# Insulator or conductor with different boundary conditions

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?

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.