  We consider a system of 2D fermions on a triangular lattice with well
separated electron and hole pockets of similar sizes, centered at certain
high-symmetry-points in the Brillouin zone. We first analyze Stoner-type
spin-density-wave (SDW) magnetism. We show that SDW order is degenerate at the
mean-field level. Beyond mean-field, the degeneracy is lifted and is either
$120^{\circ}$ "triangular" order (same as for localized spins), or a collinear
order with antiferromagnetic spin arrangement on two-thirds of sites, and
non-magnetic on the rest of sites. We also study a time-reversal symmetric
directional spin bond order, which emerges when some interactions are repulsive
and some are attractive. We show that this order is also degenerate at a
mean-field level, but beyond mean-field the degeneracy is again lifted. We next
consider the evolution of a magnetic order in a magnetic field starting from an
SDW state in zero field. We show that a field gives rise to a canting of an SDW
spin configuration. In addition, it necessarily triggers the directional bond
order, which, we argue, is linearly coupled to the SDW order in a finite field.
We derive the corresponding term in the Free energy. Finally, we consider the
interplay between an SDW order and superconductivity and charge order. For
this, we analyze the flow of the couplings within parquet renormalization group
(pRG) scheme. We show that magnetism wins if all interactions are repulsive and
there is little energy space for pRG to develop. However, if system parameters
are such that pRG runs over a wide range of energies, the system may develop
either superconductivity or an unconventional charge order, which breaks
time-reversal symmetry.
