  Bogoliubov transformations have been successfully applied in several
Condensed Matter contexts, e.g., in the theory of superconductors, superfluids,
and antiferromagnets. These applications are based on bulk models where
translation symmetry can be assumed, so that few degrees of freedom in Fourier
space can be `diagonalized' separately, and in this way it is easy to find the
approximate ground state and its excitations. As translation symmetry cannot be
invoked when it comes about nanoscopic systems, the corresponding
multidimensional Bogoliubov transformations are more complicated. For bosonic
systems it is much simpler to proceed using phase-space variables, i.e.,
coordinates and momenta. Interactions can be accounted for by the
self-consistent harmonic approximation, which is naturally developed using
phase-space Weyl symbols. The spin-flop transition in a short antiferromagnetic
chain is illustrated as an example. This approach, rarely used in the past, is
expected to be generally useful to estimate quantum effects, e.g., on phase
diagrams of ordered vs disordered phases.
