We propose that the finite size of the Kondo screening cloud, ${\ensuremath{\xi}}_{K}$, can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential $\ensuremath{\mu}$ in the system is varied at zero temperature, one should observe charge steps whose locations are at values of $\ensuremath{\mu}$ that are controlled by the Kondo effect when the system size $L$ is comparable to ${\ensuremath{\xi}}_{K}$. We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ${\ensuremath{\xi}}_{K}∕L$. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.We propose that the finite size of the Kondo screening cloud, ${\ensuremath{\xi}}_{K}$, can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential $\ensuremath{\mu}$ in the system is varied at zero temperature, one should observe charge steps whose locations are at values of $\ensuremath{\mu}$ that are controlled by the Kondo effect when the system size $L$ is comparable to ${\ensuremath{\xi}}_{K}$. We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ${\ensuremath{\xi}}_{K}∕L$. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.

arXiv:0801.4166Physical Review BMar 19, 200831 citations

Conductance of junctions of multiple interacting quantum wires and long Aharonov-Bohm-Kondo rings

Zheng Shi

In this thesis, we calculate the linear dc conductance of two types of multi-terminal interacting systems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads, and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantum dot in the Kondo regime respectively. In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, the conductance is formally given by the Landauer formula with renormalized single-particle S-matrix elements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish even in the leads, the conductance has an additional contribution dependent on the interaction strength in the leads. We calculate this additional contribution both at the first order in interaction and in the random phase approximation, and heuristically relate the FL conductance to the TLL conductance through a "contact resistance" between an FL lead and a TLL wire. In long ABK rings, where the interaction is due to spin-flip scattering at a quantum dot in the Kondo regime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form, and a connected part that can be approximately eliminated at low temperatures. For a closed long ABK ring, where the electric current is conserved in the ring, the high-temperature conductance has qualitatively different behaviors for temperatures greater than and lower than the characteristic energy scale v_F /L, where v_F is the Fermi velocity and L is the ring circumference. Meanwhile, for an open long ABK ring where electrons may leak into the side leads coupled to ring arms, as long as the ring arms have both small transmission and small reflection, the ring behaves as a two-path Aharonov-Bohm interferometer, and we predict the observation of a π/2 phase shift due to scattering off the Kondo singlet formed at low energies around the impurity spin.In this thesis, we calculate the linear dc conductance of two types of multi-terminal interacting systems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads, and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantum dot in the Kondo regime respectively. In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, the conductance is formally given by the Landauer formula with renormalized single-particle S-matrix elements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish even in the leads, the conductance has an additional contribution dependent on the interaction strength in the leads. We calculate this additional contribution both at the first order in interaction and in the random phase approximation, and heuristically relate the FL conductance to the TLL conductance through a "contact resistance" between an FL lead and a TLL wire. In long ABK rings, where the interaction is due to spin-flip scattering at a quantum dot in the Kondo regime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form, and a connected part that can be approximately eliminated at low temperatures. For a closed long ABK ring, where the electric current is conserved in the ring, the high-temperature conductance has qualitatively different behaviors for temperatures greater than and lower than the characteristic energy scale v_F /L, where v_F is the Fermi velocity and L is the ring circumference. Meanwhile, for an open long ABK ring where electrons may leak into the side leads coupled to ring arms, as long as the ring arms have both small transmission and small reflection, the ring behaves as a two-path Aharonov-Bohm interferometer, and we predict the observation of a π/2 phase shift due to scattering off the Kondo singlet formed at low energies around the impurity spin.

Jan 1, 20170 citations

Many-particle States in the Theory of the Kondo Effect

Sergei V. Maleev

Journal of Experimental and Theoretical PhysicsJan 1, 19720 citations

Impurity Quantum Monte Carlo Calculations of Persistent Currents

Peter B. Hitchcock

Recent interest in the fundamental physics of the Kondo effect has been driven by the observation of Kondo physics in mesoscopic systems such as quantum dots [1, 2] and carbon nanotubes [3], which can act analogously to magnetic impurities in a bulk metal. Highly tunable mesoscopic systems such as these present the possibility of observing the controversial Kondo length scale εK associated with the cloud of conduction electrons that screen the spin of the impurity [4]. This plays a similar role in scaling theories as the Kondo temperature Tk. One proposal [5] for detecting this length scale is to measure the finite-size dependence of persistent currents in an isolated conducting ring coupled to a quantum dot. The screening cloud should be 'trapped' in the closed system, and will not form if the size of the ring L is much smaller than εK. In particular, the current in the Kondo regime should be a universal scaling function j = L^-1 f(Φ, L/εK, T/Tk) (here Φ is the applied flux) [6, 7]. Considerable disagreement has arisen in the theoretical estimates of these persistent currents as different analytical treatments yield contradictory predictions [6, 8, 9, 10]. This thesis presents a new Quantum Monte Carlo (QMC) technique for measuring persistent currents in such systems, based on the Hirsch-Fye Impurity QMC algorithm [11] which is ideally suited to treating systems with a single impurity such as the quantum dot. The algorithm provides exact numerical results at finite temperatures. The complexity of the algorithm does not scale directly with the size of the system, making it particularly attractive for investigating a wide range of system sizes.Recent interest in the fundamental physics of the Kondo effect has been driven by the observation of Kondo physics in mesoscopic systems such as quantum dots [1, 2] and carbon nanotubes [3], which can act analogously to magnetic impurities in a bulk metal. Highly tunable mesoscopic systems such as these present the possibility of observing the controversial Kondo length scale εK associated with the cloud of conduction electrons that screen the spin of the impurity [4]. This plays a similar role in scaling theories as the Kondo temperature Tk. One proposal [5] for detecting this length scale is to measure the finite-size dependence of persistent currents in an isolated conducting ring coupled to a quantum dot. The screening cloud should be 'trapped' in the closed system, and will not form if the size of the ring L is much smaller than εK. In particular, the current in the Kondo regime should be a universal scaling function j = L^-1 f(Φ, L/εK, T/Tk) (here Φ is the applied flux) [6, 7]. Considerable disagreement has arisen in the theoretical estimates of these persistent currents as different analytical treatments yield contradictory predictions [6, 8, 9, 10]. This thesis presents a new Quantum Monte Carlo (QMC) technique for measuring persistent currents in such systems, based on the Hirsch-Fye Impurity QMC algorithm [11] which is ideally suited to treating systems with a single impurity such as the quantum dot. The algorithm provides exact numerical results at finite temperatures. The complexity of the algorithm does not scale directly with the size of the system, making it particularly attractive for investigating a wide range of system sizes.

Jan 1, 20060 citations

L

is comparable to ${\ensuremath{\xi}}_{K}$. We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ${\ensuremath{\xi}}_{K}∕L$. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.