Singularities in the paramagnetism of two-dimensional nearly magnetic itinerant-fermion systems at very low temperature: Application to degenerate two-dimensional liquid-He3films

It is shown that two-dimensional (2D) paramagnon problems exhibit strong algebraic singularities ${|q^2-4\mathrm{}{K}_F^2|}^{-\alpha }$, when some relevant momentum $q$ is close to twice the Fermi momentum $K_F$. As all $q$ values from 0 to $2K_F$ are equally relevant for the magnetic instability in 2D, the above singularities play a key role. By contrast, they are irrelevant in 3D. It is explicitly shown that standard methods to calculate the uniform static susceptibility fail in 2D due to these singular terms. The origin of the singularities are multitail ring diagrams, closed fermion loops with "tails" attached to them. These diagrams are analyzed in detail by generalizing to $d$ dimensions and dynamic tails, the method of Brovman and Kagan, developed in another context. The subsequent effects in the Ginzburg-Landau-Wilson Lagrangian describing interacting paramagnons are dramatic and render such an expansion questionable. Moreover, the nature of the magnetic instability (ferromagnetic or antiferromagnetic type), which is not well defined in mean-field theory, still remains unsolved in the presence of paramagnons since standard methods to renormalize the response functions with fluctuations fail to apply in the 2D problem. Any naive transposition from 3D to 2D of the Landau Fermi-liquid theory to compute, for instance, properties of liquid-${}_{}{}^3{}_{}{}^{}\mathrm{He}$ films are suspected to be premature—if not erroneous—at this stage.