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Tomonaga-Luttinger liquid physics in electronic quantum circuits

S. Jézouin, Mathias Albert, François D. Parmentier, A. Anthore, U. Gennser, A. Cavanna, Inès Safi A, F. Pierre

Published Jan 17, 2013• 0 citations
OpenAlex

Abstract

In one-dimensional conductors, Coulomb interactions result in correlated electronic systems called Tomonaga-Luttinger liquids (TLL). The TLL physics also applies to other many-body phenomena, providing complementary viewpoints while benefiting from the powerful TLL framework. One such phenomenon is the low-energy conductance suppression of a quantum coherent conductor embedded in a dissipative circuit, an effect called dynamical Coulomb blockade. Here we investigate the basic class of mesoscopic circuits constituted by a short single-channel quantum conductor in series with a resistance R. Remarkably, such circuits can be mapped on a TLL of interaction parameter 1/(1+Re^2/h), with an impurity. From this mapping, generalized to realistic dissipative circuits, a scaling law for the suppressed conductance is derived at R=h/e^2, and small deviations computed for R \neq h/e^2. We find that the scaling law is obeyed by our data for arbitrary quantum channels, emulated by a Ga(Al)As quantum point contact, and by the recent data of Mebrahtu et al. [Nature 488, 61 (2012)] obtained using a carbon nanotube. This demonstrates a highly tunable test-bed for TLL physics, and consolidates a recently proposed phenomenological expression for the conductance of a quantum channel in a linear circuit.

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The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures.A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity.However, this transition impedes experimental explorations of real-world TLLs.Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths.Quantum simulation can provide a powerful workaround.Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity.Measurements reveal the renormalization group "beta function" for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling.Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored.With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.

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In a one-dimensional (1D) system of interacting electrons, excitations of spin and charge travel at different speeds, according to the theory of a Tomonaga-Luttinger liquid (TLL) at low energies. However, the clear observation of this spin-charge separation is an ongoing challenge experimentally. We have fabricated an electrostatically gated 1D system in which we observe spin-charge separation and also the predicted power-law suppression of tunneling into the 1D system. The spin-charge separation persists even beyond the low-energy regime where the TLL approximation should hold. TLL effects should therefore also be important in similar, but shorter, electrostatically gated wires, where interaction effects are being studied extensively worldwide.

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In one-dimensional conductors, Coulomb interactions result in correlated electronic systems called Tomonaga-Luttinger liquids (TLL). The TLL physics also applies to other many-body phenomena, providing complementary viewpoints while benefiting from the powerful TLL framework. One such phenomenon is the low-energy conductance suppression of a quantum coherent conductor embedded in a dissipative circuit, an effect called dynamical Coulomb blockade. Here we investigate the basic class of mesoscopic circuits constituted by a short single-channel quantum conductor in series with a resistance R. Remarkably, such circuits can be mapped on a TLL of interaction parameter 1/(1+Re^2/h), with an impurity. From this mapping, generalized to realistic dissipative circuits, a scaling law for the suppressed conductance is derived at R=h/e^2, and small deviations computed for R \neq h/e^2. We find that the scaling law is obeyed by our data for arbitrary quantum channels, emulated by a Ga(Al)As quantum point contact, and by the recent data of Mebrahtu et al. [Nature 488, 61 (2012)] obtained using a carbon nanotube. This demonstrates a highly tunable test-bed for TLL physics, and consolidates a recently proposed phenomenological expression for the conductance of a quantum channel in a linear circuit.
The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures.A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity.However, this transition impedes experimental explorations of real-world TLLs.Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths.Quantum simulation can provide a powerful workaround.Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity.Measurements reveal the renormalization group "beta function" for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling.Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored.With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.
The Tomonaga-Luttinger liquid (TLL) concept is believed to generically describe the strongly correlated physics of one-dimensional systems at low temperatures.A hallmark signature in 1D conductors is the quantum phase transition between metallic and insulating states induced by a single impurity.However, this transition impedes experimental explorations of real-world TLLs.Furthermore, its theoretical treatment, explaining the universal energy rescaling of the conductance at low temperatures, has so far been achieved exactly only for specific interaction strengths.Quantum simulation can provide a powerful workaround.Here, a hybrid metal-semiconductor dissipative quantum circuit is shown to implement the analogue of a TLL of adjustable electronic interactions comprising a single, fully tunable scattering impurity.Measurements reveal the renormalization group "beta function" for the conductance that completely determines the TLL universal crossover to an insulating state upon cooling.Moreover, the characteristic scaling energy locating at a given temperature the position within this conductance renormalization flow is established over nine decades versus circuit parameters, and the out-of-equilibrium regime is explored.With the quantum-simulator quality demonstrated from the precise parameter-free validation of existing and novel TLL predictions, quantum simulation is achieved in a strong sense, by elucidating interaction regimes which resist theoretical solutions.
The eect of electronic many-body interactions in one-dimensional (1D) systems is drastically enhanced compared to higher dimensions due to the restricted phasespace, often leading to strong electronic correlations. It can be shown that due to a breakdown of the Fermi liquid theory, the ground state at low temperatures becomes a Tomonaga-Luttinger liquid (TLL). In a TLL, the strength of electronic correlations is characterized by the Luttinger parameter K, ranging between K = 0 and K = 1, where K = 1 describes non-interacting electrons. Various systems, such as single-walled carbon nanotubes, 1D atomic metal chains, and MoS 2 mirror twin boundaries, have been reported to host TLLs, in which spin-degeneracy is intact.
We discuss the conductance of quantum wires in terms of the Tomonaga-Luttinger liquid (TLL) theory. We use explicitly the charge fractionalization scheme which results from the chiral symmetry of the model. We suggest that results of the standard two-terminal (2T) conductance measurement depend on the coupling of TLL with the reservoirs and can be interpreted as different boundary conditions at the interfaces. We propose a three-terminal (3T) geometry in which the third contact is connected weakly to the bulk of TLL subjected to a large bias current. We develop a renormalization-group (RG) analysis for this problem by taking explicitly into account the splitting of the injected electronic charge into two chiral irrational charges. We study in the presence of bulk contact the leading-order corrections to the conductance for two different boundary conditions, which reproduce in the absence of bulk contact, respectively, the standard 2T source-drain (SD) conductance GSD(2)=e2/h{G}_{\mathrm{SD}}^{(2)}{=e}^{2}/hGSD(2)​=e2/h and GSD(2)=ge2/h,{G}_{\mathrm{SD}}^{(2)}{=ge}^{2}/h,GSD(2)​=ge2/h, where g is the TLL charge interaction parameter. We find that under these two boundary conditions for the end contacts the 3T SD conductance GSD(3){G}_{\mathrm{SD}}^{(3)}GSD(3)​ shows an UV-relevant deviation from the above two values, suggesting new fixed points in the Ohmic limit. Nontrivial scaling exponents are predicted as a result of electron fractionalization.
It is well known that the pristine bulk of an interacting one-dimensional system in Tomonaga-Luttinger liquid (TLL) phase shows power law suppression of quasiparticle tunneling amplitude for all values of TLL parameter ggg, in the zero-energy limit. We perform a density matrix renormalization group (DMRG) study of a fully symmetric Y junction of TLL wires and observe an anomalous enhancement of the tunneling density of states (TDOS) in the vicinity of the junction for both (a) interacting bosons case and (b) interacting fermions case, when g>1g>1g>1. We also observe suppression of TDOS for g<1g<1g<1 for both bosonic and fermionic cases. We find that the TDOS enhancements follow different power laws for bosonic and fermionic cases which suggests that these represent distinct fixed points, owing to statistical correlations that play an important role at the Y junction. Analysis of static conductance for the junction indicates that the fixed point for 1<g<31<g<31<g<3 resembles the mysterious MMM fixed point of Y junction predicted by Oshikawa, Chamon, and Affleck [J. Stat. Mech. (2006) P02008]. We also show that the TDOS enhancement spans over a length scale of $\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1}$ from the junction, for 1<g<31<g<31<g<3.
In a one-dimensional (1D) system of interacting electrons, excitations of spin and charge travel at different speeds, according to the theory of a Tomonaga-Luttinger liquid (TLL) at low energies. However, the clear observation of this spin-charge separation is an ongoing challenge experimentally. We have fabricated an electrostatically gated 1D system in which we observe spin-charge separation and also the predicted power-law suppression of tunneling into the 1D system. The spin-charge separation persists even beyond the low-energy regime where the TLL approximation should hold. TLL effects should therefore also be important in similar, but shorter, electrostatically gated wires, where interaction effects are being studied extensively worldwide.