The Fermi-liquid theory states that the low-energy properties of an interacting fermion system are similar to that of the Fermi gas. However, in contrast to the Fermi gas, where all thermodynamic quantities are analytic functions of the temperature, the next-to-leading corrections to the Fermi-liquid forms are non-analytic. In this talk, I will show that these non-analyticities are the telling sign of small-angle scattering events, which are rare in higher dimensions but becomes increasingly more probable as the dimensionality is reduced. A breakdown of the Fermi liquid in D=1 then follows naturally as the result of a progressive increase in “corrections” as the dimensionality is reduced. I will argue that the perturbation theory is ill-defined in two dimensions even for the most benign interaction: the self-energy exhibits singularity at each order. However, a re-summed theory is well-behaved and the Fermi liquid survives, albeit with some notable modifications, such as a non-Lorentzian shape of the spectral function. Finally, I will discuss a non-analyticity in the spin susceptibility which casts serious doubts on the Hertz-type scenario of a ferromagnetic quantum phase transition.

University of Florida