# Chiral symmetry in Su-Schrieffer-Heeger (SSH) model

We know that the hamiltonian SSH model in the presence of on-site potential(V) can be written on the basis of the Pauli matrix.

$$h(k)=V\sigma_0+h_x\sigma_x+h_y\sigma_y,$$

and the term V breaks the chiral symmetry by shifting the zero energy topological edge state.

So, my question is: Does the identical matrix affect topology?

In short, the $$V$$ term in your Hamiltonian can be gotten rid of by going into a suitable rotating frame. Consider the following unitary $$U(t) = \exp(-i V \sigma_0 t/\hbar)$$ transforming the Hamiltonian leads to $$\begin{eqnarray} h'(k) &=& U^{\dagger}(t) h(k) U(t) - i\hbar U^{\dagger}(t) \frac{\partial}{\partial t} U(t)\\ &=& h(k) - \hbar V\sigma_0 = h_x \sigma_x + h_y \sigma_y \end{eqnarray}$$ which is chiral symmetric, i.e. $$\sigma_z h'(k) \sigma_z = - h'(k)$$ So, choosing $$U(t)\sigma_z$$ as your unitary operator for the chiral symmetry, this should mean that also $$h(k)$$ is chiral symmetric.