Abstract
We consider a model of quantum-wire junctions where the latter are described by conformal-invariant boundary conditions of the simplest type in the multicomponent compactified massless scalar free field theory representing the bosonized Luttinger liquids in the bulk of wires. The boundary conditions result in the scattering of charges across the junction with nontrivial reflection and transmission amplitudes. The equilibrium state of such a system, corresponding to inverse temperature β and electric potential V, is explicitly constructed both for finite and for semi-infinite wires. In the latter case, a stationary nonequilibrium state describing the wires kept at different temperatures and potentials may be also constructed. The main result of the present paper is the calculation of the full counting statistics (FCS) of the charge and energy transfers through the junction in a nonequilibrium situation. Explicit expressions are worked out for the generating function of FCS and its large-deviations asymptotics. For the purely transmitting case they coincide with those obtained in the literature, but numerous cases of junctions with transmission and reflection are also covered. The large deviations rate function of FCS for charge and energy transfers is shown to satisfy the fluctuation relations and the expressions for FCS obtained here are compared with the Levitov–Lesovik formulae.
Highlights
The transport phenomena in quantum wires and, in particular, across their junctions, have attracted a lot of interest in recent times, see e.g. [16, 14]
It was realized that the boundary conformal field theories (CFTs) description of the junction of wires gives via the Green-Kubo formalism a direct access to the low temperature electric conductance of junctions [35, 39, 40] that measure small currents induced by placing different wires in slightly different external electric potentials
There is another relation of the full counting statistics (FCS) statistics that we have obtained for the junction of wires and the Levitov-Lesovik type formulae, this time for the energy transfers
Summary
The transport phenomena in quantum wires (carbon nanotubes, semiconducting, metallic and molecular nanowires, quantum Hall edges) and, in particular, across their junctions, have attracted a lot of interest in recent times, see e.g. [16, 14]. It was shown in [10, 11, 12, 19] that for some boundary defects (those with pure transmission of charge or energy), the electric and thermal conductance and the long-time asymptotics of the full counting statistics (FCS) of charge and energy transfers through the junction may be calculated for the wires initially equilibrated at different temperatures and different potentials. The calculations of FCS presented in this paper are the first ones obtained for junctions with transmission and reflection modeled by conformal boundary defects It should be mentioned, that in a different physical setup, the FCS of charge transfers across an inhomogeneous Luttinger liquid conductor connected to two leads with distinct energy distributions was obtained by a “nonequilibrium bosonization” in [24, 25, 33]. Appendix C calculates the quadratic contribution to the Levitov-Lesovik large-deviations rate function of charge transfers for free fermions
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