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B.1 Bloch Wave Functions Prefactors

To obtain CA and CB in Eq. 3.18, one can substitute Eq. 3.6 and Eq. 3.19 into the Shršodinger equation H|ψ⟩ = E|ψ⟩. Considering an A-type carbon atom at some atomic site n, and its three nearest neighbors, the Hamiltonian can be written as:

H = t|B ⟩⟨A | + t|B′ ⟩⟨A | + t|B ⟩⟨A |. N-n+1 n N -n+1 n N- n n
(B.1)

Using Eq. B.1 along with the wave functions obtained in Eq. 4.12, one obtains:

A B ECAeikxx n sin(nθ) = tCBeikxx N-n+1 sin((N - n + 1 )θ ) ′ + tC eikxxBN- n+1 sin ((N - n + 1)θ) B ikxxBN- n + tCBe sin((N - n )θ).
(B.2)

Therefore, the relation between CA and CB can be written as:

[ ( ) ik (xB′ -xA ) ECA sin(nθ) = tCB eikx(xBN-n+1-xAn) + e x N-n+1 n ] ikx(xBN -n-xAn) sin ((N - n + 1 )θ ) + e sin((N - n)θ) [ ( √ -- ) ] 3 = tCB 2cos ---kxacc sin ((N - n + 1)θ) + sin((N - n )θ) . 2
(B.3)

By employing the relation sin(x) sin(y) = (1∕2)[cos(x-y) - cos(x + y)] and using Eq. 4.22,

EC = - tC ----sin(θ)----. A B sin((N + 1)θ)
(B.4)

Analogously, for the N - n + 1th B-type carbon atom one can obtain the following relation:

[ ( ) ] √3-- ECB sin ((N - n + 1)θ) = tCA 2 cos ----kxacc sin(n θ) + sin((n - 1)θ) 2
(B.5)

which gives

EC = - tC ----sin(θ)----. B A sin((N + 1)θ)
(B.6)

From Eq. B.4 and Eq. B.6, one can find that CA = ±CB.
Also, the dispersion relation can be found by multiplying Eq. B.3 by Eq. B.5,

2 E CACB sin(nθ)sin((N - n + 1)θ)( ) √ -- = t2CACB [4 cos2 --3kxacc sin((N - n + 1)θ)sin(nθ) 2 ( √ -- ) --3- + 2cos 2 kxacc sin((N - n + 1)θ)sin ((n - 1)θ) ( √ -- ) 3 + 2cos ---kxacc sin((N - n )θ)sin(nθ) 2 + sin ((N - n )θ )sin ((n - 1 )θ )].
(B.7)

With the help of trigonometric identities and Eq. 4.22, this expression can be reformatted as

[ ( √ -- ) ( √ -- ) ]1∕2 2 3 3 E = ±t 1 + 4 cos ---kxacc + 4cos ---kxacc cos(θ) . 2 2
(B.8)

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