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arXiv:1705.08767 · Physical review. B./Physical review. B

Quantum dot at a Luttinger liquid edge

Colin Rylands, Natan Andrei

Published Sep 14, 2017• 17 citations
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Abstract

We study a system consisting of a Luttinger liquid coupled to a quantum dot on the boundary. The Luttinger liquid is expressed in terms of fermions interacting via density-density coupling, and the dot is modeled as an interacting resonant level onto which the bulk fermions can tunnel. We solve the Hamiltonian exactly and construct all eigenstates. We study both the zero- and finite-temperature properties of the system, in particular, we compute the exact dot occupation as a function of the dot energy in all parameter regimes. The system is seen to flow from weak to strong coupling for all values of the bulk interaction, with the flow characterized by a nonperturbative Kondo scale. We identify the critical exponents at the weak- and strong-coupling regimes.

DOI
10.1103/physrevb.96.115424

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We study a system consisting of a Luttinger liquid coupled to a quantum dot on the boundary. The Luttinger liquid is expressed in terms of fermions interacting via density-density coupling, and the dot is modeled as an interacting resonant level onto which the bulk fermions can tunnel. We solve the Hamiltonian exactly and construct all eigenstates. We study both the zero- and finite-temperature properties of the system, in particular, we compute the exact dot occupation as a function of the dot energy in all parameter regimes. The system is seen to flow from weak to strong coupling for all values of the bulk interaction, with the flow characterized by a nonperturbative Kondo scale. We identify the critical exponents at the weak- and strong-coupling regimes.
We study a steady-state nonequilibrium transport between two interacting helical edge states of a two-dimensional topological insulator, described by helical Luttinger liquids, through a quantum dot. For a noninteracting dot, the current is obtained analytically by including the self-energy correction to the dot's Green function. For an interacting dot, we use the equation-of-motion method to study the influence of weak on-site Coulomb interaction on the transport. We find the metal-to-insulator quantum phase transition for attractive or\mathit{or}or repulsive interactions in the leads when the magnitude of the interaction strength characterized by a charge sector Luttinger parameter KKK goes beyond a critical value. The critical Luttinger parameter Kcr{K}_{cr}Kcr​ depends on the hopping strengths between the dot and the leads, as well as the energy level of the dot with respect to the Fermi levels of the leads, ranging from the weak-interaction regime for the dot level off-resonance to the strong-interaction regime for the dot in resonance with the equilibrium Fermi level. Near the transition, there are various singular behaviors of current noise, dot density of state, and the decoherence rate (inverse of lifetime) of the dot, which are briefly discussed.
A quantum impurity attached to an interacting quantum wire gives rise to an array of new phenomena. Using the Bethe Ansatz we solve exactly models describing two geometries of a quantum dot coupled to an interacting quantum wire: a quantum dot that is (i) side coupled and (ii) embedded in a Luttinger liquid. We find the eigenstates and determine the spectrum through the Bethe Ansatz equations. Using this we derive exact expressions for the ground-state dot occupation. The thermodynamics are then studied using the thermodynamics Bethe Ansatz equations. It is shown that at low energies the dot becomes fully hybridized and acts as a backscattering impurity or tunnel junction depending on the geometry and furthermore that the two geometries are related by changing the sign of the interactions. Although remaining strongly coupled for all values of the interaction in the wire, there exists competition between the tunneling and backscattering leading to a suppression or enhancement of the dot occupation depending on the sign of the bulk interactions.
A Luttinger liquid coupled to a quantum impurity describes a large number of physical systems. The Hamiltonian consists of left- and right-moving fermions interacting among themselves via a density-density coupling and scattering off a localized transmitting and reflecting impurity. We solve exactly the Hamiltonian by means of an incoming-outgoing scattering Bethe basis which properly incorporates all scattering processes. A related model, the weak-tunneling model, wherein the impurity is replaced by a tunnel junction, is solved by the same method. The consistency of the construction is established through a generalized Yang-Baxter relation. Periodic boundary conditions are imposed and the resulting Bethe ansatz equations are derived by means of the off-diagonal Bethe ansatz approach. We derive the spectrum of the model for all coupling constant regimes and calculate the impurity free energy. We discuss the low energy behavior of the systems for both repulsive and attractive interactions.
A quantum dot coupled to two helical edge states of a two-dimensional topological insulator through electron tunnelings is studied. We show that if the electron interactions on the edge states are repulsive, with Luttinger liquid parameter K<1K<1K<1, the system reaches a stable two-channel Kondo fixed point at low temperatures. This is in contrast to the Luttinger liquid leads case in which K<1/2K<1/2K<1/2 is needed. This two-channel fixed point is described by a boundary sine-Gordon Hamiltonian with a KKK dependent boundary term. The impurity entropy, the impurity specific heat and the conductance are calculated.
We use bosonization methods to calculate the exact finite-temperature single-electron Green's function of a spinful Luttinger liquid confined by open boundaries. The corresponding local spectral density is constructed and analyzed in detail. The interplay between boundary, finite-size, and thermal effects are shown to dramatically influence the low-energy properties of the system. In particular, the well-known zero-temperature critical behavior in the bulk always crosses over to a boundary dominated regime in the vicinity of the Fermi level. Thermal fluctuations cause an enhanced depletion of spectral weight for small energies $\ensuremath{\omega}$, with the spectral density scaling as ${\ensuremath{\omega}}^{2}$ for $\ensuremath{\omega}$ much less than the temperature. Consequences for photoemission experiments are discussed.
We study a quantum phase transition that occurs in a system composed of two impurities (or quantum dots), each coupled to a different interacting (Luttinger liquid) lead. While the impurities are coupled electrostatically, there is no tunneling between them. Using a mapping of this system onto a Kondo model, we show analytically that the system undergoes a Berezinskii-Kosterlitz-Thouless quantum phase transition as a function of the Luttinger liquid parameter in the leads and the dot-lead interaction. The phase with low values of the Luttinger liquid parameter is characterized by an abrupt switch of the population between the impurities as a function of a common applied gate voltage. However, this behavior is hard to verify numerically since one would have to study extremely long systems. Interestingly, though, at the transition the entanglement entropy drops from a finite value of ln(2)\mathrm{ln}(2)ln(2) to zero. The drop becomes sharp for infinite systems. One can employ finite-size scaling to extrapolate the transition point and the behavior in its vicinity from the behavior of the entanglement entropy in moderate size samples. We employ the density matrix renormalization-group numerical procedure to calculate the entanglement entropy of systems with lead lengths of up to 480 sites. Using finite-size scaling, we extract the transition value and show it to be in good agreement with the analytical prediction.