The general formula for linear response functions (and particularly for a conductivity) is referred to as the Kubo formula or the Kubo-Greendoow formula. What is the difference between them?

In the original paper by R. Kubo, the following formula (2.23b) for the response function is derived: $$ \phi_{BA}(\omega)=-\frac{i}\hbar\int\langle[\hat{A},\hat{B}(t)]\rangle e^{i(\omega+i\delta)t}dt $$ (I have changed the original notation $e^{i\omega t}$ for time oscillations to $e^{-i\omega t}$ and taken the pure ground state for simplicity).

In the paper by D.A. Greenwood, the following formula (31) can be found: $$ \sigma=-2\pi e^2\hbar\int\sum_{mn}|v_{mn}|^2\delta(E-E_n)\delta(E-E_m)\frac{\partial f}{\partial E}dE. $$ It refers to the static limit and seems to be rather different (at least, by the initial problem statement) from the Kubo formula.


1 Answer 1


The Kubo formula for conductivity is obtained in (5.20) of the original Kubo paper: $$ \sigma_{\mu\nu}(\omega)= \int_{0}^{\infty} e^{-i\omega t} \,dt \int_{0}^{\beta} d\lambda \,\langle j_\nu (-i\hbar\lambda)\,j_\mu(t)\rangle . $$ The Kubo–Greenwood formula is a non-interacting approximation of the the Kubo formula. The derivation is explained in some detail here, where the Kubo equation above appears in (17). In the non-interacting case the current operator can be written as (23) $$ \hat{j}_x=\sum_{nm}\langle n|j_x|m\rangle c_n^\dagger c_m e^{i(\epsilon_n-\epsilon_m)t/\hbar} $$ which substituted in the Kubo equation above yields (25) $$ \sigma_{\mu\nu}(\omega)=\frac{i\hbar}V \sum_{mn} \frac{f_n- f_m} {\epsilon_m-\epsilon_n} \frac{\langle n|j_\nu|m \rangle \langle m|j_\mu|n \rangle} {\hbar(\omega + i\eta) - (\epsilon_m-\epsilon_n) }, $$ directly related for $\omega=0$ to the Kubo-Greenwood equation above.

PS I have no access to the Greenwood paper.


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