Topology in tight-binding models

OBC

PBC

We consider only one cell: \[ H = t_1 \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 1\\ 1 & 0 & 0 & 0 & 1 & 0\\ \end{pmatrix} \]

We build wave functions carrying the pseudospins \begin{align*} p_{\pm} &=& p_x \pm i p_y \\ d_{\pm} &=& d_{x^2-y^2} \pm i d_{xy} \end{align*}

From which we are able to build the operator $\sigma$ \[ \sigma = \mathbb{1}_{N_{\textrm{cluster}}}\otimes Q\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}Q^{-1} \] where $Q=(s\; p_+\; p_-\; d_+\; d_-\; f)$

We can now define a new Hamiltonian \[ H' = P\sigma P \]

From which we can split the pseudo-spins up and down: \[ H' = H_+\oplus H_- \]

And define a new topological invariant, the Spin Bott Index: \[ C_{\textrm{SB}} = \frac{C_+ - C_-}{2} \]

Reducing the size of the system:

\[ H' = H_{\textrm{Haldane}} + \sum_i w_i c_i^\dagger c_i \]