I have a confusion about the nature of topologically protected boundary states in the Chern insulator. Since the Chern insulator does not require any symmetries to be present in the system, what is the nature of the topological protection of the boundary states as opposed to other systems, where the boundary states are protected by present symmetries? I read here about intrinsic topological phases, but I am unsure whether this can be applied to the Chern insulator. Thank you in advance!
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$\begingroup$ The edge states are chiral fermions and do not need any symmetry for protection. $\endgroup$– Meng ChengCommented Jun 20, 2022 at 16:02
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2 Answers
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As far as I am aware, the Chern insulator/quantum anomalous Hall effect (QAHE) is NOT a symmetry protected topological phase (SPTP). The disspitionless nature of the edge modes is a consequence of the fact that they are chiral edge modes - electrons on one edge move in a single direction and electrons on the other edge move in the opposite direction. This ensures that electrons cannot backscatter, as there are no states on the same edge to backscatter into. This also ensures the quantization of the transverse conductance.
I would imagine that by engineering a device with a constriction, one could hybridize the edge states of a Chern insulator and open a gap in the edge state dispersion.
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(1) It is true that the Chern insulator cannot be smoothly connected to a trivial state without encountering a phase transition. This is the sense where it is called topological. However, a symmetry protected topological/trivial (SPT) phase can be connected to a trivial state if we allow symmetry-breaking perturbations.
(2) The chiral edge state is a hint for such topological nature. The fact that the chiral conformal field theory possesses a gravitational anomaly means that such an edge is ungappable, that is, it remains robust under (any) local perturbation.
(3) Technically, the Chern insulator is not "intrinsic topological phase" either. One may stack two Chern insulators with opposite Chern number and obtain a symmetry protected topological/trivial phase (known as topological insulator). So more properly the Chern insulator is called an "invertible phase". However, for intrinsic topological phases (such as fractional quantum Hall states) it is never possible to "invert" the topological order by adding an extra layer.
