Competing phases and critical behaviour in three coupled spinless Luttinger liquids

We study electronic phase competition in a strongly correlated system of three coupled spinless Luttinger liquids—one of the simplest models where topologically nontrivial chiral orders may be realized. We study the problem as a coupled sine-Gordon model, using a perturbative renormalization group (RG) approach. In contrast with counterparts with fewer fermionic species, here the scaling procedure generates off-diagonal contributions to the phase stiffness matrix, which require both rescaling as well as large rotations of the bosonic fields. These rotations, generally non-abelian in nature, introduce 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 behaviour in a variety of interaction parameter regimes where perturbative RG is possible. The phase boundaries are found to be of the Berezinskii–Kosterlitz–Thouless type, and we specify the parameter regimes where symmetry breaking between the three flavours of bosonic fields, orders involving different flavours, and chiral orders may be observed. Our approach and findings may be relevant for understanding phases and transitions at high magnetic fields in semimetals such as bismuth featuring three Fermi pockets.