Tomonaga-Luttinger liquid with reservoirs in a multiterminal geometry

We propose a formalism to describe the transport properties of a Luttinger liquid (LL) coupled to many reservoirs. We assume that the coupling of the electrodes to the LL is characterized by a set of linear relations between the external voltages and the chemical potentials of the chiral modes of the LL. We show that these linear relations completely determine linear transport in the joint system reservoirs+LL. As an illustration we consider an exactly solvable microscopic model in a multiterminal geometry for which these linear relations can be explicitly derived; in this model the Landauer-Buttiker formalism fails: if it were valid, the usual interpretation of conductance matrix elements in terms of reflection and transmission coefficients could yield negative probabilities. We then apply our formalism to a discussion of shot noise through an impurity in a LL connected to two reservoirs.