Colloquium : Theory of intertwined orders in high temperature superconductors

Understanding high temperature superconductors is a central problem in condensed matter physics. Many experiments have uncovered ordering tendencies which are responsible for the complex phase diagram of high temperature superconductors. This Colloquium discusses the interplay between different order parameters in these materials. Considering the intertwining of these orders leads to new experimentally observable consequences, shedding new light into the physics of these fascinating materials. 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 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." 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, are selectively analyzed. Several theoretical ideas concerning the origin and implications of this complexity are also summarized and critiqued.