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(adsbygoogle = window.adsbygoogle || []).push({});

##\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?

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# A A question about linear response and conductivity

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