Abstract:
We address the question of whether an anisotropic gap $d_{x^2-y^2}$ symmetry is compatible with localized states in the normal phase. The issue is important in high $T_c$ superconductors where a superconductor to insulator transition is observed in the underdoped regime, together with a number of experiments that support $d$-wave pairing. We find a reentrant behavior of superconductivity in the strongly disorder phase.

Abstract:
Making use of the extended Ginzburg Landau theory, which includes the fourth order derivative term, we study the vortex state in a magnetic field parallel to the $ c$ axis. The vortex core structure is distorted due to this higher order term, which reveals t he fourfold symmetry. Futher, this distortion gives rise to the core interaction energy, which favo rs a square lattice tilted by $45^\circ$ from the $a$ axis. The critical field of this transition is determined. The magnetization diverges at the transition. This suggests the transition is of the fi rst order.

Abstract:
We study the phase transition between the normal and non-uniform (Fulde-Ferrell-Larkin-Ovchinnikov) superconducting state in quasi two-dimensional d-wave superconductors at finite temperature. We obtain an appropriate Ginzburg-Landau theory for this transition, in which the fluctuation spectrum of the order parameter has a set of minima at non-zero momenta. The momentum shell renormalization group procedure combined with dimensional expansion is then applied to analyze the phase structure of the theory. We find that all fixed points have more than one relevant directions, indicating the transition is of the fluctuation-driven first order type for this universality class.

Abstract:
We discuss the dynamics of magnetic moments in d-wave superconductors, in particular we focus on moments induced by doping non-magnetic impurities into cuprates. The interaction of such moments with the Bogoliubov quasiparticles of the superconductors can be decribed by variants of the pseudogap Kondo model, characterized by a power-law density of states at the Fermi level. The Numerical Renormalization Group technique is employed to investigate this Kondo problem for realistic band structures and particle-hole asymmetries, both at zero and finite temperatures. In particular, we study the boundary quantum phase transition between the local-moment and the asymmetric strong-coupling phases, and argue that this transition has been observed in recent nuclear magnetic resonance experiments. We determine the spectral properties of both phases, the location of the critical point as function of Kondo coupling and doping, and discuss the quantum-critical cross-overs near this phase transition. In addition, the changes in the local density of states around the impurity are calculated as function of temperature, being relevant to scanning tunneling microscopy experiments.

Abstract:
We study the vortex state in a magnetic field parallel to the $c$ axis in the framework of the extended Ginzburg Landau equation. We find the vortex acquires a fourfold modulation proportional to $\cos(4\phi)$ where $\phi$ is the angle ${\bf r}$ makes with the $a$-axis. This term gives rise to an attractive interaction between two vortices when they are aligned parallel to $(1,1,0)$ or $(1,-1,0)$. We predict the first order vortex lattice transition at $B=H_{cr}\sim \kappa^{-1} H_{c2}(t)$ from triangular into the square lattice tilted by $45^\circ$ from the $a$ axis. This gives the critical field $H_{cr}$ a few Tesla for YBCO and Bi2212 monocrystals at low temperatures ($T\leq 10 K$).

Abstract:
Within the framework of the Ginzburg-Landau theory for $d_{x^2-y^2}+id_{xy}$-wave superconductors, we discuss the pairing state phase transition in the absence of the Zeeman coupling between the Cooper pair orbital angular momentum and the magnetic field. We find that above a temperature $T_{\ast}$, the pairing state in a magnetic field is pure $d_{x^{2}-y^{2}}$-wave. However, below $T_{\ast}$, the pairing state is $d_{x^{2}-y^{2}}+id_{xy}$-wave at low fields, and it becomes pure $d_{x^{2}-y^{2}}$-wave at higher fields. Between these pairing states there exists a field driven phase transition . The transition field increases with decreasing temperature. In the field-temperature phase diagram, the phase transition line is obtained theoretically by a combined use of a variational method and the Virial theorem. The analytical result is found to be in good agreement with numerical simulation results of the Gingzburg-Landau equations. The validity of the variational method is discussed. The difference to the case with the Zeeman coupling is discussed, which may be utilized to the detection of the Zeeman coupling.

Abstract:
We study models for superconductivity with two interactions: $V^>$ due to antiferromagnetic(AF) fluctuations and $V^<$ due to phonons, in a weak coupling approach to the high temperature superconductivity. The nature of the two interactions are considerably different; $V^>$ is positive and sharply peaked at ($\pm\pi$,$ \pm\pi$) while $V^<$ is negative and peaked at ($0,0$) due to weak phonon screening. We numerically find (a) weak BCS attraction is enough to have high critical temperature if a van Hove anomaly is at work, (b) $V^>$ (AF) is important to give d-wave superconductivity, (c) the gap order parameter $\Delta({\bf k})$ is constant(s-wave) at extremely overdope region and it changes to anisotropic s-wave as doping is reduced, (d) there exists a first order phase transition between d-wave and anisotropic s-wave gaps. These results are qualitatively in agreement with preceding works; they should be modified in the strongly underdope region by the presence of antiferromagnetic fluctuations and ensuing AF pseudogap.

Abstract:
A U(3) model proposed by Iachello for superconductivity in cuprate materials is analyzed. The model consists of s and d pairs (approximated as bosons) in a two-dimensional Fermi system with a surface. The transition occurs between a phase in which the system is a condensate of one of the bosons, and a phase which is a mixture of two types of bosons. In the current work we have investigated the validity of the Bogoliubov approximation, and we used a reduced Hamiltonian to determine a phase diagram, the symmetry of the phases and the temperature dependence of the heat capacity.

Abstract:
We present a simple theoretical explanation for a transition from d-wave to another superconducting pairing observed in the electron-doped cuprates. The d_{x^2-y^2} pairing potential Delta, which has the maximal magnitude and opposite signs at the hot spots on the Fermi surface, becomes suppressed with the increase of electron doping, because the hot spots approach the Brillouin zone diagonals, where Delta vanishes. Then, the d_{x^2-y^2} pairing is replaced by either singlet s-wave or triplet p-wave pairing. We argue in favor of the latter and discuss experiments to uncover it.

Abstract:
A lattice model for disordered d-wave superconductors in class CI is reconsidered. Near the band-center, the lattice model can be described by Dirac fermions with several species, each of which yields WZW term for an effective action of the Goldstone mode. The WZW terms cancel out each other because of the four-fold symmetry of the model, which suggests that the quasiparticle states are localized. If the lattice model has, however, symmetry breaking terms which generate mass for any species of the Dirac fermions, remaining WZW term which avoids the cancellation can derive the system to a delocalized strong-coupling fixed point.