General Rule Predicting the Hidden Induced Order Parameters and the Formation of Quartets and Patterns of Condensates

We reveal the rule with which we predicted, and verified by detailed numerical calculations on dozens of specific cases over the last decade, the emergence of hidden induced states and the formation of quartets of order parameters. The rule stipulates that if N order parameters are such that their cyclic products equal +1 or -1 then the presence of any N-1 of them will necessarily induce the missing order parameter as a hidden order. We demonstrate that the rule results from general microscopic mean field theory and is thus universally valid. Of exceptional physical interest are the cases N=4 called quartets, and all N>4 cases decompose into overlaping quartets which are the building blocks of observable patterns of condensates. Quartets may be found in two regimes, the "hierarchy" regime and the "equity" regime, and transitions between the two regimes exhibit universal characteristics. Following the rule, we predict many characteristic examples of quartets involving all types of particle-hole and superconducting condensates.