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arXiv:cond-mat/0410446 · Physical Review B

Unification of electromagnetic noise and Luttinger liquid via a quantum dot

Karyn Le Hur, Mei-Rong Li

Published Aug 9, 2005• 61 citations
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Abstract

We investigate the effect of dissipation on a small quantum dot (resonant level) tunnel coupled to a chiral Luttinger liquid (LL) with the LL parameter $K$. The dissipation stems from the coupling of the dot to an electric environment, being characterized by the resistance $R$, via Coulomb interactions. We show that this problem can be mapped onto a Caldeira-Leggett model where the (ohmic) bath of harmonic oscillators is governed by the effective dissipation strength $\ensuremath{\alpha}={(2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K})}^{\ensuremath{-}1}$ with ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}}^{\ensuremath{-}1}={K}^{\ensuremath{-}1}+2R∕{R}_{K}$ and ${R}_{K}=h∕{e}^{2}$ the quantum of resistance. Experimental consequences are discussed and the limit $K=1∕{2}^{+}$ is thoroughly studied at small $R∕{R}_{K}$ through the spin-boson-fermion model.

DOI
10.1103/physrevb.72.073305

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We investigate the effect of dissipation on a small quantum dot (resonant level) tunnel coupled to a chiral Luttinger liquid (LL) with the LL parameter KKK. The dissipation stems from the coupling of the dot to an electric environment, being characterized by the resistance RRR, via Coulomb interactions. We show that this problem can be mapped onto a Caldeira-Leggett model where the (ohmic) bath of harmonic oscillators is governed by the effective dissipation strength \ensuremathα=(2K\ifmmode~\else˜\fi)\ensuremath−1\ensuremath{\alpha}={(2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K})}^{\ensuremath{-}1}\ensuremathα=(2K\ifmmode~\else˜\fi​)\ensuremath−1 with K\ifmmode~\else˜\fi\ensuremath−1=K\ensuremath−1+2R∕RK{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}}^{\ensuremath{-}1}={K}^{\ensuremath{-}1}+2R∕{R}_{K}K\ifmmode~\else˜\fi​\ensuremath−1=K\ensuremath−1+2R∕RK​ and RK=h∕e2{R}_{K}=h∕{e}^{2}RK​=h∕e2 the quantum of resistance. Experimental consequences are discussed and the limit K=1∕2+K=1∕{2}^{+}K=1∕2+ is thoroughly studied at small R∕RKR∕{R}_{K}R∕RK​ through the spin-boson-fermion model.
We investigate a one dimensional quantum fluid coupled to a dissipative bath. The quantum fluid is captured by the canonical Luttinger liquid; the bath is given by the model of Caldeira and Leggett, i.e. a tower of oscillators coupled linearly to the fluid density, ρρρ. The bath can be integrated out exactly, producing an effective interaction for the fluid that is nonlocal in time; we argue that the form corresponding to Ohmic dissipation is generic. Compared to previous works, we compute correlation functions for this minimal model without approximation, including at finite temperature T>0T>0T>0. From these and a Kubo calculation, we conclude that arbitrary dissipation destroys the perfect conductivity of the Luttinger liquid via Zeno localization, even in the absence of a spatial potential; from RG analysis of harmonic terms, we also find that the open Luttinger liquid is significantly more prone to localization by such potentials, in contrast to the usual intuition that baths make systems less localized.
It is well known that dissipation usually induces decoherence in quantum systems. However, recently it has been employed to engineer zero-energy Majorana modes on the boundary of a one-dimensional fermionic chain. In this thesis, by employing the Caldeira-Leggett theory of quantum dissipation, we study the equivalence between a particle on a one-dimensional lattice coupled to a thermal bath, and the massless Luttinger liquid with an impurity on the boundary in a closed system. In particular, the Luttinger liquid with a single impurity at the boundary of a semi-infinite chain is bosonized into the boundary sine-Gordon theory. Here, a zero-energy Majorana mode appears at the boundary when the Luttinger parameter K=1/2. The equivalence is shown to apply also in the case of helical Luttinger liquids, which play an important role in topological insulators. This equivalence is then generalized to the full sine-Gordon theory, such that massive Luttinger liquids can be mapped to a single-particle one-dimensional tunneling Hamiltonian with a dissipative Ohmic Caldeira-Leggett bath.
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 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.
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 calculate the finite-frequency current noise of a nonequilibrium resonance-level quantum dot close to a dissipative quantum phase transition of the Kosterlitz-Thouless (KT) type between a de-localized phase for weak dissipation and a localized phase for strong dissipation. The resonance-level is coupled to two spinless fermionic baths with a finite bias voltage and an Ohmic boson bath representing the dissipative environment. The system is equivalent to an effective anisotropic Kondo model out of equilibrium. To compute the finite-frequency noise, we combine two recently developed Functional Renormalization Group (FRG) approaches in Refs.[17,22] and in Ref.[23]. The nonequilibrium current noise at zero-temperature and finite frequencies shows a singular dip in the de-localized phase for the magnitude of frequencies equal to the bias voltage; while the dip is smeared out as the system moves to the localized phase. The corresponding peak-to-dip crossover is found in the AC conductance for the magnitude of frequencies equal to the bias voltage. The relevance and applications of our results for the experiments and for tunnelings between Fractional Quantum Hall Edge (FQHE) states and chiral Luttinger liquids are discussed.