A non-Fermi liquid state without time-reversal and parity symmetries arises
when a chiral Fermi surface is coupled with a soft collective mode in two space
dimensions. The full Fermi surface is described by a direct sum of chiral patch
theories, which are decoupled from each other in the low energy limit. Each
patch includes low energy excitations near a set of points on the Fermi surface
with a common tangent vector. General patch theories are classified by the
local shape of the Fermi surface, the dispersion of the critical boson, and the
symmetry group, which form the data for distinct universality classes. We prove
that a large class of chiral non-Fermi liquid states exist as stable critical
states of matter. For this, we use a renormalization group scheme where low
energy excitations of the Fermi surface are interpreted as a collection of
(1+1)-dimensional chiral fermions with a continuous flavor labelling the
momentum along the Fermi surface. Due to chirality, the Wilsonian effective
action is strictly UV finite. This allows one to extract the exact scaling
exponents although the theories flow to strongly interacting field theories at
low energies. In general, the low energy effective theory of the full Fermi
surface includes patch theories of more than one universality classes. As a
result, physical responses include multiple universal components at low
temperatures. We also point out that, in quantum field theories with extended
Fermi surface, a non-commutative structure naturally emerges between a
coordinate and a momentum which are orthogonal to each other. We show that the
invalidity of patch description for Fermi liquid states is tied with the
presence of UV/IR mixing associated with the emergent non-commutativity. On the
other hand, UV/IR mixing is suppressed in non-Fermi liquid states due to UV
insensitivity, and the patch description is valid.
