Conductance of a dissipative quantum dot: Nonequilibrium crossover near a non-Fermi-liquid quantum critical point

Gu Zhang, E. Novais, and Harold U. Baranger

Phys. Rev. B 104, 165423 – Published 25 October 2021

Abstract

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.

Received 30 July 2021
Revised 1 October 2021
Accepted 11 October 2021

DOI:https://doi.org/10.1103/PhysRevB.104.165423

Physics Subject Headings (PhySH)

Dissipative dynamics Kondo effect Nonequilibrium statistical mechanics Open quantum systems Quantum criticality Quantum kinetic theory Quantum phase transitions Quantum transport

Nanostructures Quantum dots Strongly correlated systems

Bethe ansatz Bosonization Luttinger liquid model Nonequilibrium Green's function Renormalization group

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Gu Zhang^1,2,*, E. Novais ^3,†, and Harold U. Baranger ^1,‡

¹Department of Physics, Duke University, Durham, North Carolina 27708-0305, USA
²Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
³Centro de Ciíncias Naturais e Humanas, Universidade Federal do ABC, Santo Andrè, SP 09210-580, Brazil

^*zhanggu217@gmail.com
^†eduardo.novais@ufabc.edu.br
^‡baranger@phy.duke.edu

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Vol. 104, Iss. 16 — 15 October 2021

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Images

Figure 1
The setup of the dissipative resonant level (DRL) model: two leads, source (S), and drain (D), couple to a quantum dot (QD) with hopping strengths $t_{S}$ and $t_{D}$ , respectively. The dissipative environment is represented by an ohmic impedance $R$ . Bias $V$ is applied between the two leads, and the quantum dot energy level is tuned by the backgate voltage $V_{gate}$ .
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Figure 2
(a) Schematic RG flow of the system with three fixed points marked in red. Coupling strength near the decoupled-level fixed point (with Hamiltonian given in Appendix pp1) increases with the RG flow, driving the system towards the strong coupling regime (indicated by the blue arrow) described by Eq. (10). Although the system being close to the quantum critical point, in this regime, the system asymmetry or detuning becomes important. At lower energy scales, the system enters the crossover regime (the red arrow), and flows towards the cut-wire fixed point at which the conductance vanishes. All three fixed points can be distinguished by their impurity entropy. (b) The corresponding flow in the phase diagram. The system begins to enter the crossover regime near the quantum critical boundary.
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Figure 3
Comparison between experiment (dots) [98] and theory (lines) [99] for $r = 0.75$ [yielding an exponent of $r / (1 + r) = 0.43$ ]. (a) Nonequilibrium result at zero temperature [more precisely, $T = 0.002 V$ (theory) and $T \approx 0.05 V$ (experiment)]. (b) Equilibrium result (linear response) for finite temperature. [(c) and (d)] Corresponding plots for the “backscattering conductance” $1 - G h / e^{2}$ . (e) All of the data and theoretical results in a single plot. The constant $a = π Γ {(1 + r)}^{1 / r} \approx 2.8$ is the relation between the bias and temperature scales. For the theoretical curves, we obtain $A^{'} / V_{gate} \approx 0.398$ by fitting the equilibrium data in (b), thus leaving no fit parameters for the excellent agreement in the nonequilibrium case. Although equivalent in the two limits, in the crossover regime the temperature and bias curves differ.
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Figure 4
(a) Differential conductance and (b) “backscattering current” for the $r = 1$ case as a function of $T$ or $V$ . The finite $V$ and finite $T$ curves correspond to the limits $V ≫ T$ (where $T = 10^{- 7} V$ ) and $T ≫ V$ (where $V = 10^{- 7} T$ ), respectively. We choose to present the “backscattering current” in (b) because the leading-order conductance is constant when the deviation is small [109].
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Figure 5
Physical picture behind the difference between temperature-induced and bias-induced crossover. The peak in the renormalized LDOS (black line) is detuned from the resonance condition by the renormalized gate voltage ${\tilde{V}}_{gate}$ . The width of the energy window contributing to transport (indicated by the blue and red arrows) is associated with temperature in equilibrium (top row) or bias voltage out of equilibrium (bottom row). [(a) and (c)] The peak in the LDOS is within the energy window. [(b) and (d)] The LDOS peak is outside the window. The fact that the window for applied bias is a sharp rectangle makes the nonequilibrium crossover sharper.
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Figure 6
(a) Conductance and (b) “backscattering current” as functions of the normalized detuning $λ / \sqrt{max (T, V / π)}$ , for the Toulouse point $r = 1$ .
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