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arXiv:1805.00996 · Physical Review Letters

Transconducting Transition for a Dynamic Boundary Coupled to Several Luttinger Liquids

B. Horovitz, Thierry Giamarchi, Pierre Le Doussal

Published Oct 17, 2018• 8 citations
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Abstract

We study a dynamic boundary, e.g., a mobile impurity, coupled to N independent Tomonaga-Luttinger liquids (TLLs) each with interaction parameter K. We demonstrate that for N≥2 there is a quantum phase transition at K≥1/2, where the TLL phases lock together at the particle position, resulting in a nonzero transconductance equal to e^{2}/Nh. The transition line terminates for strong coupling at K=1-(1/N), consistent with results at large N. Another type of a dynamic boundary is a superconducting (or a Bose-Einstein condensate) grain coupled to N≥2 TLLs; here the transition signals also the onset of a relevant Josephson coupling.

DOI
10.1103/physrevlett.121.166803

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We study a dynamic boundary, e.g., a mobile impurity, coupled to N independent Tomonaga-Luttinger liquids (TLLs) each with interaction parameter K. We demonstrate that for N≥2 there is a quantum phase transition at K≥1/2, where the TLL phases lock together at the particle position, resulting in a nonzero transconductance equal to e^{2}/Nh. The transition line terminates for strong coupling at K=1-(1/N), consistent with results at large N. Another type of a dynamic boundary is a superconducting (or a Bose-Einstein condensate) grain coupled to N≥2 TLLs; here the transition signals also the onset of a relevant Josephson coupling.
Quantum impurity problems in Tomonaga-Luttinger liquids (TLLs) are reviewed with emphasis on their analogy to the Kondo problem in Fermi liquids. First, the problem of a static impurity in a spinless TLL is considered, which is related to the model studied in the context of the macroscopic quantum coherence. In the low-energy limit the TLL is essentially cut into two pieces when interaction is repulsive. The orthogonality catastrophe in a TLL is then discussed. Finally, the Kondo effect of a spin-1/2 impurity in a one-dimensional repulsively interacting electron liquids (a spinful TLL) is reviewed. Regardless of the sign of the exchange coupling, the impury spin is completely screened in the ground state. The leading low-temperature contributions to thermodynamic quantities come from boundary contributions of a bulk leading irrelevant operator.
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.
We study various realizations of collective coordinates, e.g., the position of a particle, the charge of a Coulomb box, or the phase of a Bose or a superconducting condensate, coupled to Luttinger liquids with N flavors. We find that for a Luttinger parameter (1/2)<K<1 there is a phase transition from a delocalized phase into a phase with a periodic potential at strong coupling. In the delocalized phase the dynamics is dominated by an effective mass, i.e., diffusive in imaginary time, while on the transition line it becomes dissipative. At K=(1/2) there is an additional transition into a localized phase with no diffusion at zero temperature.
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 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.
The dissipation induced by a metallic gate on the low-energy properties of interacting 1D electron liquids is studied. As a function of the distance to the gate, or the electron density in the wire, the system can undergo a quantum phase transition from a Tomonaga-Luttinger liquid to two kinds of dissipative phases, one of them with a finite spatial correlation length. We also define a dual model, which describes an attractive one-dimensional metal with a Josephson coupling to a dirty metallic lead.