# Winding number for the Su-Schrieffer-Heeger (SSH) model

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

• The winding number depends on the ration of $t_1$ an $t_2$. If $|t_1|>|t_2|$, both have zero winding numbers. Apr 2, 2015 at 2:14
• @MengCheng ok you are right but my question was why we only calculate winding number of $t_1+t_2exp^{-ika}$ and not of whole $h(k)$ Apr 2, 2015 at 2:33
• The Hamiltonian has a symmetry, which basically says the diagonal terms must be zero. Within this space of Hamiltonians, there are two classes distinguished by this winding number. In a sense, this is the "winding number of the whole $h(k)$, whatever that means. Apr 2, 2015 at 3:09

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.
You can show by yourself that if $|t_1|<|t_2|$ such a winding number is 1.