  We study electronic phase competition in a system of three coupled spinless
Luttinger liquids using abelian bosonization, together with a perturbative
renormalization group (RG) analysis. The scaling procedure generates
off-diagonal contributions to the phase stiffness matrix, which require both
rescaling as well as large rotations of the fields. These rotations, generally
non-abelian in nature, are important for correctly obtaining the dominant
electronic orders and critical behavior in different parameter regimes. They
generate a coupling between different interaction channels even at the
tree-level order in the coupling constant scaling equations. We study competing
phases in this system, taking into account the aforementioned rotations, and
determine its critical behavior in a variety of interaction parameter regimes
where perturbative RG is possible. The phase boundaries are found to be of the
Berezinskii-Kosterlitz-Thouless (BKT) type, and we specify the parameter
regimes where valley-symmetry breaking, chiral orders, and restoration of
$C_{3}$ symmetry may be observed. We discuss experimental systems where our
approach and findings may be relevant.
