| The theoretical study of critical phenomena at phase transitions is one of the most challenging problems in condensed matter physics. As has been shown by renormalization group (RG) theory, critical fluctuations can cause drastic changes in the nature of the phase transition which have been observed experimentally. However, there are many discussions about the reasons for these changes. Since physically different phenomena, such as additional interactions, or the dependance of system parameters upon temperature or critical fluctuations, can result in similar effects, it is very difficult to conclusively ascribe critical fluctuations as the real reason for some striking effects in phase transitions. Unfortunately, RG theory is not a big help in settling this dispute. This theory can only determine the critical asymptotics. For the case of qualitative effects this theory does not work very well and can only make intuitive predictions which very often totally contradict the picture obtained from mean field theory. Consequently, hoping to find the true description of critical phenomena at phase transitions, we study various important systems through an alternative approach. We use an exactly solvable model that takes into account fluctuation interactions partially. It is, therefore, conceptually somewhere between mean field and RG theories. The advantage of the model is that it is exactly solvable, and can therefore provide us with the description of phase transitions within the whole range of variation of thermodynamical quantities. In systems with coupled order parameters the model finds the first order phase transition induced by fluctuations. This type of transition is replaced by a second order when fluctuations are suppressed. Systems with two interacting order parameters which additionally are coupled to two random fields exhibit second order transitions. Finally the model is applied to the case of a random field coupled to an order parameter in d dimension and proves similar critical behavior with the pure case of ({dollar}d-2{dollar}) dimension. At the end we use the fundamental formalism of RG in systems described by the most arbitrary symmetry Hamiltonian in order to construct an exact RG equation which contains no redundant operators. |
