Chiral Y junction of Luttinger liquid wires at weak coupling: Lines of stable fixed points

Abstract

We calculate the conductances of a Y-junction setup of Luttinger liquid wires threaded by a magnetic flux, allowing for a different interaction strength ${g}_{3}\ensuremath{\ne}g$ in the third wire. The scattering matrix and the matrix of conductances are parametrized by three variables. For these we derive coupled renormalization group (RG) equations, up to second order in the interaction, within the scattering state formalism. The fixed point structure of these equations is analyzed in detail. For a repulsive interaction ($g,{g}_{3}>0$) there is only one stable fixed point, corresponding to the complete separation of the wires. For an attractive interaction ($g<0$ and/or ${g}_{3}<0$) four fixed points are found, whose stability depends on the interaction strength. For special values of the interaction parameters (a) $g=0$, ${g}_{3}<0$ or (b) ${g}_{3}+{g}^{2}/2=0$, $g<0$ we find whole lines of stable fixed points. We conjecture that lines of fixed points appear generically at interfaces in the interaction coupling constant space separating two regions with different stable fixed points. We also find that in certain regions of the $g$-${g}_{3}$ plane the RG flow is towards a fixed point without chirality, implying that the effect of the magnetic flux is completely screened.