Bose-Einstein condensation (BEC) occurs. This is a macroscopic quantum phenomenon characterized by spontaneous quantum coherence persisting over macroscopic length and time scales. In dilute atomic gases this phenomenon was realized experimentally for cold atoms. Several quantum spin systems in solids, which show a magnetic-field induced transition, are expected to also show condensation above or below a certain critical field [@2; @7; @12]. Studies have shown that the magnetic system can be mapped non-locally onto a set of weakly interacting bosons on a lattice.

The sorting phase can be described as a BEC of bosonic quasi-particles, in which the magnetic field acts to preserve the number of bosons. Therefore, the tuning parameter to induce condensation in spin-ordered systems is not the temperature, but the magnetic field. For particles as well as magnons, a macroscopic number of bosons condense into a single quantum statethe state of lowest energy. The quantum coherence of Bose-Einstein condensation dates back to the prediction of Einstein, based on Bose’s work, in 1924.

In the diluted BEC the macroscopic wave function is directly connected with the microscopic energy levels, providing a complete description of these phenomena in terms of the Gross-Pitaevskii equation. The concept of a coherent macroscopic 