We study mesoscopic fluctuations in a system in which there is a continuous connection between two distinct Fermi liquids, asking whether the mesoscopic variation in the two limits is correlated.The particular system studied is an Anderson impurity coupled to a finite mesoscopic reservoir described by the random matrix theory, a structure which can be realized using quantum dots.We use the slave boson mean-field approach to connect the levels of the uncoupled system to those of the strong-coupling Nozières' Fermi liquid.We find strong but not complete correlation between the mesoscopic properties in the two limits and several universal features.
We study mesoscopic fluctuations in a system in which there is a continuous connection between two distinct Fermi liquids, asking whether the mesoscopic variation in the two limits is correlated.The particular system studied is an Anderson impurity coupled to a finite mesoscopic reservoir described by the random matrix theory, a structure which can be realized using quantum dots.We use the slave boson mean-field approach to connect the levels of the uncoupled system to those of the strong-coupling Nozières' Fermi liquid.We find strong but not complete correlation between the mesoscopic properties in the two limits and several universal features.
We study the Anderson impurity problem in a mesoscopic setting, namely the ``Anderson box,'' in which the impurity is coupled to finite reservoir having a discrete spectrum and large sample-to-sample mesoscopic fluctuations. Note that both the weakly coupled and strong coupling Anderson impurity problems are characterized by a Fermi-liquid theory with weakly interacting quasiparticles. We study how the statistical fluctuations in these two problems are connected, using random matrix theory and the slave boson mean-field approximation (SBMFA). First, for a resonant level model such as results from the SBMFA, we find the joint distribution of energy levels with and without the resonant level present. Second, if only energy levels within the Kondo resonance are considered, the distributions of perturbed levels collapse to universal forms for both orthogonal and unitary ensembles for all values of the coupling. These universal curves are described well by a simple Wigner-surmise-type toy model. Third, we study the fluctuations of the mean-field parameters in the SBMFA, finding that they are small. Finally, the change in the intensity of an eigenfunction at an arbitrary point is studied, such as is relevant in conductance measurements. We find that the introduction of the strongly coupled impurity considerably changes the wave function but that a substantial correlation remains.
This thesis contributes to the field of transport through quantum dots. These devices allow for a controlled study of quantum transport and fundamental physical effects, like the Kondo effect. In this thesis we will focus on dots that are well described by generalized Anderson impurity models, where the discrete levels of the quantum dot are tunnel-coupled to fermionic reservoirs. The model parameters, like level energy and width, can be tuned in experiments. Therefore these systems constitute a valuable arena for testing experiment against theory and vice versa. In order to describe these strongly correlated systems, we employ the numerical renormalization group method. This allows us to address both longstanding questions concerning experimental results and new physical phenomena in these fundamental models. This thesis consists of three major projects. The first and most extensive one is concerned with the phase of the transmission amplitude through a quantum dot. Measurements of many-electron quantum dots with small level spacing reveal universal phase behaviour, a result not fully understood for almost 10 years. Recent experiments have seen that, contrarily, for dots with only a few electrons, i.e. large level spacing, the phase depends on the mesoscopic dot parameters. Analyzing a multi-level Anderson model, we show that the generic feature of the two regimes can be reproduced in the regime of overlapping levels or well separated levels, respectively. Thereby the universal character follows from Fano-type antiresonances of the renormalized single-particle levels. Moderate temperature supports the universal character. In the mesoscopic regime, we also investigate the effect of Kondo correlations on the transmission phase. In a second project we analyze a quantum dot coupled to a superconducting reservoir. In contrast to previous belief, the energy resolution of our method is not restricted by the energy scale of the superconducting gap, leading to new insights into the method. The high resolution allows us to resolve sharp peaks in the spectral function that emerge for a certain regime of parameters. A third project deals with a quantum dot coupled to two independent channels, a system known to exhibit non-Fermi liquid behaviour. We investigate the existence of the non-Fermi liquid regime when driving the system out of the Kondo regime by emptying the dot. We find that the extent of the non-Fermi liquid regime strongly depends on the mechanisms that couple impurity and reservoirs but prevent mixing of the latter.
\n We study the problem of two single-level quantum dots side-coupled to a single site of a one-channel quantum wire, modelling the dots as repulsive Anderson impurities. We study the model within a perturbative approach restricting ourselves to the particle-hole symmetric case and show that it has marginal Fermi liquid properties. Further, we show that the perturbative approach also captures the “singular Fermi liquid physics” characteristic of the underscreened Kondo model expected in the strong-coupling limit. We also predict an underscreened charge Kondo effect for the corresponding model with attractive Anderson interaction.\n
We construct and analyze a lattice generalization of the Yukawa-Sachdev-Ye-Kitaev model, where spinful fermions experience onsite, random, all-to-all interactions with an Einstein bosonic mode, and random intersite coherent hopping. We obtain the exact self-consistent numerical solution of the model at mean-field level, and analytical approximations, for all values of fermion-boson coupling and hopping, under the spin-singlet ansatz and at particle-hole symmetry, both in the normal and superconducting states, thus tracing the entire phase diagram. In the normal state, the competition between hopping and coupling leads to crossovers between Fermi-liquid and non-Fermi-liquid states, as reflected by the fermionic and bosonic spectral functions and the normal-state entropy. We calculate the finite phase stiffness of the superconducting state through the equilibrium electromagnetic response. Furthermore, we study the critical temperature Tc, as well as the spectral functions, the quasiparticle weight, the gap, and the condensation energy in the superconducting state. At weak coupling, we retrieve a disordered generalization of Bardeen-Cooper-Schrieffer theory. At strong coupling, asymptotically Tc saturates but the stiffness decreases, which suggests strong superconducting fluctuations. Tc is maximum in the single-dot limit, while the stiffness peaks exactly at the crossover between non-Fermi-liquid and Fermi-liquid phases. We discover that the quasiparticle weight, the stiffness, and the condensation energy, are all correlated as a function of coupling, reminiscent of the correlations observed in high-temperature cuprate superconductors.
We construct and analyze a lattice generalization of the Yukawa-Sachdev-Ye-Kitaev model, where spinful fermions experience onsite, random, all-to-all interactions with an Einstein bosonic mode, and random intersite coherent hopping. We obtain the exact self-consistent numerical solution of the model at mean-field level, and analytical approximations, for all values of fermion-boson coupling and hopping, under the spin-singlet ansatz and at particle-hole symmetry, both in the normal and superconducting states, thus tracing the entire phase diagram. In the normal state, the competition between hopping and coupling leads to crossovers between Fermi-liquid and non-Fermi-liquid states, as reflected by the fermionic and bosonic spectral functions and the normal-state entropy. We calculate the finite phase stiffness of the superconducting state through the equilibrium electromagnetic response. Furthermore, we study the critical temperature Tc, as well as the spectral functions, the quasiparticle weight, the gap, and the condensation energy in the superconducting state. At weak coupling, we retrieve a disordered generalization of Bardeen-Cooper-Schrieffer theory. At strong coupling, asymptotically Tc saturates but the stiffness decreases, which suggests strong superconducting fluctuations. Tc is maximum in the single-dot limit, while the stiffness peaks exactly at the crossover between non-Fermi-liquid and Fermi-liquid phases. We discover that the quasiparticle weight, the stiffness, and the condensation energy, are all correlated as a function of coupling, reminiscent of the correlations observed in high-temperature cuprate superconductors.
We argue through a combination of slave-boson mean-field theory and the Bethe ansatz that the ground state of closely spaced double quantum dots in parallel coupled to a single effective channel are Fermi liquids. We do so by studying the dots' conductance, impurity entropy, and spin correlation. In particular, we find that the zero-temperature conductance is characterized by the Friedel sum rule, a hallmark of Fermi-liquid physics, and that the impurity entropy vanishes in the limit of zero temperature, indicating that the ground state is a singlet. This conclusion is in opposition to a number of numerical renormalization-group studies. We suggest a possible reason for the discrepancy.