Many-body theory of magnetic susceptibility of electrons in solids

We present a theory of the total magnetic susceptibility ($\chi$) of interacting electrons in solids. We have included the effects of both the lattice potential and electron-electron interaction and constructed in $\overrightarrow{\mathrm{k}}$ space, using the Bloch representation, the effective one-particle Hamiltonian and the equation of motion of the Green's function in the presence of a magnetic field. We have used a finite-temperature Green's-function formalism where the thermodynamic potential $\Omega$ is expressed in terms of the exact one-particle propagator $G$ and have derived a general expression for $\chi$ by assuming the self-energy to be independent of frequency. We have calculated the many-body effects on orbital (${\chi }_o$), spin (${\chi }_s$), and spin-orbit (${\chi }_{\mathrm{so}}$) contributions to $\chi$. If we make simple approximations for the self-energy, our expression for ${\chi }_o$ reduces to the earlier results. If we make drastic assumptions while solving the matrix integral equations for the field-dependent part of the self-energy, our expression for ${\chi }_s$ is equivalent to the earlier results for exchange-enhanced spin susceptibility but with the $g$ factor replaced by the effective $g$ factor, a result which has been intuitively used but not yet rigorously derived. An important aspect of our work is the careful analysis of exchange and correlation effects on ${\chi }_{\mathrm{so}}$, the contribution to susceptibility from the effect of spin-orbit coupling on the orbital motion of Bloch electrons. Although ${\chi }_{\mathrm{so}}$ is of the same order of magnitude as ${\chi }_s$ for some metals and semiconductors, its contribution has been hitherto completely ignored in all the many-body theories of magnetic susceptibility. We have also shown that if we neglect electron-electron interactions our expression for $\chi$ agrees with the well-known results for noninteracting Bloch electrons.