Abstract:
The magnetic flux dependence of order parameter and supercurrent is studied in a hollow d-wave superconducting cylinder. It is shown that the existence of line nodal quasiparticles in a pure $d_{x^2-y^2}$ pairing state gives rise to an $hc/e$ periodicity in the order parameter and a first-order quantum phase transition for a large system size. We demonstrate that the flux periodicity in the supercurrent is sensitive to the detailed electronic band structure and electron filling factor. In particular, we find that, in cooperation with the increase of the cylinder circumference, the $hc/2e$ periodicity can be restored significantly in the supercurrent by avoiding the particle-hole symmetry point. A similar study of a $d_{x^2-y^2}+id_{xy}$ pairing state verifies the peculiarity of unconventional superconductors with nodal structure.

Abstract:
A new theoretical model is presented to study the nanoscale electronic inhomogeneity in high-$T_c$ cuprates. In this model, we argue that the randomly distributed out-of-plane interstitial oxygen dopants induces locally the off-diagonal (i.e., hopping integral) disorder. This disorder modulates the superexchange interaction resulting from a large-$U$ Hubbard model, which in turns changes the local pairing interaction. The microscopic self-consistent calculations shows that the large gap regions are registered to the locations of dopants. Large gap regions exhibit small and broader coherence peaks. These results are qualitatively consistent with recent STM observations on optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.

Abstract:
Both the differential tunneling conductance and the surface local density of states (LDOS) of a [110]-oriented high-temperature superconductor in the pseudogap (PG) regime are studied theoretically. As a competing candidate for the mechanism of PG state, the charge-density wave (CDW), spin-density wave (SDW), $d$-density wave (DDW), and d-wave superconducting (DSC) orderings show distinct features in the tunneling conductance. For the CDW, SDW, and DSC orderings, the tunneling conductance approaches the surface LDOS as the barrier potential is increased. For the DDW ordering, we show for the first time that there exist midgap states at the [110] surface, manifesting themselves as a sharp zero-energy peak in the LDOS, as in the case of DSC ordering. However, due to the particle-hole pair nature of the DDW state, these states do not carry current, and consequently the one-to-one correspondence between the tunneling conductance and the surface LDOS is absent.

Abstract:
For two coupled Anderson impurities, two energy scales are present to characterize the evolution from local moment state of the impurities to either of the inter-impurity singlet or the Kondo singlet ground states. The high energy scale is found to deviate from the single-ion Kondo temperature and rather scales as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction when it becomes dominant. We find that the scaling behavior and the associated physical properties of this scale are consistent with those of a coherence scale defined in heavy fermion systems.

Abstract:
We propose that a composite fermion operator $f_{i\sigma}(2n_{i{\bar \sigma}}-1)$ could have coherent excitations, where $f_{i\sigma}$ is the fermion operator for interacting electrons and $n_{i{\bar \sigma}}$ is the number operator of the opposite spin. In the two-impurity Anderson model, it is found that the excitation of this composite fermion has a pseudogap in the Kondo regime, and has a finite spectral weight in the regime where the excitation of the regular fermion $f_{i\sigma}$ has a pseudogap. In the latter regime, the self-energy of $f_{i\sigma}$ is found to be singular near Fermi energy. We argue that this composite fermion could develop a Fermi surface with Fermi liquid behaviors but "hidden" from charge excitations in lattice generalizations. We further illustrate that this type of excitations is essential in addressing the pseudogap state and unconventional superconductivity.

Abstract:
We study the intersite correlation effects on the Kondo dynamics of two Anderson impurities. Two characteristic energy scales are present associated with the splitting of the uniform and staggered spin susceptibilities. The high energy scale has similar properties as the coherence scale in heavy fermion materials while the low energy scale characterizes the competing orders between Kondo singlet state and inter-impurity singlet state. A singlet superconductivity instability is found near the quantum critical point and is associated with the singular staggered spin susceptibility.

Abstract:
In the two-impurity Anderson model, the inter-impurity spin exchange interaction favors a spin singlet state between two impurities leading to the breakdown of the Kondo effect. We show that a local uniform magnetic field can delocalize the quasiparticles to restore the Kondo resonance. This transition is found to be continuous, accompanied by not only the divergence of the staggered (antiferromagnetic) susceptibility, but also the divergence of the uniform spin susceptibility. This may imply that the magnetic field induced quantum phase transitions in Kondo systems are in favor of the local critical type.

Abstract:
We study the non-local superconducting pairing of two interacting Anderson impurities, which has an instability near the quantum critical point from the competition between the Kondo effect and an antiferromagnetic inter-impurity spin exchange interaction. As revealed by the dynamics over the whole energy range, the superconducting pairing fluctuations acquire considerable strength from an energy scale much higher than the characteristic spin fluctuation scale while the low energy behaviors follow those of the staggered spin susceptibility. We argue that the glue to the superconducting pairing is not the spin fluctuations, but rather the effective Coulomb interaction. On the other hand, critical spin fluctuations in the vicinity of quantum criticality are also crucial to a superconducting pairing instability, by preventing a Fermi liquid fixed point being reached to keep the superconducting pairing fluctuations finite at low energies. A superconducting order, to reduce the accumulated entropy carried by the critical degrees of freedom, may arise favorably from this instability.

Abstract:
We study a scheme for electrical detection, using electron spin resonance, of coherent vibrations in a molecular single electron level trapped near a conduction channel. Both equilibrium spin-currents and non-equilibrium spin- and charge currents are investigated. Inelastic side-band anti-resonances corresponding to the vibrational modes appear in the electron spin resonance spectrum.

Abstract:
We study the phase diagram of a superconducting ring threaded by an Aharonov-Bohm flux and an in-plane magnetic Zeeman field. The simultaneous presence of both the external flux and the in-plane magnetic field leads to the competition between the Fulde-Ferrell (FF) phase and the Larkin-Ovchinnikov (LO) phase. Using the Bogoliubov-de Gennes equation, we investigate the spacial profile of the order parameter. Both the FF phase and the LO phase are found to exist stably in this system. The phase boundary is determined by comparing the free energy. The distortion of the phase diagrams due to the mesoscopic effect is also studied.