(a) Hamiltonian in the sublattice space. (i) By performing the Fourier transformations |n, A) = keink...

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(a) Hamiltonian in the sublattice space. (i) By performing the Fourier transformations |n, A) = Σkeink |k, A) and (n, B| = √eink (k, B| (with N the total number of sites), show that the Hamiltonian can be expressed as  = Σ ħ(k), where h(k) = (v + we-ik) |k, A) (k, B| + h.c., (10) Useful formula: Ene-in(k-k') = √k,k'- (ii) Show that, in the (2-D) sublattice space spanned by the base kets {k, A), k, B)}, the Hamiltonian (k) can be written as h(k)=d(k).ô, (11) where [= (r, y, z)] is the vector of Pauli matrices, d, (k) = v + w cos (k), dy (k)= w sin (k), and dz (k)= 0. Hint: Write the Hamiltonian h (k) as a 2 × 2 matrix with matrix elements (k, i| h (k) |k, i'), where i(i') = A or B. (b) Find the energy bands and tune the band gap. (i) By diagonalizing h(k), show that its energy eigenvalues of h (k) are given by E = ±d (k)| with |d(k)| the magnitude of the vector d(k). (ii) For three different cases, namely, 1) v>w, 2) v = w, and 3) v<w, sketch 1) the trajectory of the tip of the vector d(k) in the d-dy plane as we vary the parameter k from 0 to 27, and 2) the dispersion relations E+(k)-vs-k for k € [-, π]. (c) (d) Show that, by applying two successive rotations - one by π/2 about the y-axis followed by another about the z-axis by ok (where tanok = dy/d₂]) - to the eigenvector of ô₂ with eigenvalue +1 (i.e., the column vector (1,0) in matrix form), the eigenvector of h(k) corresponding to eigenvalue E+ (k) can be written as 1 e-ion/2 |u(k))} = √/2 (eion/2 ) The electric polarization of the system may be calculated via Pe (13) where the closed loop, C, represents the trajectory of the tip of the vector d(k) as we vary the parameter k from 0 to 27. Show that the polarization is given by (14) 21 Jo :{ d dk (u(k)|i|u(k)), dk Pe= = -e/2, v<w, (12) v>w. (a) Hamiltonian in the sublattice space. (i) By performing the Fourier transformations |n, A) = Σkeink |k, A) and (n, B| = √eink (k, B| (with N the total number of sites), show that the Hamiltonian can be expressed as  = Σ ħ(k), where h(k) = (v + we-ik) |k, A) (k, B| + h.c., (10) Useful formula: Ene-in(k-k') = √k,k'- (ii) Show that, in the (2-D) sublattice space spanned by the base kets {k, A), k, B)}, the Hamiltonian (k) can be written as h(k)=d(k).ô, (11) where [= (r, y, z)] is the vector of Pauli matrices, d, (k) = v + w cos (k), dy (k)= w sin (k), and dz (k)= 0. Hint: Write the Hamiltonian h (k) as a 2 × 2 matrix with matrix elements (k, i| h (k) |k, i'), where i(i') = A or B. (b) Find the energy bands and tune the band gap. (i) By diagonalizing h(k), show that its energy eigenvalues of h (k) are given by E = ±d (k)| with |d(k)| the magnitude of the vector d(k). (ii) For three different cases, namely, 1) v>w, 2) v = w, and 3) v<w, sketch 1) the trajectory of the tip of the vector d(k) in the d-dy plane as we vary the parameter k from 0 to 27, and 2) the dispersion relations E+(k)-vs-k for k € [-, π]. (c) (d) Show that, by applying two successive rotations - one by π/2 about the y-axis followed by another about the z-axis by ok (where tanok = dy/d₂]) - to the eigenvector of ô₂ with eigenvalue +1 (i.e., the column vector (1,0) in matrix form), the eigenvector of h(k) corresponding to eigenvalue E+ (k) can be written as 1 e-ion/2 |u(k))} = √/2 (eion/2 ) The electric polarization of the system may be calculated via Pe (13) where the closed loop, C, represents the trajectory of the tip of the vector d(k) as we vary the parameter k from 0 to 27. Show that the polarization is given by (14) 21 Jo :{ d dk (u(k)|i|u(k)), dk Pe= = -e/2, v<w, (12) v>w.

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a Hamiltonian in the sublattice space i Performing the Fourier transformation we have hk n eikn hn n eikn un An wn Bn where hn is the Hamiltonian in t... View the full answer