  The electronic phase diagrams of many highly correlated systems, and in
particular the cuprate high temperature superconductors, are complex, with many
different phases appearing with similar-sometimes identical-ordering
temperatures even as material properties, such as a dopant concentration, are
varied over wide ranges. This complexity is sometimes referred to as "competing
orders." However, since the relation is intimate, and can even lead to the
existence of new phases of matter such as the putative "pair-density-wave," the
general relation is better thought of in terms of "intertwined orders." We
selectively analyze some of the experiments in the cuprates which suggest that
essential aspects of the physics are reflected in the intertwining of multiple
orders-not just in the nature of each order by itself. We also summarize and
critique several theoretical ideas concerning the origin and implications of
this complexity.
