A procedure for studying the properties of the phase diagram is to understand the transitions that appear when the boundaries separating the phases are crossed [@7; @20; @21; @22]. When a border is crossed vertically to enter the ordered phase a thermal phase transition results, driven by thermal fluctuations. The magnetic and thermal energies compete. At temperatures near zero, by contrast, there are no thermal fluctuations and a quantum transition results, driven by quantum fluctuations. The magnetic energy and the zero-point energies compete. This second-order quantum transition to the BEC phase is free of interactions. However, including even week interactions into the description of the condensation has proven to be a great challenge, which was overcome by introducting effective interactions [@7; @10; @20; @21]. These interactions, well-approximated by a contact pseudo-potential, are central to arriving at the Gross-Pitaevskii equation. Understanding the behavior of the phase transition near zero temperature is understanding the quantum behavior of the system. The transition to the BEC state is characterized by specific characteristics of the thermodynamic properties.

We studied BEC in magnons because condensation occurs at temperatures that are much more accessible than the condensation of particles.

![\[fig2\] The phase diagram from the normal to 