two) where the band gap is zero. That gives rise to conducting surface states at the surface of 3D TIs. We will now explore these conducting states, known as [*edge states*]{} in 1D, that appear in the SSH model.

A finite SSH chain
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Till now we looked at the solutions of a 1D SSH chain with periodic boundaries (i.e. a ring). To answer the question posed in the last section on what physical consequence does the winding number has let us look at the version of the SSH chain with open boundaries. The problem is not trivial - since there is no translational invariance we cannot make use of the Fourier transformation to diagonalise the Hamiltonian. We, therefore, feed the Hamiltonian into our computers and explicitly solve the Hamiltonian for its eigenvalues and eigenfunctions.

But before this, let us first try to look at the extreme cases and guess what could be the solutions. There are two extreme cases - $w=0$,$v\ne 0$ and $v=0$,$w\ne 0$. These are called the dimerised limits as this would correspond to breaking the chain into dimers. The first case, $w=0$, for a chain with $N=10$ looks something like what is shown in FIG. \[openssh1\],

