# What is the difference between the Kubo and Kubo-Greenwood formulas?

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.

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.