General formula for the magnetoresistance on the basis of Fermi liquid theory

The general expression for the magnetoresistance (MR) due to the Lorentz force is derived by using the Fermi liquid transport theory based on the Kubo formula. The obtained gauge-invariant expression is exact for any strength of the interaction, as for the most singular term with respect to $1/{\gamma }_k^{*}$ $({\gamma }_k^{*}$ being the quasiparticle damping rate). By virtue of the exactness, the conservation laws are satisfied rigorously in the present expression, which is indispensable for avoiding unphysical solutions. Based on the derived expression, we can calculate the MR within the framework of the Baym-Kadanoff-type conserving approximation, by including all the vertex corrections required by the Ward identity. The present expression is significant especially for strongly correlated systems because the current vertex corrections will be much important. On the other hand, if we drop all the vertex corrections in the formula, we get the MR of the relaxation time approximation (RTA), which is commonly used because of its simplicity. However, the RTA is dangerous because it may give unphysical results owing to the violation of conservation laws. In conclusion, the present work enables us to study the MR in terms of the conserving approximation, which is highly demanded in strongly correlated electrons such as high-$T_c$ superconductors, organic metals, and heavy fermion systems.