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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

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  • $\begingroup$ The winding number depends on the ration of $t_1$ an $t_2$. If $|t_1|>|t_2|$, both have zero winding numbers. $\endgroup$ – Meng Cheng Apr 2 '15 at 2:14
  • $\begingroup$ @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 $h(k)$ $\endgroup$ – user48826 Apr 2 '15 at 2:33
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    $\begingroup$ 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. $\endgroup$ – Meng Cheng Apr 2 '15 at 3:09
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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.

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Winding number in this model by definition is the winding of (pseudo-)spin polarization of the filled band as momentum goes over the Brillouin zone. It is not obtained by computing the phase winding of the matrix element in one block.

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

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