No abstract

We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g2, and subject to a potential barrier of arbitrary strength, as a function of temperature. We map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. First the renormalization group (RG) equation for weak interaction is derived in the current operator language both using the operator product expansion and the equation of motion method. To access the strong coupling regime, two methods of deducing the RG equation from perturbation theory, based on the scaling hypothesis and on the Callan-Symanzik formulation, are discussed. The important role of scale independent terms is emphasized. The latter depend on the regulaization scheme used (length versus temperature cutoff). Analyzing the perturbation theory in the fermionic representation, the diagrams contributing to the renormalization group β-function are identified. A universal part of the β-function is given by a ladder series and summed to all orders in g2. First non-universal corrections beyond the ladder series are discussed and are shown to differ from the exact solutions obtained within conformal field theory which use a different regularization scheme. The RG equation for the temperature dependent conductance is solved analytically. Our result agrees with known limiting cases.

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 g3 = g in the third wire. We employ the fermionic representation in the scattering 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 Yjunction) is analyzed in detail. For repulsive interaction (g, g3 > 0) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction (g < 0 and/or g3 < 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 = g3, at variance with the results of Oshikawa et al. The pair tunneling phenomenon conjectured by the latter authors is not found by us.