  We elaborate how to construct interweaving chiral spirals in (2+1)
dimensions, defined as a superposition of chiral spirals oriented in different
directions. We divide a two-dimensional Fermi sea into distinct wedges,
characterized by the opening angle 2Theta and depth Q ~ pF, where pF is the
Fermi momentum. In each wedge, the energy is lowered by forming a single chiral
spiral. The optimal values for Theta and Q are chosen by balancing this gain in
energy versus the cost of deforming the Fermi surface (which dominates at large
Theta) and patch-patch interactions (dominant at small Theta). Using a
non-local four-Fermi interaction model, we estimate the gain and cost in energy
by expanding in terms of 1/Nc (where Nc is the number of colors), lqcd/Q, and
Theta. Due to a form factor in our non-local model, at small 1/Nc the mass gap
(chiral condensate) is large, and the interaction among quarks and the
condensate local in momentum space. Consequently, interactions between
different patches are localized near their boundaries, and it is simple to
embed many chiral spirals. We identify the dominant and subdominant terms at
high density and categorize formulate an expansion in terms of lqcd/Q or Theta.
The kinetic term in the transverse directions is subdominant, so that
techniques from (1+1)-dimensional systems can be utilized. To leading order in
1/Nc and lqcd/Q, the total gain in energy is ~ pF lqcd^2 with Theta ~
(lqcd/pF)^{3/5}. Since Theta decreases with increasing pF, there should be
phase transitions associated with the change in the wedge number. We also argue
the effects of subdominant terms at lower density where the large-Nc
approximation is more reliable.
