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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.
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We develop the renormalization group theory of the conductances of N-lead junctions of spinless Luttinger-liquid wires as functions of bias voltages applied to N independent Fermi-liquid reservoirs. Based on the perturbative results up to second order in the interaction we demonstrate that the conductances obey scaling. The corresponding renormalization group $beta$ functions are derived up to second order.
We study charge transport through $N$-lead junctions ($Ngeq 3$) of spinless Luttinger liquid wires with bias voltages applied to Fermi-liquid reservoirs. In particular, we consider a Y junction, which is a setup characteristic of the tunneling experiment. In this setup, the strength of electron-electron interactions in one of the arms (tunneling tip) is different from that in the other two arms (which form together the main wire). For a generic single-particle $S$ matrix of the junction, we find that the bias voltage $V$ applied---even symmetrically---to the main wire generates a current proportional to $|V|$ in the tip wire. We identify two mechanisms of this nonequilibrium-induced emergent chirality in a setup characterized by the time-reversal and parity symmetric Hamiltonian of the junction. These are: (i) the emergence of an effective magnetic flux, which breaks time-reversal symmetry, and (ii) the emergence of parity-breaking asymmetry of the setup, both proportional to the interaction strength and the sign of the voltage. The current in the tip wire generated by mechanism (i) is reminiscent of the Hall current in the linear response of a system the Hamiltonian of which breaks time-reversal symmetry; however, in the absence of any magnetic field or a local magnetic moment. Similarly, mechanism (ii) can be thought of as an emergent photogalvanic effect; however, in the presence of inversion symmetry within the main wire. The nonequilibrium chirality implies a rectification of the current in the tip when the main wire is biased by $it ac$ voltage.
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 applied 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-way junction of spinless Luttinger-liquid wires as functions of bias voltages applied to three independent Fermi-liquid reservoirs. In particular, we consider the setup that is characteristic of a tunneling experiment, in which the strength of electron-electron interactions in one of the arms of the junction (tunneling tip) is different from that in the other two arms (which together form a main wire). The scaling dependence of the two independent conductances on bias voltages is determined within a fermionic renormalization-group approach in the limit of weak interactions. The solution shows that, in general, the conductances scale with the bias voltages in an essentially different way compared to their scaling with the temperature $T$. Specifically, unlike in the two-terminal setup, the nonlinear conductances cannot be generically obtained from the linear ones by simply replacing $T$ with the corresponding bias voltage or the largest one. Remarkably, a finite tunneling bias voltage prevents the interaction-induced breakup of the main wire into two disconnected pieces in the limit of zero $T$ and a zero source-drain voltage.
We consider theoretically the transport in a one-channel spinless Luttinger liquid with two strong impurities in the presence of dissipation. As a difference with respect to the dissipation free case, where the two impurities fully transmit electrons at resonance points, the dissipation prevents complete transmission in the present situation. A rich crossover diagram for the conductance as a function of applied voltage, temperature, dissipation strength, Luttinger liquid parameter K and the deviation from the resonance condition is obtained. For weak dissipation and 1/2<K<1, the conduction shows a non-monotonic increase as a function of temperature or voltage. For strong dissipation the conduction increases monotonically but is exponentially small.
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