We study non-Fermi liquid states that arise at the quantum critical points
associated with the spin density wave (SDW) and charge density wave (CDW)
transitions in metals with twofold rotational symmetry. We use the dimensional
regularization scheme, where a one-dimensional Fermi surface is embedded in
$3-\epsilon$ dimensional momentum space. In three dimensions, quasilocal
marginal Fermi liquids arise both at the SDW and CDW critical points : the
speed of the collective mode along the ordering wavevector is logarithmically
renormalized to zero compared to that of Fermi velocity. Below three
dimensions, however, the SDW and CDW critical points exhibit drastically
different behaviors. At the SDW critical point, a stable anisotropic non-Fermi
liquid state is realized for small $\epsilon$, where not only time but also
different spatial coordinates develop distinct anomalous dimensions. The
non-Fermi liquid exhibits an emergent algebraic nesting as the patches of Fermi
surface are deformed into a universal power-law shape near the hot spots. Due
to the anisotropic scaling, the energy of incoherent spin fluctuations disperse
with different power laws in different momentum directions. At the CDW critical
point, on the other hand, the perturbative expansion breaks down immediately
below three dimensions as the interaction renormalizes the speed of charge
fluctuations to zero within a finite renormalization group scale through a
two-loop effect. The difference originates from the fact that the vertex
correction anti-screens the coupling at the SDW critical point whereas it
screens at the CDW critical point.
