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arXiv:1002.4038 · Physical Review B

Nonequilibrium occupation number and charge susceptibility of a resonance level close to a dissipative quantum phase transition

Chung‐Hou Chung, K. V. P. Latha

Published Aug 30, 2010• 3 citations
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Abstract

Based on the recent paper [Phys. Rev. Lett. 102, 216803 (2009)], we study the nonequilibrium occupation number ${n}_{d}$ and charge susceptibility $\ensuremath{\chi}$ of a resonance level close to dissipative quantum phase transition of the Kosterlitz-Thouless (KT) type between a delocalized 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 bosonic bath representing the dissipative environment. The system is equivalent to an effective anisotropic Kondo model out of equilibrium. Within the nonequilibrium renormalization-group approach, we calculate nonequilibrium magnetization $M$ and spin susceptibility $\ensuremath{\chi}$ in the effective Kondo model, corresponding to $2{n}_{d}\ensuremath{-}1$ and $\ensuremath{\chi}$ of a resonance level, respectively. We demonstrate the smearing of the KT transition in the nonequilibrium magnetization $M$ as a function of the effective anisotropic Kondo couplings, in contrast to a perfect jump in $M$ at the transition in equilibrium. In the limit of large bias voltages, we find $M$ and $\ensuremath{\chi}$ at the KT transition and in the localized phase show deviations from the equilibrium Curie-law behavior. As the system gets deeper in the localized phase, both ${n}_{d}\ensuremath{-}1/2$ and $\ensuremath{\chi}$ decrease more rapidly to zero with increasing bias voltages.

DOI
10.1103/physrevb.82.085120

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Based on the recent paper [Phys. Rev. Lett. 102, 216803 (2009)], we study the nonequilibrium occupation number nd{n}_{d}nd​ and charge susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ of a resonance level close to dissipative quantum phase transition of the Kosterlitz-Thouless (KT) type between a delocalized 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 bosonic bath representing the dissipative environment. The system is equivalent to an effective anisotropic Kondo model out of equilibrium. Within the nonequilibrium renormalization-group approach, we calculate nonequilibrium magnetization MMM and spin susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ in the effective Kondo model, corresponding to 2nd\ensuremath−12{n}_{d}\ensuremath{-}12nd​\ensuremath−1 and \ensuremathχ\ensuremath{\chi}\ensuremathχ of a resonance level, respectively. We demonstrate the smearing of the KT transition in the nonequilibrium magnetization MMM as a function of the effective anisotropic Kondo couplings, in contrast to a perfect jump in MMM at the transition in equilibrium. In the limit of large bias voltages, we find MMM and \ensuremathχ\ensuremath{\chi}\ensuremathχ at the KT transition and in the localized phase show deviations from the equilibrium Curie-law behavior. As the system gets deeper in the localized phase, both nd\ensuremath−1/2{n}_{d}\ensuremath{-}1/2nd​\ensuremath−1/2 and \ensuremathχ\ensuremath{\chi}\ensuremathχ decrease more rapidly to zero with increasing bias voltages.
The dynamical decoherence rate and charge susceptibility of a nonequilibrium quantum dot close to a dissipative quantum phase transition are calculated. The setup concerns a resonance-level quantum dot coupled to two spinless fermionic baths with a finite bias voltage and an ohmic bosonic bath representing a dissipative environment. The system is equivalent to an anisotropic Kondo model. As dissipation strength increases, the system at zero temperature and zero bias exhibits a quantum phase transition of the Kosterlitz-Thouless (KT) type between a conducting delocalized phase and an insulating localized phase. Within the nonequilibrium frequency-dependent renormalization group (RG) approach, the finite bias crossover in dynamical decoherence rate and charge susceptibility close to the transition are addressed. The dynamical decoherence rate is found to increase with increasing frequency. In the delocalized phase, it shows a singularity at frequencies equal to positive or negative bias voltage. As the system cross overs to the localized phase, the decoherence rate at low frequencies gets progressively smaller and the singular feature is gradually smeared out, leading to a single linear frequency dependence. The dynamical charge susceptibility at low frequencies shows a dip-to-peak crossover across the transition. Relevance of these results to the experiments is discussed.
The resonant-level model represents a paradigmatic quantum system which serves as a basis for many other quantum impurity models. We provide a comprehensive analysis of the nonequilibrium transport near a quantum phase transition in a spinless dissipative resonant-level model, extending earlier work [Phys. Rev. Lett 102, 216803 (2009)]. A detailed derivation of a rigorous mapping of our system onto an effective Kondo model is presented. A controlled energy-dependent renormalization-group approach is applied to compute the nonequilibrium current in the presence of a finite bias voltage VVV. In the linear-response regime $V\ensuremath{\rightarrow}0$, the system exhibits as a function of the dissipative strength a localized-delocalized quantum transition of the Kosterlitz-Thouless (KT) type. We address fundamental issues of the nonequilibrium transport near the quantum phase transition: Does the bias voltage play the same role as temperature to smear out the transition? What is the scaling of the nonequilibrium conductance near the transition? At finite temperatures, we show that the conductance follows the equilibrium scaling for V<TV<TV<T, while it obeys a distinct nonequilibrium profile for V>TV>TV>T. We furthermore provide different signatures of the transition in the finite-frequency current noise and ac conductance via a recently developed functional renormalization group (FRG) approach. The generalization of our analysis to nonequilibrium transport through a resonant level coupled to two chiral Luttinger liquid leads, generated by fractional quantum Hall edge states, is discussed. Our work on the dissipative resonant level has direct relevance to experiments on a quantum dot coupled to a resistive environment, such as H. Mebrahtu et al., [Nature (London) 488, 61 (2012)].
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.
We study the competition between Kondo physics and dissipation within an Anderson model of a magnetic impurity level that hybridizes with a metallic host and is also coupled, via the impurity charge, to the displacement of a bosonic bath having a spectral density proportional to ${\ensuremath{\omega}}^{s}$. As the impurity-bath coupling increases from zero, the effective Coulomb interaction between two electrons in the impurity level is progressively renormalized from its repulsive bare value until it eventually becomes attractive. For weak hybridization, this renormalization in turn produces a crossover from a conventional spin-sector Kondo effect to a charge Kondo effect. At particle-hole symmetry, and for sub-Ohmic bath exponents 0<s<10<s<10<s<1, further increase in the impurity-bath coupling results in a continuous zero-temperature transition to a broken-symmetry phase in which the ground-state impurity occupancy ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}$ acquires an expectation value ${⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}⟩}_{0}\ensuremath{\ne}1$. The response of the impurity occupancy to a locally applied electric potential features the hyperscaling of critical exponents and $\ensuremath{\omega}∕T$ scaling that are expected at an interacting critical point. The numerical values of the critical exponents suggest that the transition lies in the same universality class as that of the sub-Ohmic spin-boson model. For the Ohmic case s=1s=1s=1, the transition is instead of Kosterlitz-Thouless type. Away from particle-hole symmetry, the quantum phase transition is replaced by a smooth crossover but signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies.
We study a spinless level that hybridizes with a fermionic band and is also coupled via its charge to a dissipative bosonic bath. We consider the general case of a power-law hybridization function $\ensuremath{\Gamma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{r}$, with $r\ensuremath{\geqslant}0$, and a bosonic-bath spectral function $B(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$, with $s\ensuremath{\geqslant}\ensuremath{-}1$. For r<1r<1r<1 and $\mathrm{max}(0,2r\ensuremath{-}1)<s<1$, this Bose-Fermi quantum impurity model features a continuous zero-temperature transition between a delocalized phase, with tunneling between the impurity level and the band, and a localized phase, in which dissipation suppresses tunneling in the low-energy limit. The phase diagram and the critical behavior of the model are elucidated using perturbative and numerical renormalization-group techniques, between which there is excellent agreement in the appropriate regimes. For r=0r=0r=0, this model's critical properties coincide with those of the spin-boson and Ising Bose-Fermi Kondo models, as expected from bosonization.
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