Nonlinear Evolution Equations and Dynamical Systems

Continuous phase transitions between disordered and ordered phase states are accompanied by spontaneous symmetry breaking. The Landau theory of phase transitions, which describes this phenomenon, is based on the use of the local order parameters with non-zero value in the states with the decreased sym-metry. However this theory cannot describe all continuous phase transitions between quantum-ordered states. This happens because in the systems with developed quantum fluctuations, phase transitions between disordered phases may take place at zero temperature without symmetry breaking. In this phe-nomenon, the initial and the final topologically ordered states are a special case of the order, which is characterized only by gapped excitations. The study of topologically ordered states and quantum phase transitions be-tween them is important for classifying various phase states in low-dimensional systems, where the role of quantum fluctuations is significant. In this case new types of ordering of strongly correlated spin degrees of freedom may be based on the employ of topological features of dynamics of excitations in low-dimensiona