Insulator or conductor with different boundary conditions

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 $E_n=pm 2cos k$ 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.