R}\end{aligned}$$ where we define Berry potential as, $$\begin{aligned}
{\bf A}_n({\bf R}) = i \bra{n({\bf R})} \nabla_{\bf R} \ket{n({\bf R})} \end{aligned}$$ From quantum mechanics, we know that by multiplying the states by an overall [*global*]{} phase factor, $$\begin{aligned}
\ket{n({\bf R})} \rightarrow \ket{\tilde{n}({\bf R})}  = e^{i\chi({\bf R})} \ket{n({\bf R})} \end{aligned}$$ the dynamics of the system does not change - this is known as [*gauge invariance*]{}. However, we see that the Berry Potential ${\bf A}_n({\bf R})$, is not a gauge invariant quantity. Under gauge transformation it changes as $$\begin{aligned}
{\bf A}_n({\bf R}) \rightarrow {\bf A}_n({\bf R}) - \frac{\partial}{\partial {\bf R}} \chi({\bf R})\end{aligned}$$ Cosequently, Berry phase will change by, $-\int_{\mathcal{C}} \frac{\partial}{\partial {\bf R}} \chi({\bf R}) d{\bf R} = \chi({\bf R_0}) - \chi({\bf R_t})$. Therefore, We can have Berry phase to be gauge invariant as long as the path $\mathcal{C}$ is closed. This quantity shows up in many areas of physics, one classic example is the Aharonov-Bohm effect [@ABeffect].

The SSH Model {#sec:sshmodel}
=============

We now move onto a slightly more complicated version of the previous Hamiltonian where instead of a single site unit cell we have a two-site unit cell, as shown in FIG. \[sshfig\], as if the lattice has been distorted due to 