Critical pheneomena in random systems

Critical phenomena in pure systems are in principal more or less well understood. Theory has reached a high quantitative level. Comparison with computer simulations and experiment verifies results obtained with nowadays standard methods. The main field of research goes to more complicated systems with multicritical behavior and the study of crossover phenomena. However there is a class of systems which are less understood, namely impure systems. One class of such systems are those with quenched (so to say fixed) impurities. These impurities may be defects, substitutional molecules, vacancies or other irregularities in the pure system. The main question with respect to critical phenomena are: what is the effect of the presence of impurities? Is the phase transition of the pure system changed? Does it remain second order and if so does it belong to the same universality class? Concerning the asymptotic critical behavior this question has been answered by the so called Harris criterium which states that in pure systems with non diverging specific heat, the asymptotic critical behavior is unchanged (characterized by the properties around the stable fixed point of the pure system). But this is not the whole story. Usually the experimental accessible region is not the asymptotic region but a region showing some crossover behavior, either from the background region to asymptotics or from some region around an unstable fixed point to the asymptotics. This holds for both statics and dynamics. Therefore a main topic in our project will be the study of a (non-trivial) asymptotic critical behavior in disordered systems for the cases when it differs from that of the corresponding pure system as well as the detailed analysis of the effective critical behavior of random site-disorder and random anisotropy systems. In dynamics besides the relaxational model a coupling to the energy density will be considered. The calculations for this model will be a complicated task as already the pure model constitutes a quite non trivial dynamic model for which the correct field theoretic functions have been found only recently (in the FWF project PHY15247).

Beneath the most intriguing questions of modern condensed matter physics one should certainly name the problem of influence of structural disorder on criticality. Taken that a non-disordered ("ideal") system possesses a phase transition and critical behavior governed e.g. by some scaling laws, will these universal laws be altered by the disorder of structure induced into the system? Although this question addresses properties of matter in a narrow region of a phase diagram in the vicinity of a phase transition point, the answer on it is of a great importance both for fundamental reasons (description of criticality arising in different systems ranging from high-energy physics to cosmology) as well as due to applications of structurally-disordered materials in modern technologies. The goal of this project was to achieve a better understanding of complex critical behavior that occurs in systems with structural disorder. The properties of the disordered systems and the characteristics of the phase transitions were analysed using combination of several state-of-the-art methods, including renormalization group approach, extensive Monte Carlo (MC) simulations, and analytic methods within statistical physics. Those questions of criticality and structural disorder are not only restricted to condensed matter systems but concern more general complex systems, where methods known from statistical physics can be applied. Particular systems we analysed include diluted and amorphous magnets, polymers, and superconductors. In our research, we got quantitative description of universal asymptotic critical behavior which arises in close vicinity of the critical point as well as we extended existing theoretical methods to describe non-universal effective criticality. It is this last behavior that is typically observed in experiments and MC simulations, therefore its description is of particular interest. Moreover, our research concerned not only the static critical behavior but it is also addressed to dynamic phenomena, arising as the critical point is approached. An example of such phenomenon is given by the critical slowing down and leads e.g. to singular behavior of transport coefficients. Due to the universality of critical phenomena, application of similar techniques allowed us to shed light on appearance of scaling laws that govern critical behavior in different systems. Particular results, obtained in our research include: analysis of an impact of extended impurities on effective critical dynamics; description of critical slowing down in random anisotropy and random site magnets; enhancement of the critical slowing down caused by extended defects; explanation of the scaling behavior of polymers in porous medium and of multi-component polymers of complex topology; study of the influence of frustrations on non-collinear ordering; origin of scaling laws in complex systems, critical dynamics of superconductors.