  We determine the phase diagram of copper nitrate Cu(NO$_3$)$_2\cdot$2.5D$_2$O
in the context of quantum phase transitions and novel states of matter. We
establish this compound as an ideal candidate to study quasi-1D Luttinger
liquids, 3D Bose-Einstein-Condensation of triplons, and the crossover between
1D and 3D physics. Magnetocaloric effect, magnetization, and neutron scattering
data provide clear evidence for transitions into a Luttinger liquid regime and
a 3D long-range ordered phase as function of field and temperature. Theoretical
simulations of this model material allow us to fully establish the phase
diagram and to discuss it in the context of dimerized spin systems.
