# DC conductivity, linear response, “spectral representation”

I am trying to derive the DC electrical conductivity using the pertubation theory in Interaction picture and linear response theory. If working in a energy eigen basis and using the density matrix, the Fourier transform of the susceptibility can be written as $\chi {(\omega )_{ij}} = i\int_0^\infty {dt{e^{i\omega t}}\sum\limits_n {\left\langle n \right|{e^{ - \beta {{\hat H}_0}}}[{{\hat J}_j}(0),{{\hat J}_i}(t)]\left| n \right\rangle } }$

Then use $I = \sum\limits_n {\left| n \right\rangle \left\langle n \right|}$ and ${{\hat J}_i}(t) = {e^{i{{\hat H}_0}t}}{{\hat J}_i}(0){e^{ - i{{\hat H}_0}t}}$ . We can rewrite ${\sum\limits_n {\left\langle n \right|{e^{ - \beta {{\hat H}_0}}}[{{\hat J}_j}(0),{{\hat J}_i}(t)]\left| n \right\rangle } }$ as

$\sum\limits_{m,n} {{e^{ - {E_m}\beta }}\left[ {\left\langle m \right|{{\hat J}_i}\left| n \right\rangle \left\langle n \right|{{\hat J}_j}\left| m \right\rangle {e^{i({E_m} - {E_n})t}} - \left\langle m \right|{{\hat J}_j}\left| n \right\rangle \left\langle n \right|{{\hat J}_i}\left| m \right\rangle {e^{i({E_n} - {E_m})t}}} \right]}$

so

${\chi _{ij}}(\omega ) = - i\int_0^\infty {dt{e^{i\omega t}}} \sum\limits_{m,n} {{e^{ - {E_m}\beta }}\left[ {\left\langle m \right|{{\hat J}_i}\left| n \right\rangle \left\langle n \right|{{\hat J}_j}\left| m \right\rangle {e^{i({E_m} - {E_n})t}} - \left\langle m \right|{{\hat J}_j}\left| n \right\rangle \left\langle n \right|{{\hat J}_i}\left| m \right\rangle {e^{i({E_n} - {E_m})t}}} \right]}$

I see in literature, to make the integral converge, a complex frequency $\omega + i\varepsilon$ is used to make the integrand vanish in $+ \infty$, therefore

$\begin{array}{l} {\chi _{ij}}(\omega + i\varepsilon ) = \sum\limits_{m,n} {{e^{ - {E_m}\beta }}\left[ {\frac{{\left\langle m \right|{{\hat J}_i}\left| n \right\rangle \left\langle n \right|{{\hat J}_j}\left| m \right\rangle }}{{\omega + i\varepsilon + {E_m} - {E_n}}} - \frac{{\left\langle m \right|{{\hat J}_j}\left| n \right\rangle \left\langle n \right|{{\hat J}_i}\left| m \right\rangle }}{{\omega + i\varepsilon + {E_n} - {E_m}}}{e^{i({E_n} - {E_m})t}}} \right]} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sum\limits_{m,n} {\frac{{\left\langle m \right|{{\hat J}_i}\left| n \right\rangle \left\langle n \right|{{\hat J}_j}\left| m \right\rangle }}{{\omega + i\varepsilon + {E_m} - {E_n}}}} ({e^{ - {E_m}\beta }} - {e^{ - {E_n}\beta }}) \end{array}$

My question is, since I am interested in the DC conductivity, I expect that finally I can remove ${i\varepsilon }$ in above expression when $\omega \to 0$. But I am not sure how to do this since the integral is not well defined if there is no imaginary part in frequency.

On the other hand, if I just simply set $\omega + i\varepsilon = 0$, I seem to be able to find the correct form of conductivity formula used in the quantum hall effect.

Can somebody help?