Insulator or conductor with different boundary conditions

Question

I'm studying the 1-D SSH model. It's a toy model for a topological insulator. Here's the reference I'm using. If the hopping amplitudes $v$ and $w$ are equal, then with periodic boundary conditions we have a conductor. If however, we take open boundary conditions, a plot of the eigenvalues shows that there is a band gap with the same values of $v$ and $w$. What is going on?

Answer 1

Are you refering to fig 1.4 in the paper you cite as evidence for an enegy gap at $u=v=1$? If so, you need to realize that in the that plot the authors are using a very small (10 site) system. The gaps between the eigenvalues are therefore due to the finite-size discreteness of the eigenvalues $E_n=\pm 2\cos k$ with $k$ something like $k=(n+1/2)\pi/N$ for $N$ sites. There will only be an exact zero energy $E=2 \cos \pi/2$ state when $N$ is odd. $N=10$ is even in their plot. As $N$ becomes larger the spaces between the $N=1$ eigenvalues will fill in, and there will be no gap at zero energy in the $u=v$ case.