The first formula indeed follows from the second formula if we let $\omega\to0$. To see that, expand the fractions as
$$ \frac1{\pm\hbar\omega + E^a - E^b} = \frac1{E^a-E^b}\left(1 \mp \frac{\hbar\omega}{E^a-E^b}\right) + \mathcal O(\omega^2)$$
to obtain $\sigma_{xy} = \sigma^1 + \sigma^2$ as the sum of a potentially divergent term
$$ \sigma^1 = \frac{-ie^2}{V\omega} \sum_{a,b} f(E^a) \frac{\langle a|v_x|b \rangle \langle b|v_y|a \rangle + \langle a|v_y|b \rangle \langle b|v_x|a \rangle}{E^a - E^b} $$
and a term that looks like the first formula
$$ \sigma^2 = \frac{-ie^2\hbar}{V} \sum_{a,b} f(E^a) \frac{- \langle a|v_x|b \rangle \langle b|v_y|a \rangle + \langle a|v_y|b \rangle \langle b|v_x|a \rangle}{(E^a - E^b)^2} .$$
To see that the first term vanishes instead of diverging, we have to use the Heisenberg equation of motion $v_x = \frac{d}{dt}x = [H_0,x]$ which gives
$$ \langle a | v_x | b \rangle = \langle a | H_0 x - x H_0 | b \rangle = (E^a-E^b) \langle a | x | b \rangle $$
and thus
$$ \langle a|v_x|b \rangle \langle b|v_y|a \rangle + \langle a|v_y|b \rangle \langle b|v_x|a \rangle = (E^a-E^b) (\langle a|x|b \rangle \langle b|v_y|a \rangle - \langle a|v_y|b \rangle \langle b|x|a \rangle) .$$
The factors $(E^b-E^b)$ cancel and the remaining sum over $b$ becomes a sum over the identity $\sum_b |b\rangle\langle b| = 1$. Thus, we arrive at
$$ \sigma^1 = \frac{-ie^2}{V\omega} \sum_{a,b} f(E^a) \left(\langle a|xv_y - v_yx |a \rangle \right) = 0 .$$
since the commutator $[x,v_y]$ vanishes.
To see that the second term is correct, we have to get the summation indices right. To do that, we have to rearrange the summation to obtain
$$ \sigma^2 = \frac{ie^2\hbar}{V} \sum_{a,b} (f(E^a)-f(E^b))\frac{\langle a|v_x|b \rangle \langle b|v_y|a \rangle}{(E^a - E^b)^2} .$$
In the limit $T\to0$, the difference of Fermi-Dirac distributions $f(E^a)-f(E^b)$ will be equal to
- $1$ if $E^a < E_F < E^b$
- $-1$ if $E^b < E_F < E^a$
- $0$ otherwise
Using this and rearranging the summation again gives the Kubo formula in the first form.
The TKNN (bulk) and Büttiker (edge) explanations for the quantized Hall conductance correspond to different geometries.
In the TKNN theory, the "sample" consists of a torus closed on itself and therefore has no edges at all. In this case the electric potential is uniform, and the electric field is due to the time derivative of the vector potential (it lasts only as long as one varies the magnetic flux inside the torus). In this case, the Hall current is truly a bulk current.
Büttiker, on the other hand, considers a Hall bar with different electrochemical potentials on each side. If one (as does Büttiker) assumes that the electrostatic potential is uniform within the central region of the bar, and rises on the sides, then one finds that the current flows along the edges (in opposite directions), with more current flowing along one edge than along the other because of the different chemical potentials.
In a more realistic description, the electrostatic potential is not uniform in the bulk of the bar, so that the current flow takes place both along the edges and within the bulk of the bar. In any case, the net current is completely independent of the actual profile of the electrostatic potential accross the hall bar, and depends only upon the chemical potential difference. That is why why Büttuker and TKNN obtain the same answer for the (quantized) total current.
A nice discussion of this question is given by Yoshioka.
Best Answer
There is indeed a long history of people expressing doubt regarding Kubo's formula, see, for example, the discussion here.
To me, the central idea behind Kubo's relation is that of ``matching''. We have a microscopic quantum many-body theory that, in principle, can be used to compute non-equilibrium properties of a system. In practice, however, these calculations are too difficult to be practical (and they contain "too much information" -- we don't actually want to know exact many-body wave functions). We therefore make use of the fact that the low energy, long distance limit is described by a simpler hydrodynamic theory. This theory can be established on general grounds, but it does contain effective parameters, transport coefficients like conductivities, viscosities, etc.
The questions is whether we can microscopically compute these parameters without having to derive the full hydrodynamic theory from the microscopic one.
The answer is yes: Identify a simple correlation function that can be computed in both the microscopic and the effective theory, and require that they match. What Kubo suggests is to look at the linear response to an arbitrarily weak external perturbation. This does not imply that the real response always has to be linear. Indeed, fluid dynamics is a complicated non-linear theory. Solving the equations of fluid dynamics automatically reproduces the correct non-linear response. All we ask is that in the limit in which fluid dynamics can be linearized, it matches the microscopic theory.