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2

The zero energy states are localized at the boundary, and their wave functions decay exponentially in the bulk. So they are boundary modes (or edge states), which do not count as the bulk states, and do not contribute to the bulk gap closing. The only way to close the bulk gap in the SSH model is to tune $\delta t$ to zero. As long as $\delta t\neq 0$, the ...

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A topological invariant in condensed matter systems is a number that doesn't change under a smooth deformation of the Hamiltonian. I like to think this as stretching a material that would change for instance the hopping constants. Now the way that one can take such a number may vary and I don't think there is a limitation in the definition neither that there ...

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Your statement itself is not quite right. What is not conserved is the chiral current, namely the current of fermions at one of the Weyl nodes. The physics can be understood essentially in one-dimensional version of the Weyl metal: consider a 1D electron gas. There are two Fermi points, and the low-energy theory is given by two "Weyl fermions" in 1D with ...

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In simplest terms, the presence of sub-gap zero energy localized modes (Majorana modes) makes a superconductor topological. A superconducting ground state is just a bunch of Cooper pairs and the BdG Hamiltonian describes excitations above the ground state. If the excitation spectrum has these localized modes then it is a topological superconductor otherwise ...

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The difference is obvious: In the second figure, the blue and red line connect the valence and conduction bands. These are actually surface states. So regardless of where your chemical potential lies, there will be low-energy excitations on the surface, i.e. the surface is conducting. This is what happens in a topological insulator. In the first figure, you ...

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The TKNN (bulk) and Büttiker (edge) explanations for the quantized Hall conductance correspond to different geometries. In the TKNN theory, the "sample" consists of a torus closed on itself and therefore has no edges at all. In this case the electric potential is uniform, and the electric field is due to the time derivative of the vector potential (it lasts ...

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