  We study the spontaneous non-magnetic time-reversal symmetry breaking in a
two-dimensional Fermi liquid without breaking either the translation symmetry
or the U(1) charge symmetry. Assuming that the low-energy physics is described
by fermionic quasiparticle excitations, we identified an "emergent" local
$U(1)^N$ symmetry in momentum space for an $N$-band model. For a large class of
models, including all one-band and two-band models, we found that the
time-reversal and chiral symmetry breaking can be described by the $U(1)^N$
gauge theory associated with this emergent local $U(1)^N$ symmetry. This
conclusion enables the classification of the time-reversal symmetry-breaking
states as types I and II, depending on the type of accompanying spatial
symmetry breaking. The properties of each class are studied. In particular, we
show that the states breaking both time-reversal and chiral symmetries are
described by spontaneously generated Berry phases. We also show examples of the
time-reversal symmetry-breaking phases in several different microscopically
motivated models and calculate their associated Hall conductance within a
mean-field approximation. The fermionic nematic phase with time-reversal
symmetry breaking is also presented and the possible realizations in strongly
correlated models such as the Emery model are discussed.
