of vacancies requires a quantum mechanical description. They hop from site to site with a certain rate $\nu_v$, leading to a band of states $\varepsilon(\vec{k})$, much like the electron motion in the tight binding approximation. For $^4$He the vacancies are obviously bosons, since the creation operator for a vacancy is the annihilation operator for a $^4$He particle, which is a boson [@And2]. The fact that two vacancies cannot occupy the same lattice site has to be incorporated as a hard-core potential for the hopping bosons. So the simplest model for vacancies in $^4$He is that of a gas of hard-core bosons on a lattice. In reality the strain fields around the vacancies produce a more complicated interaction between them than the simple on-site exclusion. However, we consider the hard-core boson approximation as a sufficiently realistic description of the vacancy motion to leave out these further refinements, in order not to complicate the model too much. The hard-core bosons on a lattice are known to be equivalent to a spin-$\frac{1}{2}$ quantum system with an interaction of the XY type [@M+M]. It is however not this analogy that is exploited in this paper. The reason is that the vacancy system is in 