This article is a review of recent developments in the phenomenological description of unconventional superconductivity.Starting with the BCS theory of superconductivity with anisotropic Cooper pairing, the authors explain the group-theoretical derivation of the generalized Ginzburg-Landau theory for unconventional superconductivity. This is used to classify the possible superconducting states in a system with given crystal symmetry, including strong-coupling effects and spin-orbit interaction. On the basis of the BCS theory the unusual low-temperature properties and the (resonant) impurity scattering effects are discussed for superconductors with anisotropic pairing. Using the Ginzburg-Landau theory, the authors study several bulk properties of such superconductors: spontaneous lattice distortion, upper critical magnetic field, splitting of a phase transition due to uniaxial stress. Two possible mechanisms for ultrasound absorption are discussed: collective modes and damping by domain-wall motion. The boundary conditions for the Ginzburg-Landau theory are derived from a correlation function formulation and by grouptheoretical methods. They are applied to a study of the Josephson and proximity effects if unconventional superconductors are involved there. The magnetic properties of superconductors that break time-reversal symmetry are analyzed. Examples of current and magnetic-field distributions close to inhomogeneities of the superconducting order parameter are given and their physical origin is discussed. Vortices in a superconductor with a multicomponent order parameter can exhibit various topo1ogical structures. As examples the authors show fractional vortices on domain walls and nonaxial vortices in the bulk. Furthermore, the problem of the possible coexistence of a superconducting and a magnetically ordered phase in an unconventional superconductor is analyzed. The combination of two order parameters that are almost degenerate in their critical temperature is considered with respect to the phase-transition behavior and effects on the lower and upper critical fields. Because heavy-fermion superconductors -which are possible realizations of unconventional superconductivity -have been the main motivation for the phenomenological studies presented here, the authors compare the theoretical results with the experimental facts and data. In particular, they emphasize the intriguing features of the compound UPt3 and consider in detail the alloy U& Th Be/3.

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The two dimensional Heisenberg model for SrCu 2 ͑BO 3 ͒ 2 has an exact dimer ground state which was proven by Shastry and Sutherland almost twenty years ago. The critical value of the quantum phase transition from the dimer state to the Néel ordered state is determined. Analysis of the experimental data shows that SrCu 2 ͑BO 3 ͒ 2 has the dimer ground state but its coupling constants are close to the transition point, which leads to an unusual temperature dependence of the susceptibility. The almost localized nature of the triplet excitations explains the plateaus observed in the magnetization curve.[S0031-9007(99)09090-0]

Theory of spin¯uctuations for itinerant magnetism and its application to high temperature superconductivity are reviewed. After a brief introduction to the whole subject the developments of the self-consistent renormalization theory of spin¯uctuations are summarized with particular emphasis on critical properties at the quantum phase transitions. Most of the anomalous properties in the normal state of high-T c cuprates are understood as due to the critical behaviours for the two dimensional antiferromagnetic metals. By analysing the nuclear magnetic relaxation rate and the T -linear term of resistivity, the set of parameters to specify the spin¯uctuations are determined. It is shown that by using the parameters thus obtained one can describe other quantities as well, e.g. optical conductivity. Then we proceed to the theory of superconductivity by the spin¯uctuation mechanism. After some discussion on the weak coupling treatments, the strong coupling theory is reviewed. It is shown that the set of parameters determined by the normal state properties of the high-T c cuprates just give a transition temperature of the right order of magnitude. Among the parameters, the most sensitive one for T c is the frequency spread of the spin¯uctuations. This fact enables us to present a possible uni® ed picture of the antiferromagnetic spin¯uctuation-induced superconductors, including heavy fermion superconductors and organic superconductors. This point of view may be con® rmed to a certain extent by microscopic calculations based on the¯uctuation exchange approximation for the two-dimensional Hubbard models representing not only the cuprates but also organic and trellis lattice compounds. The review is concluded with some discussions on future problems, e.g. the pseudo spin-gap in the under-doped region.

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