Nonanalytic Corrections to the Landau Diamagnetic Susceptibility

We analyze potential non-analytic terms in the Landau diamagnetic susceptibility, ${\chi }_{dia}$, at a finite temperature $T$ and/or in-plane magnetic field $H$ in a two-dimensional (2D) Fermi liquid. To do this, we express the diamagnetic susceptibility as ${\chi }_{dia}=(e\mathrm{/}c{)}^2{\lim }_{Q\to 0}{\mathrm{\Pi }}_{\perp }^{JJ}(Q)\mathrm{/}{Q}^2$, where ${\mathrm{\Pi }}_{\perp }^{JJ}$ is the transverse component of the static current-current correlator, and evaluate ${\mathrm{\Pi }}_{\perp }^{JJ}(Q)$ for a system of fermions with Hubbard interaction to second order in Hubbard $U$ by combining self energy, Maki-Thompson, and Aslamazov-Larkin diagrams. We find that at $T=H=0$, the expansion of ${\mathrm{\Pi }}_{\perp }^{JJ}(Q)\mathrm{/}{Q}^2$ in $U$ is regular, but at a finite $T$ and/or $H$, it contains $U^{2}T$ and/or $U^2\mathrm{\mid }H\mathrm{\mid }$ terms. Similar terms have been previously found for the paramagnetic Pauli susceptibility. We obtain the full expression for the non-analytic $\delta {\chi }_{dia}(H,T)$ when both $T$ and $H$ are finite, and show that the $H\mathrm{/}T$ dependence is similar to that for the Pauli susceptibility.