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arXiv:cond-mat/0211474 · Physical Review Letters

Resonant Tunneling between Luttinger Liquids: A Solvable Case

A. Komnik, Alexander O. Gogolin

Published Jun 19, 2003• 56 citations
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Abstract

We discuss the conductance of a Luttinger liquid interrupted by a quantum dot containing a single resonant level. Using bosonization and refermionization methods, we find a mapping to a Kondo-type problem which possesses a nontrivial Toulouse-type solvable point. At this point, we obtain an analytic expression for the nonlinear current-voltage characteristics and analyze the differential conductance and the width of the resonance peak as functions of bias and gate voltages, temperature, and barrier asymmetry. We also determine the exact scaling function for the linear conductance.

DOI
10.1103/physrevlett.90.246403

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We discuss the conductance of a Luttinger liquid interrupted by a quantum dot containing a single resonant level. Using bosonization and refermionization methods, we find a mapping to a Kondo-type problem which possesses a nontrivial Toulouse-type solvable point. At this point, we obtain an analytic expression for the nonlinear current-voltage characteristics and analyze the differential conductance and the width of the resonance peak as functions of bias and gate voltages, temperature, and barrier asymmetry. We also determine the exact scaling function for the linear conductance.
We study point contact tunneling between two leads of a Tomonaga-Luttinger liquid through two degenerate resonant levels in parallel. This is one of the simplest cases of a quantum junction problem where the Fermi statistics of the electrons plays a nontrivial role through the Klein factors appearing in bosonization. Using a mapping to a ``generalized Coulomb model'' studied in the context of the dissipative Hofstadter model, we find that any asymmetry in the tunneling amplitudes from the two leads grows at low temperatures, so that ultimately there is no conductance across the system. For the symmetric case, we identify a nontrivial fixed point of this model; the conductance at that point is generally different from the conductance through a single resonant level.
We study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. For a range of repulsive interactions, we find that for symmetric barriers there exist resonances with a universal peak conductance 2g∗e2/h2g^* e^2/h2g∗e2/h that are controlled by a non-trivial intermediate fixed point. This fixed point is also a quantum critical point separating symmetry-protected topological phases. By tuning the system through resonance, all SPT phases can be accessed. For a particular interaction strength with Luttinger parameters gρ=1/3g_ρ=1/3gρ​=1/3 and gσ=1g_σ=1gσ​=1, we show that the problem is equivalent to a two channel SU(3)SU(3)SU(3) Kondo problem(SU(3)2SU(3)_2SU(3)2​ CFT). At the Toulouse limit, both problems can be mapped to a quantum Brownian motion model on a Kagome lattice, which in turn is related to the quantum Brownian motion on a honeycomb lattice and the three-channel SU(2)SU(2)SU(2) Kondo problem(SU(2)3SU(2)_3SU(2)3​ CFT). "Level-rank duality" in the quantum Brownian motion model relating SU(2)kSU(2)_kSU(2)k​ CFT to SU(k)2SU(k)_2SU(k)2​ CFT is also explored. Utilizing the boundary conformal field theory, the on-resonance conductance of our resonant tunneling problem is calculated as well as the scaling dimension of the leading relevant operator. This allows us to compute the scaling behavior of the resonance line-shape as a function of temperature.
The resonant tunneling through a double-barrier structure in a one-dimensional system of interacting spinless fermions is studied. The conductance is calculated as a function of gate voltage and temperature. It is shown that even in Luttinger liquids the line shape of resonances is almost the same as that of the noninteracting Fermi liquid provided that the temperature is not too low. The temperature dependence, on the other hand, is drastically changed by the interaction, and the height of the conductance peak is a nonmonotonic function of temperature for weak repulsive interaction.
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.
We have calculated the two-probe Landauer conductance of a one-channel quantum wire containing a Luttinger liquid and connected to two noninteracting leads through tunnel barriers. The tunneling conductance shows broad resonances as a function of the bias voltage, this being a manifestation of the spin-charge separation. In the limit of zero bias and zero barrier, the tunneling conductance reduces to the ballistic contact value e2h\frac{{e}^{2}}{h}he2​ per channel.
We find the nonlinear conductance of a dissipative resonant level in the nonequilibrium steady state near its quantum critical point. The system consists of a spin-polarized quantum dot connected to two resistive leads that provide ohmic dissipation. We focus on the crossover from the strong-coupling, non-Fermi-liquid regime to the weak-coupling, Fermi-liquid ground state, a crossover driven by the instability of the quantum critical point to hybridization asymmetry or detuning of the level in the dot. We show that the crossover properties are given by tunneling through an effective single barrier described by the boundary sine-Gordon model. The nonlinear conductance is then obtained from thermodynamic Bethe ansatz results in the literature, which were developed to treat tunneling in a Luttinger liquid. The current-voltage characteristics are thus found for any value of the resistance of the leads. For the special case of lead resistance equal to the quantum resistance, we find mappings onto, first, the two-channel Kondo model and, second, an effectively noninteracting model from which the nonlinear conductance is found analytically. A key feature of the general crossover function is that the nonequilibrium crossover driven by applied bias is different from the crossover driven by temperature---we find that the nonequilibrium crossover is substantially sharper. Finally, we compare to experimental results for both the bias and temperature crossovers: the agreement is excellent.