It is known that one can find the Hall conductivity $\sigma_{xy}$ from a lattice model (in a magnetic field, say) using the TKNN formula (PRL 49 405-408 (1982)), i.e. by summing the Chern numbers for bands below the Fermi energy $E_F$: $$\sigma_{xy}=\frac{e^2}{h}\sum_{E_n<E_F}C_n$$ $C_n$ can be straightforwardly computed from the eigenstates.

If one were to cause the bands in such a model to broaden (by adding an external potential, say), could the longitudinal conductivity $\sigma_{xx}$ be computed in a straightforward way in this model?


$\sigma_{xx}$ can be computed via the usual Kubo formula for linear response. You can find that for example in Bruus and Flensberg I suppose.

However, as long as you fill the system up to a spectral gap, even broadening the bands (while keeping $E_F$ in a gap) will lead to $\sigma_{xx}=0$. Indeed, this is part of what characterizes the IQHE. When $\sigma_{xx}\neq0$, the Chern number should not be well defined.

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  • $\begingroup$ Hall conductivity should still be well defined even if $E_F$ lies within a band and can be found by integrating the Berry curvature over occupied states. Indeed, the "Chern number" in such a case will not be an integer. $\endgroup$ – zeta Feb 6 at 18:13
  • $\begingroup$ I disagree with that. The Fermi projection doesn’t have sufficient off-diagonal decay in the position basis to give a finite result for the trace formula of the Chern number (the double commutator formula) when $E_F$ is in the middle of a band. $\endgroup$ – PPR Feb 6 at 18:17
  • $\begingroup$ If one takes the formula from the paper referenced in the question and integrates only over states below $E_F$ within a band (in k-space), one gets a finite answer. Are you saying the answer should be infinite? $\endgroup$ – zeta Feb 6 at 19:07
  • $\begingroup$ What do you mean by "states below $E_F$" if $E_F$ cuts states in their middle (at least some of them) as it passes within a band? $\endgroup$ – PPR Feb 7 at 5:20

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