The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the a
pplied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $Lambda$, which equals the bias voltage $V$, if $V$ is the largest energy scale compared to temperature $T$ and inverse wire length $L^{-1}$, and interpolates between these quantities in the crossover regimes.
We calculate the conductances of a three-terminal junction set-up of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength g_3 != g in the third wire. We employ the fermionic representation in the sc
attering state picture, allowing for a direct calculation of the linear response conductances, without the need of introducing contact resistances at the connection points to the outer ideal leads. The matrix of conductances is parametrized by three variables. For these we derive coupled renormalization group (RG) equations, by summing up infinite classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y-junction) is analyzed in detail. For repulsive interaction (g,g_3>0) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction (g<0 and/or g_3<0) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong coupling domain. We find new fixed points not discussed before, even at the symmetric line g=g_3, at variance with the results of Oshikawa et al. The pair tunneling phenomenon conjectured by the latter authors is not found by us.
We calculate the conductance of a system of two spinless Luttinger liquid wires with different interaction strengths g_1, g_2, connected through a short junction, within the scattering state formalism. Following earlier work we formulate the problem
in current algebra language, and calculate the scale dependent contribution to the conductance in perturbation theory keeping the leading universal contributions to all orders in the interaction strength. From that we derive a renormalization group (RG) equation for the conductance. The analytical solution of the RG-equation is discussed in dependence on g_1, g_2. The regions of stability of the two fixed points corresponding to conductance G=0 and G=1, respectively, are determined.
