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arXiv:1101.3723 · Physical Review B

Capacitance of a resonant level coupled to Luttinger liquids

Moshe Goldstein, Richard Berkovits

Published Oct 18, 2010• 14 citations
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Abstract

In this Rapid Communication, we study the differential capacitance of a single-level quantum dot attached to quantum wires in the Luttinger-liquid phase or to fractional quantum-Hall edges, by both tunneling and interactions. We show that logarithmic or even power-law divergence of the capacitance at low temperatures may result in a substantial region of the parameter space (including the $\ensuremath{\nu}=1/3$ filling). This behavior is a manifestation of generalized two-channel Kondo physics and can be used to extract Luttinger-liquid parameters from thermodynamic measurements.

DOI
10.1103/physrevb.82.161307

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In this Rapid Communication, we study the differential capacitance of a single-level quantum dot attached to quantum wires in the Luttinger-liquid phase or to fractional quantum-Hall edges, by both tunneling and interactions. We show that logarithmic or even power-law divergence of the capacitance at low temperatures may result in a substantial region of the parameter space (including the \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 filling). This behavior is a manifestation of generalized two-channel Kondo physics and can be used to extract Luttinger-liquid parameters from thermodynamic measurements.
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.
We prove an exact duality between the side-coupled and embedded geometries of a single level quantum dot attached to a quantum wire in a Luttinger liquid phase by a tunneling term and interactions. This is valid even in the presence of a finite bias voltage. Under this relation the Luttinger liquid parameter g goes into its inverse, and transmittance maps onto reflectance. We then demonstrate how this duality is revealed by the transport properties of the side-coupled case. Conductance is found to exhibit an antiresonance as a function of the level energy, whose width vanishes (enhancing transport) as a power law for low temperature and bias voltage whenever g > 1, and diverges (suppressing transport) for g < 1. On-resonance transmission is always destroyed, unless g is large enough.
Luttinger-liquid behavior of one-dimensional correlated electron systems is characterized by power-law scaling of a variety of physical observables with exponents determined by a single interaction dependent parameter KKK. We suggest a setup to study Luttinger-liquid behavior in quantum wires which allows us to determine KKK from two independent measurements: transport through a quantum dot embedded in the wire and the charge on the dot. Consistency of the two KKK's for a single probe would provide strong experimental evidence for the Luttinger-liquid paradigm.
We study resonant tunneling in a Luttinger liquid with a double barrier enclosing a dot region. Within a microscopic model calculation the conductance GGG as a function of temperature TTT is determined over several decades. We identify parameter regimes in which the peak value Gp(T){G}_{p}(T)Gp​(T) shows distinctive power-law behavior. For intermediate dot parameters, Gp{G}_{p}Gp​ behaves in a nonuniversal way.
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 generalize the hierarchical equations of motion method to study electron transport through a quantum dot or molecule coupled to one-dimensional interacting leads that can be described as Luttinger liquids. Such leads can be realized, for example, by quantum wires or fractional quantum Hall edge states. In comparison to noninteracting metallic leads, Luttinger liquid leads involve many-body correlations and the single-particle tunneling density of states shows a power-law singularity at the chemical potential. Using the generalized hierarchical equations of motion method, we assess the importance of the singularity and the next-to-leading order many-body correlations. To this end, we compare numerically converged results with second- and first-order results of the hybridization expansion that is inherent to our method. As a test case, we study transport through a single-level quantum dot or molecule that can be described by an Anderson impurity model. Cotunneling effects turn out to be most pronounced for attractive interactions in the leads or repulsive ones if an excitonic coupling between the dot and the leads is realized. We also find that an interaction-induced negative differential conductance near the Coulomb blockade thresholds is slightly suppressed as compared to a first-order and/or rate equation result. Moreover, we find that the two-particle (nnn-particle) correlations enter as a second-order (nnn-order) effect and are, thus, not very pronounced at the high temperatures and parameters that we consider.