  Materials with correlated electrons often respond very strongly to external
or internal influences, leading to instabilities and states of matter with
broken symmetry. This behavior can be studied theoretically either by
evaluating the linear response characteristics, or by simulating the ordered
phases of the materials under investigation. We developed the necessary tools
within the dynamical mean-field theory (DMFT) to search for electronic
instabilities in materials close to spin-state crossovers and to analyze the
properties of the corresponding ordered states. This investigation, motivated
by the physics of LaCoO$_3$, led to a discovery of condensation of spinful
excitons in the two-orbital Hubbard model with a surprisingly rich phase
diagram. The results are reviewed in the first part of the article. Electronic
correlations can also be the driving force behind structural transformations of
materials. To be able to investigate correlation-induced phase instabilities we
developed and implemented a formalism for the computation of total energies and
forces within a fully charge self-consistent combination of density functional
theory and DMFT. Applications of this scheme to the study of structural
instabilities of selected correlated electron materials such as Fe and FeSe are
reviewed in the second part of the paper.
