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We discuss shape (Pomeranchuk) instabilities of the Fermi surface of atwo-dimensional Fermi system using bosonization. We consider in detail thequantum critical behavior of the transition of a two dimensional Fermi fluid toa nematic state which breaks spontaneously the rotational invariance of theFermi liquid. We show that higher dimensional bosonization reproduces thequantum critical behavior expected from the Hertz-Millis analysis, and verifythat this theory has dynamic critical exponent $z=3$. Going beyond thisframework, we study the behavior of the fermion degrees of freedom directly,and show that at quantum criticality as well as in the the quantum nematicphase (except along a set of measure zero of symmetry-dictated directions) thequasi-particles of the normal Fermi liquid are generally wiped out. Instead,they exhibit short ranged spatial correlations that decay faster than anypower-law, with the law $|x|^{-1} \exp(-\textrm{const.} |x|^{1/3})$ and weverify explicitely the vanishing of the fermion residue utilizing thisexpression. In contrast, the fermion auto-correlation function has the behavior$|t|^{-1} \exp(-{\rm const}. |t|^{-2/3})$. In this regime we also find that, atlow frequency, the single-particle fermion density-of-states behaves as$N^*(\omega)=N^*(0)+ B \omega^{2/3} \log\omega +...$, where $N^*(0)$ is largerthan the free Fermi value, N(0), and $B$ is a constant. These results confirmthe non-Fermi liquid nature of both the quantum critical theory and of thenematic phase.Comment: 20 pages, 2 figures, 1 table; new version with minor changes; new  subsection 3C2 added with an explicit calculation of the quasiparticle  residue at the nematic transition; minor typos corrected, new references;  general beautification of the text and figure

Nonperturbative behavior of the quantum phase transition to a nematic Fermi fluid