Unveiling Order behind Complexity: Coexistence of Ferromagnetism and Bose-Einstein Condensation

We present an algebraic framework for identifying the order parameter and the possible phases of quantum systems that is based on identifying the local dimension $N$ of the quantum operators and using the SU(N) group representing the generators of generalized spin-particle mappings. We illustrate this for $N$=3 by presenting for any spatial dimension the exact solution of the bilinear-biquadratic $S$=1 quantum Heisenberg model at a high symmetry point. Through this solution we rigorously show that itinerant ferromagnetism and Bose-Einstein condensation may coexist.