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arXiv:1501.07433 · Nuclear Physics B

Nonequilibrium transport through quantum-wire junctions and boundary defects for free massless bosonic fields

Krzysztof Gawȩdki, Clément Tauber

Published Apr 20, 2015• 12 citations
arXivHTMLPDF

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.

DOI
10.1016/j.nuclphysb.2015.04.014

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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.
The thesis, consisting of three parts, is focusing on different physical problems that are related to two dimensional Conformal Field Theory (CFT).The first part deals with nonequilibrium transport properties across a junction of quantum wires. Three models are studied. The first one describes the wires by a free compactified bosonic field, seen as the bosonization of the Luttinger liquid of electrons. The junction of the wires is modeled by a boundary condition that ensures nontrivial scattering of the charges between the wires. Combining canonical quantization and functional integral, we compute exactly the current correlation functions in equilibrium, but also in a nonequilibrium stationary state, as well as the full counting statistics of charge and energy between the wires set at different temperatures and potentials. The two other models of quantum wire junction are based on Wess-Zumino-Witten theory (WZW). In the first one, the junction is described by a “cyclic brane” and in the second, by a “coset brane”. The results in the first case are as complete as for the free field, but the charges are fully transmitted from one wire to the next one. In the second case, the scattering is nontrivial, but the model turns out to be difficult to solve.The second part of the thesis studies the global gauge anomalies in “coset” models of CFT, realized as gauged WZW theories. The (almost) complete classification of such anomalies, that lead to some inconsistent coset models, is presented. It is based on Dynkin classification of subalgebras of simple Lie algebras.Finally, the third part of the thesis describes the geometric construction of index from unitary operator families obtained from valence band projectors of a two-dimensional time-reversal invariant topological insulator. The index is related on one hand to the square root of the Wess-Zumino amplitude of such a family, and, on the other hand, it reproduces the Kane-Mele invariant of the insulator. The last identification requires a nontrivial argument that uses a new gauge anomaly of WZW models with boundary.The three parts of the thesis use similar geometrical tool of CFT, that permits to obtain several original results. The unity in the method, as well as the topic of anomalies, builds a bridge between the different components of the manuscript.
Nonequilibrium bosonization technique is used to study current fluctuations of interacting electrons in a single-channel quantum wire representing a Luttinger liquid (LL) conductor. An exact expression for the time resolved full counting statistics of the transmitted charge is derived. It is given by the Fredholm determinant of the counting operator with a time-dependent scattering phase. The result has a form of counting statistics of noninteracting particles with fractional charges, induced by scattering off the boundaries between the LL wire and the noninteracting leads.
We propose and investigate an exactly solvable model of non-equilibrium Luttinger liquid on a star graph, modeling a multi-terminal quantum wire junction. The boundary condition at the junction is fixed by an orthogonal matrix , which describes the splitting of the electric current among the leads. The system is driven away from equilibrium by connecting the leads to heat baths at different temperatures and chemical potentials. The associated non-equilibrium steady state depends on and is explicitly constructed. In this context, we develop a non-equilibrium bosonization procedure and compute some basic correlation functions. Luttinger liquids with general anyon statistics are considered. The relative momentum distribution away from equilibrium turns out to be the convolution of equilibrium anyon distributions at different temperatures. Both the charge and heat transport are studied. The exact current–current correlation function is derived and the zero-frequency noise power is determined.
Inhomogeneities and junctions in wires are natural sources of scattering, and hence resistance. A conducting fixed point usually requires an adiabatically smooth system. One notable exception is ``healing,'' which has been predicted in systems with special symmetries, where the system is driven to the homogeneous fixed point. Here we present theoretical results for a different type of conducting fixed point which occurs in inhomogeneous wires with an abrupt jump in hopping and interaction strength. We show that it is always possible to tune the system to an unstable conducting fixed point which does not correspond to translational invariance. We analyze the temperature scaling of correlation functions at and near this fixed point and show that two distinct boundary exponents appear, which correspond to different effective Luttinger liquid parameters. Even though the system consists of two separate interacting parts, the fixed point is described by a single conformally invariant boundary theory. We present details of the general effective bosonic field theory including the mode expansion and the finite size spectrum. The results are confirmed by numerical quantum Monte Carlo simulations on spinless fermions. We predict characteristic experimental signatures of the local density of states near junctions.
We propose and investigate an exactly solvable model of non-equilibriumLuttinger liquid on a star graph, modeling a multi-terminal quantum wirejunction. The boundary condition at the junction is fixed by an orthogonalmatrix S, which describes the splitting of the electric current among theleads. The system is driven away from equilibrium by connecting the leads toheat baths at different temperatures and chemical potentials. The associatednon-equilibrium steady state depends on S and is explicitly constructed. Inthis context we develop a non-equilibrium bosonization procedure and computesome basic correlation functions. Luttinger liquids with general anyonstatistics are considered. The relative momentum distribution away fromequilibrium turns out to be the convolution of equilibrium anyon distributionsat different temperatures. Both the charge and heat transport are studied. Theexact current-current correlation function is derived and the zero-frequencynoise power is determined.
We study junctions of single-channel spinless Luttinger liquids using bosonization. We generalize earlier studies by allowing the junction to be superconducting and find new charge nonconserving low-energy fixed points. We establish the existence of $g\ensuremath{\leftrightarrow}1/g$ duality (where ggg is the Luttinger liquid parameter) between the charge-conserving (normal) junction and the charge nonconserving (superconducting) junction by evaluating and comparing the scaling dimensions of various operators around the fixed points in both the normal and superconducting sectors of the theory. For the most general two-wire junction, we show that there are two conformally invariant one-parameter families of fixed points which are also connected by a duality transformation. We also show that the stable fixed point for the two-wire superconducting junction corresponds to the situation where the crossed Andreev reflection (an incoming electron is transmitted as an outgoing hole) is perfect between the wires. For the three-wire junction, we study, in particular, the superconducting analogs of the chiral ${\mathbsf{D}}_{\mathbsf{P}}$ and the disconnected fixed points obtained earlier in the literature in the context of charge-conserving three-wire junctions. We show that these fixed points can be stabilized for g<1g<1g<1 (repulsive electrons) within the superconducting sector of the theory which makes them experimentally relevant.