  We examine a two-dimensional Fermi liquid with a Fermi surface which touches
the Umklapp surface first at the 4 points $(\pm \pi/2, \pm \pi/2)$ as the
electron density is increased. Umklapp processes at the 4 patches near $(\pm
\pi/2, \pm\pi/2)$ lead the renormalization group equations to scale to strong
coupling resembling the behavior of a 2-leg ladder at half-filling. The
incompressible character of the fixed point causes a breakdown of Landau theory
at these patches. A further increase in density spreads the incompressible
regions so that the open Fermi surface shrinks to 4 disconnected segments. This
non-Landau state, in which parts of the Fermi surface are truncated to form an
insulating spin liquid, has many features in common with phenomenological
models recently proposed for the cuprate superconductors.
