Effect of TRS potential on Topological insulator (QSH)

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Discussion Overview

The discussion revolves around the effects of time-reversal symmetry (TRS) potential on topological insulators, specifically in the context of the quantum spin Hall effect (QSHE). Participants explore the implications of staggered sublattice potentials and their relationship to the robustness of topological insulator phases.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes a paradox regarding the robustness of topological insulators against potentials that do not break TRS, referencing the Kane-Mele model where a staggered sublattice potential leads to a trivial insulator phase.
  • Another participant explains that a staggered sublattice potential can open a gap and discusses the critical value at which this gap closes, leading to a phase transition to a trivial insulator.
  • A further clarification is made regarding the Haldane model and its relation to the Kane-Mele model, suggesting that the Haldane mass introduces gaps of opposite signs at specific points, influencing the system's behavior.
  • One participant questions whether the observed gap aligns with expectations of the robustness of the topological insulator phase against interactions.
  • A correction is provided regarding the nature of the Haldane mass and its competition with the staggered sublattice potential, emphasizing the significance of gap closure in determining the phase transition.
  • Interactions in the QSHE are discussed, particularly how impurities that do not break TRS do not lead to backscattering due to the chiral nature of the states.

Areas of Agreement / Disagreement

Participants express differing views on the implications of staggered sublattice potentials and their effects on the robustness of topological insulators. The discussion remains unresolved regarding the relationship between the gap and the expected robustness of the TI phase against interactions.

Contextual Notes

Participants reference specific models and terms (e.g., Haldane mass, staggered sublattice potential) that may require further clarification or context for full understanding. The discussion includes assumptions about the behavior of the system under various conditions, which are not fully resolved.

mohsen2002
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Hi every body,

I faced a paradox. The topological insulator is robust against a potential that does not breaks the TRS.
But in the original work of Kane-Mele (PRL 95, 146802), the "staggered sublattice potential" that does not breaks the TRS,, makes zigzag ribbon trivial insulator (figure 1 in the PRL).
Is there any explanation?
 
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The staggered sublattice potential can open a gap. Think of boron nitride. Because the A and B sublattice are in equivalent so it is gapped. If the gap becomes large enough in the QSHE we have a phase transition, the gap closes and are now not inverted so we have a trivial insulator. So there can be a staggered sublattice potential but at a critical value it will close the gap due to spin orbit and the system becomes a trivial insulator.

Also think of the Haldane model. The Kane mele model is just two copies of the Haldane model, one for each spin. The sigma z term in the Haldane model is proportional to a Haldane mass, and there is a region where you can have a chiral edge state also due to the gap closing and reopening.
 
Thank you for your reply. I can understand the gap, but is it match with our expectation of robustness of TI phase against interactions?
 
A correction to the above, the Haldane mass is actually proportional to sigma z Sz. In the QSHE this mass would correspond to sigma z tau z s z. The staggered sublattice potential is just proportional to sigma z and produces the gap of the same size at K and K'

The Haldane mass is a different kind of mass term than from the sublattice potential as it introduces gaps of opposite signs at the points K and K' (location of the Dirac points in graphene). If the sublattice potential is too large, the bands will not invert since the gaps at K and K' will have the same sign even with the Haldane mass. The transition comes when the gap closes. So basically they are competing.

Usually when you think of interactions in the QSHE, you could think of something like an impurity somewhere or something that could cause backscattering. If such a thing does not break time reversal, backscattering cannot happen since the states are chiral.
 
Thank you so much for helpful discussion and your time.
 

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