The Hamiltonian for SSH model can be written as
$h(k)=\begin {pmatrix}0&t_1+t_2exp^{-ika}\\t_1+t_2 exp^{ika}&0
\end{pmatrix}$
for finding the topological invariant Why we only calculate the winding number of either $t_1+t_2 exp^{-ika}$ or $t_1+t_2 exp^{-ika}$ where as both of these matrix terms have opposite widing numbers 1 and -1 respectively
Best Answer
The reason we calculate the winding number of only one block instead of calculating the winding number of the whole Hamiltonian is:
1) The winding number of the whole Hamiltonian will always be zero due to the chiral symmetry. 2) The winding number of just one block is a topologically meaningful quantity in the sense that changing the values of $t_1$ and $t_2$ continuously such that the spectral gap remains open will not change the winding number.
Since we are exactly looking for such topologically meaningful quantities, we compute the winding number of just one block.
In addition, one could also relate a physical meaning to this quantity in the form of electric polarization.