Why does edge states emerge in SSH model

Question

In reading Girvin and Yang’s “Modern condensed matter Physics” p146, I came across the following argument.

In the traditional SSH model, if we consider a system with open boundary conditions, particle-hole symmetry and an odd number of atoms. Due to particle-hole symmetry it is guaranteed that there must exist one state with exactly zero energy. If the system is dimerized, then there is a gap in the bulk and the zero mode must live on one of the boundaries (and decay exponentially into the bulk).

Now my question is why we consider the energy gap is located in the “bulk”, which is the space location of the material. Do we assume that the system will behave differently on the boundary?

Answer 1

"There is a gap in the bulk" -- this means that if you consider energy eigenstates that have a significant support in the bulk, you will not find such states with energy in the gap. You can also define a local density of states, and then you will find that this local density of states is zero in the bulk. You can still have eigenstates of the Hamiltonian with energy in the bulk gap, but their wavefunctions will not extend deep into the bulk. So yes, the system is different on the boundary than on the bulk. The behavior of most solid state systems is dominated by the bulk, so we often neglect boundary effects. For topological insulators this is different.