We have developed a constrained-occupation local-density-functional technique to calculate individual model Hamiltonian parameters for states of arbitrary localization. The method generates numerical Wannier functions for the relevant states, and by decoupling these functions, permits direct calculation of the individual parameters. We have tested this method for the cuprate ${\mathrm{La}}_2$ ${\mathrm{CuO}}_4$, and obtained values for the Cu(3${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$) and O(2${\mathit{p}}_{\mathrm{\sigma }}$) parameters in general agreement with previous empirical and theoretical estimates. These calculations provide even-handed treatment of both Cu(3d) and the more extended O(2p) states, and yield the important difference ${\mathrm{\varepsilon }}_{\mathit{p}\left(\mathrm{\sigma }\right)}$-${\mathrm{\varepsilon }}_{\mathit{d}(\mathit{x}}^2$-${\mathit{y}}^2$) consistent with expectations based on the superexchange frequency. We also report parameter values for the Cu(3${\mathit{d}}_{3\mathit{z}}^2$-${\mathit{r}}^2$) states, and for compatible O(2p) states, which may be of use in theoretical attempts to confirm the symmetry of the doped-in carriers.

Whatever its microscopic origins, it seems likely that the superconductivity in the copper-oxides can be described by an appropriate Ginzburg-Landau theory. The possible order parameters and free energies are then determined by the symmetry breaking of the phase transition alone. The group-theoretically allowed states are enumerated here for singlet and triplet superconductors in orthorhombic and tetragonal crystals. As well as broken gauge, time reversal and rotational symmetries, broken translational symmetry states are also investigated. Experiments that might distinguish the various possible states are reviewed, concentrating especially on experiments where the outcomes are determined by the symmetry properties of the order parameter alone. These include Josephson effects, transition splitting, spontaneous strain and magnetism, critical and Gaussian fluctuations, collective modes and vortices. Current experimental results on the high-T_c materials are discussed where relevant, although in many cases these must be regarded as provisional.

The low-temperature behavior of the a-b plane penetration depth, ${\lambda }_{\mathit{a}\mathit{b}}$, is a probe of the pairing state in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathrm{\delta }}$. A group-theoretic analysis shows that in orthorhombic or tetragonal crystals all singlet pairing states other than ‘‘s wave’’ would lead to Δ${\lambda }_{\mathit{a}\mathit{b}}$(T)==${\lambda }_{\mathit{a}\mathit{b}}$(T)-${\lambda }_{\mathit{a}\mathit{b}}$(0)∼T. In contrast, for an isotropic system, there are combinations of singlet pairing states and field directions that would give rise to Δλ(T)∼${\mathit{T}}^3$. We reanalyze the surface impedance data of Fiory et al. [Phys. Rev. Lett. 61, 1419 (1988)], and show that these data exhibit neither a BCS temperature dependence nor a linear temperature dependence at low temperature, but instead follow Δ${\lambda }_{\mathit{a}\mathit{b}}$(T)∼${\mathit{T}}^2$. This behavior is probably not intrinsic, and possible explanations are discussed.

We have investigated the superconducting state of the noncentrosymmetric compound ${\mathrm{Re}}_{6}\mathrm{Zr}$ using magnetization, heat capacity, and muon-spin relaxation or rotation ($$\mu${}\mathrm{SR}$) measurements. ${\mathrm{Re}}_{6}\mathrm{Zr}$ has a superconducting transition temperature, ${T}_{c}=6.75\ifmmode\pm\else\textpm\fi{}0.05\text{ }\text{ }\mathrm{K}$. Transverse-field $$\mu${}\mathrm{SR}$ experiments, used to probe the superfluid density, suggest an $s$-wave character for the superconducting gap. However, zero and longitudinal-field $$\mu${}\mathrm{SR}$ data reveal the presence of spontaneous static magnetic fields below ${T}_{c}$ indicating that time-reversal symmetry is broken in the superconducting state and an unconventional pairing mechanism. An analysis of the pairing symmetries identifies the ground states compatible with time-reversal symmetry breaking.

Muon spin rotation and relaxation experiments on the centrosymmetric intermetallic superconductor LaNiGa2 are reported. The appearance of spontaneous magnetic fields coincides with the onset of superconductivity, implying that the superconducting state breaks time reversal symmetry, similarly to noncentrosymmetric LaNiC2. Only four triplet states are compatible with this observation, all of which are nonunitary triplets. This suggests that LaNiGa2 is the centrosymmetric analogue of LaNiC2. We argue that these materials are representatives of a new family of paramagnetic nonunitary superconductors.