We consider a one-channel coherent conductor with a good transmission embedded into an Ohmic environment, the impedance of which is equal to the quantum of resistance
Rq=h/e2 below the
RC frequency. This choice is motivated by the mapping of this problem to a Tomonaga-Luttinger liquid with one impurity, the interaction parameter of which corresponds to the specific value
K=1/2, allowing for a refermionization procedure. The ``new'' fermions have an energy-dependent transmission amplitude, which incorporates the strong correlation effects and yields the exact dc current and zero-frequency noise through expressions similar to those of the scattering approach. We recall and discuss these results for our present purpose. Then, we compute the finite-frequency differential conductance and the finite-frequency nonsymmetrized noise. Contrary to intuitive expectation, both can not be expressed within the scattering approach for the new fermions, even though they are still determined by the transmission amplitude. Even more, the finite-frequency conductance obeys an exact relation in terms of the dc current, which is similar to that derived perturbatively with respect to weak tunneling within the Tien-Gordon theory, and extended recently to arbitrary strongly interacting systems coupled eventually to an environment and/or with a fractional charge. We also show that the emission excess noise vanishes exactly above
eV, even though the underlying Tomonaga-Luttinger liquid model corresponds to a many-body correlated system. Our results apply for all ranges of temperature, voltages, and frequencies below the
RC frequency, and they allow us to explore fully the quantum regime.
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.
We study the out-of-equilibrium transport in a Tomonaga-Luttinger liquid containing a weak or a tunneling barrier coupled to an arbitrary electromagnetic environment. This applies as well to a coherent one-channel noninteracting conductor with a transmission coefficient close to one or to zero. We derive formal expressions for the current and finite-frequency (FF) noise at arbitrary voltages, temperatures, and frequency-dependent impedance $Z(\ensuremath{\omega})$ in the regimes of weak and strong backscattering. We show that these two regimes are no longer related by duality at finite frequency. We then carry explicit computations of the nonlinear conductance and FF noise when $Z(\ensuremath{\omega})$ describes a harmonic oscillator such as an
LC circuit or a cavity.
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.
Nonequilibrium bosonization technique is used to study current fluctuations of interacting electrons in a single-channel quantum wire representing a Luttinger liquid (LL) conductor. An exact expression for the time resolved full counting statistics of the transmitted charge is derived. It is given by the Fredholm determinant of the counting operator with a time-dependent scattering phase. The result has a form of counting statistics of noninteracting particles with fractional charges, induced by scattering off the boundaries between the LL wire and the noninteracting leads.
Interacting Tomonaga-Lüttinger liquid with impurity for interaction constant
K=1/2: Thermopower investigation, entropy variation and heat capacity densities associated with thermoelectric particle transport
We investigate thermoelectric and thermodynamic properties of a Tomonaga-Luttinger liquid with interaction parameter
K=1/2 in the presence of a localized impurity. Using bosonization and refermionization, we derive an exact expression for the thermopower in the nonlinear regime and obtain the corresponding Seebeck coefficient in linear response. At low temperature, the Seebeck coefficient follows a Mott-like relation governed by the energy dependence of the transmission coefficient and is found to be closely related to the entropy per particle. This connection allows us to derive using Kelvin formula the thermopower contribution to the entropy density variation and the charge-carriers contribution to the heat capacity density. Both quantities exhibit Mott-like behavior in the low-temperature regime. We apply our results to a fractional quantum Hall quantum point contact and to a one-channel quantum conductor coupled to an Ohmic environment. Our results demonstrate the close connection between thermoelectric transport and thermodynamic properties in interacting one-dimensional quantum systems.