Interweaving chiral spirals

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 2Θ 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 Θ and Q are chosen by balancing this gain in energy versus the cost of deforming the Fermi surface (which dominates at large Θ) and patch-patch interactions (dominant at small Θ). 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), ΛQCD/Q, and Θ. 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 ΛQCD/Q or Θ. 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 ΛQCD/Q, the total gain in energy is ∼pFΛQCD2 with Θ∼(ΛQCD/pF)3/5. Since Θ 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.