Simplified models with a minimal number of degrees of freedom have been a fruitful tool for gaining understanding of the essence of topological-insulator physics without the ballast of complications typical for band structures of these narrow-gap materials. The Bernevig-Hughes-Zhang model is such a minimal model, designed for describing the quantum spin Hall effect in HgTe quantum wells, more precisely, the transition between a trivial and topological phase as a function of the quantum well thickness. Since symmetry is central to the idea of quantum anomalies (e.g., the parity anomaly) in topological insulators, it is essential to identify them properly in these minimal models. In this work, we use dimensional reduction of three-dimensional point groups to the two-dimensional point groups appropriate for these quasi-two-dimensional minimal models, using the models for thin and thick layers of HgTe as examples. We determine the actions of crystal and time reversal operations on the basis states and the associated conserved quantum numbers, which aid the interpretation of the physics. While the usual effective models are invariant under the parity operation $z\to -z$, this symmetry is broken by electric fields along the $z$ direction. We identify parity-odd contributions such as the well-known Rashba-type term in these effective models and show which quantum numbers remain conserved.

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• Crystal symmetry
• Electronic structure
• Topological insulators
• Topological materials
• Discrete symmetries in condensed matter
• k dot p method