1D Topological Hamiltonians

Lattice Hamiltonian (Generalized):

$$ H = \sum_n \left[ \Delta_n(c_{n,A}^\dagger c_{n,A} - c_{n,B}^\dagger c_{n,B}) + \left( v_n c_{n,A}^\dagger c_{n,B} + w_n c_{n,B}^\dagger c_{n+1,A} + t_A c_{n,A}^\dagger c_{n+1,A} + t_B c_{n,B}^\dagger c_{n+1,B} + u_1 c_{n,A}^\dagger c_{n+1,B} + u_2 c_{n,B}^\dagger c_{n+2,A} + h.c. \right) \right] $$

N Cells: PBC: v (Intra): w (Inter): u1 (3NN A-B): u2 (3NN B-A): tA (NNN A): tB (NNN B): Δ (Stagger): DW in Hoppings DW in Staggering

Dirac Hamiltonian Discretization:

$$ H_D \approx m(x)\sigma_x + i v_F \sigma_y \partial_x $$

Discretized naively via: $v(x) = v_F + m(x)/2$ and $w(x) = v_F - m(x)/2$.

N Cells: PBC: Profile: m0 (Mass): vF (Velocity): χ (DW Width): Δ (Constant):

Local Sublattice Polarization: Calculated using $P_{A/B}^n = \frac{\sum_{\langle i, j \rangle} \left| |\psi_i^{(n)}|^2 - |\psi_j^{(n)}|^2 \right|}{\sum_{\langle i, j \rangle} \left| |\psi_i^{(n)}|^2 + |\psi_j^{(n)}|^2 \right|}$, where $\langle i, j \rangle$ denotes intra-cell A and B pairs.

Bulk Bloch Hamiltonian

For the uniform lattice with parameters $v, w, t_A, t_B, u_1, u_2$ and constant staggering $\Delta$, the momentum-space Hamiltonian $H(k) = \vec{d}(k) \cdot \vec{\sigma} + d_0(k)\mathbb{I}$ is given by:

$$ d_0(k) = (t_A + t_B)\cos(ka) $$

$$ d_x(k) = v + (w + u_1)\cos(ka) + u_2\cos(2ka) $$

$$ d_y(k) = (w - u_1)\sin(ka) + u_2\sin(2ka) $$

$$ d_z(k) = \Delta + (t_A - t_B)\cos(ka) $$