Failure of Fermi Liquid Theory in 2-D: A Signal from Perturbation Theory

We study the perturbative correction to the ground state energy eigenvalue of a 2-dimensional dilute fermi gas with weak short-range two body repulsion. From the structure of the energy shift we infer the presence of an induced two body long range repulsive interaction $ {\displaystyle {1\over{r^2}} }$ among the constituent electrons indicating a potential instability of the fermi liquid ground state.