Recently, I am learning the topological insulator, I learned about SSH model, and found out that the edge states of topological insulator always have zero energy, but on the other side,we define edge states as local states. I can’t see the connection between them?
Can we simply derive the zero energy by ‘localization’ condition? (Because on textbook we seem to search the edge simply by letting the energy equals zero and solve it.)
Being topological in this case means that the wave function of the system is locally the exact same as the vacuum wave function, but globally may have a different structure. For example, the fields described by the state may have a non-trivial winding number but be equivalent to the vacuum state if you look at the fields any simply connected subdomain. Because the Hamiltonian is a local operator (in the bulk), that means it shares the same energy as the vacuum, and hence has zero energy.
Another way to think about it is that bulk time evolution by a local operator can only deform the state in a continuous way, but the topologically protected states are unchanged under continuous deformations of the state, as long as the boundary conditions are held fixed. Invariant under time translations = zero energy by the Schrodinger equation.
Note: I'm more familiar with the concept of edge states in high energy settings such as TQFTs, and not in condensed matter like the SSH model, but I believe the key physics is actually the same. The big idea is that small wiggles of the fields don't change the state, and therefore the subset of time translations which keeps the boundary conditions fixed don't change the state, which implies a vanishing energy by the Schrodinger equation.
