity, σ = ( σ$_{x}$, σ$_{y}$ ) is the vector of the Pauli matrices and σ ∗ its complex con- jugate. The two component field is Ψ i = ( c$_{iA}$, c$_{iB}$ ) with c$_{i,A}$$_{(}$$_{B}$$_{)}$ creation field operator over the A(B) sublattice The index i = 1 , 2 refers to states with momentum near K , K ′ . The gauge vector A ( r ) = ( A$_{x}$, A$_{y}$ ) were induced by the rippling and is given by: ψ$_{0}$$_{,}$ k ( x ) = N √ L$_{y}$L$_{x}$ ( a$_{0}$e F ( x ) b$_{0}$e − F ( x ) ) (8) 2 2 A$_{x}$ = 0 A$_{y}$ = − α 4 ( ∂$_{x}$h ) 2 = − α 4 ( h$_{0}$q ) $^{2}$( cos ( 4 π τ x ) + 1 2 ) ≡ ˆ A$_{y}$ + k$_{0}$ (5) 0 0,05 0,1 0,15 h$_{0}$/ 0,2 0,4 0,6 0,8 1 v$_{F}$ /v$_{0}$ | k$_{0}$| / K$_{y}$  = 72 a a$_{0}$ and b$_{0}$ satisfying a 0 + b 0 = 1. In (7) the Bloch function fulfill ψ$_{n,k}$$_{x}$ ( x + τ $_{2}$) = ψ$_{n,k}$$_{x}$ ( x ) and satisfy H ( k ) ψ$_{n,}$ k = E$_{n,}$ k ψ$_{n,}$ k , with H ( k ) given by: k$_{0}$ = − α $_{8}$( h$_{0}$q ) 2 is the k = 0 Fourier component of A$_{y}$ and q = 2 π τ the wave vector of the rippling. Note that as the time reversal symmetry is not broken by the rippling, this implies that the coupling of Ψ 1 with A ( r ) have different sign than the one of Ψ $_{2}$. This fact will be important in the study of the nanoribbons undertaken in the next Section. The Dirac equation in a magnetic field corresponding to the one particle problem generated by eq. (4) at i = 1 could be exactly solved for the zero energy eigenstate. By following similar steps as in Ref. 14 we find the following two degenerated eigenfunctions: As H$_{0}$ ( k ) ψ$_{0}$$_{,}$ k ( x ) = 0 we identify H$_{0}$ ( k ) as the unper- turbed Hamiltonian. The approximated near-zero energy eigenstates could be obtained by diagonalizing H$_{int}$ in the subspace of the degenerated zero energy states. H$_{int}$ proyected onto this subspace is a 2 × 2 matrix called h and given by: H ( k ) = H$_{0}$ ( k ) + H$_{int}$ ( k ) H$_{0}$ ( k ) = v$_{F}$ σ ∗ · ( − i ∇ + ˆ A ( r )) H$_{int}$ ( k ) = v$_{F}$ σ ∗ · ( k + k$_{0}$ ) (9) h = ˆ v$_{F}$ ( 0 k$_{x}$ + i ( k$_{y}$ + k$_{0}$ ) k$_{x}$ − i ( k$_{y}$ + k$_{0}$ ) 0 ) = = ˆ v$_{F}$ σ ∗ · ( k + k$_{0}$ ) (10) E$_{1}$ ( k$_{x}$, k$_{y}$ ) = ± ˆ v$_{F}$ √ k 2 x + ( k$_{y}$ + k$_{0}$ ) 2 (11) ˆ v$_{F}$ = N $^{2}$v$_{F}$ is a renormalized Fermi velocity. The eigen- values of (10) are: The low energy eigenstates of (4) can be studied by de- generated perturbation theory over the zero energy states Ψ$_{+0}$( x, y $^{) and Ψ}$− $_{0}$( x, y ). The potential A$_{y}$ is periodic with period τ $_{2}$. Therefore the solutions of (4) fulfill the We see that the main effects of the corrugation are: 1. DP sited in K have been shifted to K − k$_{0}$ . Ψ$_{+0}$( x, y ) = N √ L$_{y}$L$_{x}$ e − ik$_{0}$y ( e F ( x ) 0 ) $^{Ψ}$− $_{0}$( x, y ) = N √ L$_{y}$L$_{x}$ $_{e}$− ik$_{0}$y ( 0 $_{e}$− F ( x ) ) (6) α 2 with F ( x ) = $_{8}$h $_{0}$q sin(2 qx ) and L$_{x,y}$ the system length in both direction. N is a normalization constant given by N = [ I$_{0}$ ( αh 2 $_{0}$q 4 )] − 1 2 with I$_{0}$ the modified Bessel function of zero order.