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arXiv:0808.0849 · Europhysics Letters (EPL)

Interacting resonant level coupled to a Luttinger liquid: Universality of thermodynamic properties

M. Goldstein, Y. Weiss, Richard Berkovits

Published Jun 1, 2009• 29 citations
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Abstract

We investigate a model of a single resonant level coupled to the edge of a quantum wire in the Luttinger-liquid phase or to the middle of a chiral Luttinger liquid via both tunneling and a contact interaction. Utilizing the Yuval-Anderson approach, we map this model onto a classical 1D Coulomb gas in which all the details of both the interactions in the lead and the level-lead interaction enter only through the corresponding Fermi-edge singularity exponent, which we explicitly evaluate using the Bethe ansatz solution for a particular model of the lead. Thus the population, dynamical capacitance and level entropy are universal in the sense of being equal for models with interactions differing in magnitude and even in sign. We demonstrate this to hold quantitatively using density matrix renormalization group calculations. Since the Coulomb gas description is of the single-channel Kondo type, we infer that the universality we found implies that Luttinger-liquid physics has no qualitative effect on these properties, in contrast with perturbative results.

DOI
10.1209/0295-5075/86/67012

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We study a model of resonant point-contact tunneling of a single Luttinger-liquid lead, using the boundary state formulation. The model is described by a single chiral Fermion in one-dimensional space with one point contact interaction at the origin. It is the simplest model of this type. By folding the infinite line, we map the model onto a nonchiral model on the half infinite line and transcribe the point-contact tunneling interaction into a boundary interaction. The tunneling interaction yields a nonlocal effective action at the boundary. We explicitly construct the boundary state and evaluate the current correlation functions. The contact interaction is shown to act on the lower frequency modes more strongly.
We study a system consisting of a Luttinger liquid coupled to a quantum dot on the boundary. The Luttinger liquid is expressed in terms of fermions interacting via density-density coupling, and the dot is modeled as an interacting resonant level onto which the bulk fermions can tunnel. We solve the Hamiltonian exactly and construct all eigenstates. We study both the zero- and finite-temperature properties of the system, in particular, we compute the exact dot occupation as a function of the dot energy in all parameter regimes. The system is seen to flow from weak to strong coupling for all values of the bulk interaction, with the flow characterized by a nonperturbative Kondo scale. We identify the critical exponents at the weak- and strong-coupling regimes.
We study a model of a quantum Hall system with the electrons confined to a narrow, linear channel. This system is mapped to a one-dimensional (1D) system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. The mapping is studied at the microscopic level, and we show in detail how the form of the 1D Luttinger liquid is detemined by interactions with particles at the edges and in the bulk of the quantum Hall system. We focus, in particular, on how the parameters are renormalized by the interactions between the two edges and show that the velocity parameter of the current is not modified. The paper is Paper I in a sequence of two papers, where Paper II [M. Horsdal and J. M. Leinaas, Phys. Rev. B 76, 195322 (2007)] gives a derivation of the two-dimensional electron correlation function.
We study a Luttinger liquid (LL) coupled to a generic environment consisting of bosonic modes with arbitrary density-density and current-current interactions. The LL can be either in the conducting phase and perturbed by a weak scatterer or in the insulating phase and perturbed by a weak link. The environment modes can also be scattered by the imperfection in the system with arbitrary transmission and reflection amplitudes. We present a general method of calculating correlation functions under the presence of the environment and prove the duality of exponents describing the scaling of the weak scatterer and of the weak link. This duality holds true for a broad class of models and is sensitive to neither interaction nor environmental modes details, thus it shows up as the universal property. It ensures that the environment cannot generate new stable fixed points of the renormalization group flow. Thus, the LL always flows toward either conducting or insulating phase. Phases are separated by a sharp boundary which is shifted by the influence of the environment. Our results are relevant, for example, for low-energy transport in (i) an interacting quantum wire or a carbon nanotube where the electrons are coupled to the acoustic phonons scattered by the lattice defect; (ii) a mixture of interacting fermionic and bosonic cold atoms where the bosonic modes are scattered due to an abrupt local change of the interaction; (iii) mesoscopic electric circuits.
We study a quantum phase transition that occurs in a system composed of two impurities (or quantum dots), each coupled to a different interacting (Luttinger liquid) lead. While the impurities are coupled electrostatically, there is no tunneling between them. Using a mapping of this system onto a Kondo model, we show analytically that the system undergoes a Berezinskii-Kosterlitz-Thouless quantum phase transition as a function of the Luttinger liquid parameter in the leads and the dot-lead interaction. The phase with low values of the Luttinger liquid parameter is characterized by an abrupt switch of the population between the impurities as a function of a common applied gate voltage. However, this behavior is hard to verify numerically since one would have to study extremely long systems. Interestingly, though, at the transition the entanglement entropy drops from a finite value of ln(2)\mathrm{ln}(2)ln(2) to zero. The drop becomes sharp for infinite systems. One can employ finite-size scaling to extrapolate the transition point and the behavior in its vicinity from the behavior of the entanglement entropy in moderate size samples. We employ the density matrix renormalization-group numerical procedure to calculate the entanglement entropy of systems with lead lengths of up to 480 sites. Using finite-size scaling, we extract the transition value and show it to be in good agreement with the analytical prediction.