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:
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.
- Why does an energy level crossing the Fermi energy mean the gap has closed?
- Is what we mean by "gap" actually different to the band gap?