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This online course on topology in condensed matter states the following:

We say that two gapped quantum systems are topologically equivalent if their Hamiltonians can be continuously deformed into each other without ever closing the energy gap.

This deformation of a Hamiltonian $H$ into $H'$ is given by

$$ H(\alpha) = \alpha H' + (1-\alpha)H, $$

where $\alpha \in [0,1]$ is some parameter that governs the deformation. The example provided in the course is that $H(\alpha)$ describes a quantum dot in contact with some probe, with Fermi level fixed at $E_F = 0$. This means that all negative energy levels of the quantum dot are occupied and all positive energy levels are unoccupied. The energy levels of the system as $\alpha$ is varied are plotted below:

enter image description here

It is stated that when an energy level crosses zero, the gap closes. I do not understand why. My interpretation of the gap is the difference in energy between the highest occupied energy level and the lowest unoccoupied energy level, sometimes called the band gap. I would assume the gap would close if these two levels touched at some point, but in the diagram above this does not happen---there is always a band gap between these energy levels.

I can see that at the point where a level crosses zero energy, the ground state of the system (where all negative energy states are occupied) will suddenly change, because the occupied energy level with negative energy will suddenly become a positive energy level, indicating it is no longer occupied. However, I do not see where the gap closing is coming from.

My questions

  1. Why does an energy level crossing the Fermi energy mean the gap has closed?
  2. Is what we mean by "gap" actually different to the band gap?
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I think the confusion here arises from the fact that people say "band gap" to refer to a very specific band gap. As you correctly noticed, the gaps between the different bands have not disappeared as we varied $\alpha$ and they shouldn't.

However, what people usually mean when they speak about a band gap is the gap between the lowest state in the conduction band and the highest state in the valence band. Now the conduction band is defined as the band closest and above the Fermi-level, whereas the valence band is the band that is closest and below the Fermi-level. So what is meant by "having a band gap" is whether the Fermi-level is in that gap, because that is what determines the properties of materials. If the Fermi-level goes from being in the gap to being in a conduction band, we say that the gap has closed. Yes, there are other gaps between energy levels, but they are not important.

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