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(Oh my god, why can't my post show normally? Some sentences just disappear, but I can still see them while editing. It's very weird. Can someone help me? I've reported.)

I know only partially filled bands result in current density, and I think

I know only partially filled bands result in current density, and I think

**there must be not only 2 partially filled bands**, but "**several**" partially filled bands. For each partially filled band, its Ohm's law is $$\mathbf{E}=\tilde{\rho}_{\text{n}}\:\mathbf{j}_{\text{n}},$$where the ##\tilde{\rho}_{\text{n}}## has the form:$$\tilde{\rho}_n=\begin{pmatrix}\rho_{\text{n}}&-R_{\text{n}}H\\R_{\text{n}}H&\rho_{\text{n}}\end{pmatrix}$$Furthermore, the total induced current is given by##\mathbf{E}=\tilde{\rho}\cdot\mathbf{j}##,with$$\tilde{\rho}=\left(\sum\tilde{\rho}_{\text{n}}^{-1}\right)^{-1}$$In Ashcroft & Mermin Solid State Physics textbook, it's said the Hall coefficient and magnetoresistance by only two bands are:$$R=\frac{R_1\rho_2^2+R_2\rho_1^2+R_1R_2(R_1+R_2)H^2}{(\rho_1+\rho_2)^2+(R_1+R_2)^2H^2}$$$$\rho=\frac{\rho_1\rho_2(\rho_1+\rho_2)+(\rho_1R_2^2+\rho_2R_1^2)H^2}{(\rho_1+\rho_2)^2+(R_1+R_2)^2H^2}$$Why do we need these two formulas if it's almost impossible to have only**two**partially filled bands? Thank you very much!
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