Just a Little Thought on TKNN

Thouless, Kohmoto, Nightingale and den Nijs in 1982 wrote a landmark paper relating the Hall conductivity to the Chern number (also now known as the TKNN invariant).

It is well-known that the Quantized Hall Effect (QHE) is an extremely robust phenomenon of topological origin (pdf!). One can think of the Hall conductivity as measuring the total Chern number of the occupied Landau levels. What baffles me about the TKNN result is that despite the robustness and topological character of the QHE, the authors are able to use linear response theory.

This must mean that second and higher order responses must somehow be exponentially suppressed and that the response is perfectly linear. I have not come across a proof of this in the literature, though it may very well be an (boneheaded!) oversight on my part. This line of questioning also applies to the Quantum Spin Hall Effect and the Quantum Anomalous Hall Effect.


2 responses to “Just a Little Thought on TKNN

  1. I’m not in a position to confidently comment on this (I just started reading through the QHE related literature). But regarding linear response, I think it merely states that the applied external field (e.g. electric field in QHE) is weak and treating it up to linear order will suffice. Considering higher order contributions, we have:

    J = (sigma) E + (sigma_prime) E^2 + O(E^3)

    Where (sigma) is ‘defined’ as the conductivity tensor. Theferore whenever we talk about conductivity we already assume linear response. It is what theorists calculate and experimentalists measure, because perturbations are almost always weak enough and we are in linear response regime.
    I strongly recommend taking a look at Xiao-Gang Wen’s book. There you can find some interesting remarks about linear response.

    Finally, what I understood from the post is the possibility of vanishing (sigma_prime) in the context of QHE, which is an interesting thing to check out. My guess is the calculation isn’t feasible analytically and one should resort to numerical methods.


    • Theorists definitely calculate the linear response. The question is really about what experimentalists measure. They do not necessarily measure a linear response. They measure the current induced by an electric field. For a small electric field, one would expect less “contamination” from higher order corrections, but they are still present. I’m just surprised that one cannot see any “contamination” in actual experiments in the quantum hall systems in that they are so precise.


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