Thermodynamics of type II superconductors in electromagnetic field based on the GinzburgLandau theory is presented. The Abrikosov flux lattice solution is derived using an expansion in a parameter characterizing the "distance" to the superconductor -normal phase transition line. The expansion allows a systematic improvement of the solution. The phase diagram of the vortex matter in magnetic field is determined in detail. In the presence of significant thermal fluctuations on the mesoscopic scale (for example in high Tc materials) the vortex crystal melts into a vortex liquid. A quantitative theory of thermal fluctuations using the lowest Landau level approximation is given. It allows to determine the melting line and discontinuities at melt, as well as important characteristics of the vortex liquid state. In the presence of quenched disorder (pinning) the vortex matter acquires certain "glassy" properties. The irreversibility line and static properties of the vortex glass state are studied using the "replica" method. Most of the analytical methods are introduced and presented in some detail. Various quantitative and qualitative features are compared to experiments in type II superconductors, although the use of a rather universal Ginzburg -Landau theory is not restricted to superconductivity and can be applied with certain adjustments to other physical systems, for example rotating Bose -Einstein condensate.

The quantum mechanics of a system of charged particles interacting with a magnetic field on Riemann surfaces is studied. We explicitly construct the wave functions of ground states in the case of a metric proportional to the Chern form of the θ-bundle, and the wave functions of the Landau levels in the case of the the Poincaré metric. The degeneracy of the the Landau levels is obtained by using the Riemann-Roch theorem. Then we construct the Laughlin wave function on Riemann surfaces and discuss the mathematical structure hidden in the Laughlin wave function. Moreover the degeneracy of the Laughlin states is also discussed.

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The influence of recently discovered topological transition between type I and type II Weyl semimetals on superconductivity is considered. A set of Gorkov equations for weak superconductivity in Weyl semi-metal under topological phase transition is derived and solved. The critical temperature and superconducting gap both have spike in the point the transition point as function of the tilt parameter of the Dirac cone determined in turn by the material parameters like pressure. The spectrum of superconducting excitations is different in two phases: the sharp cone pinnacle is characteristic for a type I, while two parallel almost flat bands, are formed in type II. Spectral density is calculated on both sides of transition demonstrate different weight of the bands. The superconductivity thus can be used as a clear indicator for the topological transformation. Results are discussed in the light of recent experiments.

Flux line lattice in type II superconductors undergoes a transition into a "disordered" phase such as vortex liquid or vortex glass, due to thermal fluctuations and random quenched disorder. We quantitatively describe the competition between the thermal fluctuations and the disorder using the Ginzburg-Landau approach. The following T-H phase diagram of YBCO emerges. There are just two distinct thermodynamical phases, the homogeneous and the crystalline one, separated by a single first order transition line. The line, however, makes a wiggle near the experimentally claimed critical point at 12 T. The "critical point" is reinterpreted as a (noncritical) Kauzmann point in which the latent heat vanishes and the line is parallel to the T axis. The magnetization, the entropy, and the specific heat discontinuities at melting compare well with experiments.

Recent measurements of fluctuation diamagnetism in high-temperature superconductors show distinct features above and below T c , which can not be explained by simple Gaussian fluctuation theory. Self-consistent calculation of magnetization in layered high-temperature superconductors, based on the Ginzburg-Landau-LawrenceDoniach model and including all Landau levels is presented. The results agree well with the experimental data in a wide region around T c , including both the vortex liquid below T c and the normal state above T c . The Gaussian fluctuation theory significantly overestimates the diamagnetism for strong fluctuations. It is demonstrated that the intersection point of magnetization curves appears in the region where the lowest Landau level contribution dominates and magnetization just below T c is nonmonotonic. Our calculation supports the phase disordering picture of fluctuations above T c .

A new systematic calculation of magnetization and specific heat contributions of vortex liquids and solids is presented. We develop an optimized perturbation theory for the Ginzburg-Landau description of thermal fluctuations effects in the vortex liquids. The expansion is convergent in contrast to the conventional high temperature expansion which is asymptotic. In the solid phase we calculate the first two orders which are already quite accurate. The results are in good agreement with existing Monte Carlo simulations and experiments. Limitations of various nonperturbative and phenomenological approaches are noted. In particular, we show that there is no exact intersection point of the magnetization curves.

Higher than the lowest Landau level contributions to magnetization and specific heat of superconductors are calculated using Ginzburg -Landau equations approach. Corrections to the excitation spectrum around solution of these equations (treated perturbatively) are found. Due to symmetries of the problem leading to numerous cancellations the range of validity of the LLL approximation in mean field is much wider then a naive range and extends all the way down to. Moreover the contribution of higher Landau levels is significantly smaller compared to LLL than expected naively. We show that like the LLL part the lattice excitation spectrum at small quasimomenta is softer than that of usual acoustic phonons. This enhanses the effect of fluctuations. The mean field calculation extends to third order, while the fluctuation contribution due to HLL is to one loop. This complements the earlier calculation of the LLL part to two loop order.1