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It seems that Kubo formula is widely adopted to calculate conductivity, or at least Hall conductivity [for example, in the famous paper by TKNN: PRL 49 405-408 (1982)].

However, the derivation of Kubo formula (for example, see Sec. 4.3 in http://www.damtp.cam.ac.uk/user/tong/kintheory/four.pdf) relies on linearizing the evolution operator $$ \overline{\exp}[-\mathrm{i}\int_{0}^{t}V(t')\mathrm{d}t']\approx 1-\mathrm{i}\int_{0}^{t}V(t')\mathrm{d}t', $$ where $\overline{\exp}$ denotes ordered exponential and $V$ is a Hermitian operator (typically perturbative Hamiltonian in the interaction picture in the derivation of Kubo formula).

The approximation above is indeed valid for small $V$ and small $t$. However, in my opinion, if $t$ is large enough, the approximation above loses validity no matter how small $V$ is. Therefore, Kubo formula should only be applicable for a small time scale.

In transport experiments, the measured system usually enters steady state and the time scale might be quite large. In this case, What justifies the application of Kubo formula in the theory of conductivity?

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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.

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  • $\begingroup$ Any references for such a view point? This sounds plausible but more rigor could be helpful. $\endgroup$ – Chong Wang Sep 10 '19 at 14:38
  • $\begingroup$ I'm not sure if there are rigorous arguments, and in 3+1 d there are not that many exact solutions either. I think the best evidence is i) theories where you can compute transport coefficients using both the Kubo relation and solutions of the Boltzmann equation (see text books like Mahan), ii) theories where you have the Kubo formula and direct derivations of hydro (these days the best example is AdS/CFT, see text books on that, such as Erdmenger or Zaanen). $\endgroup$ – Thomas Sep 11 '19 at 2:53

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