three dimensions. The case of the hcp lattice, which applies to real solid $^4$He, will be treated elsewhere.

The hard-core Bose lattice gas
==============================

The vacancies are represented by Bose creation and annihilation operators $b_i^{\dagger}, b_i$ obeying the usual Bose commutation relations. The Hamiltonian for the vacancies is given by $${\cal H} = -t \sum_{\langle i,j \rangle} ( b_i^{\dagger}b_j +
b_j^{\dagger}b_i) +(\Delta + ct ) \sum_i b_i^{\dagger} b_i
+\frac{U}{2} \sum_i b_i^{\dagger} b_i^{\dagger} b_i b_i .
\label{ham}$$ Here the transfer integral $t$ is equal to $h\nu_v$, and the hops take place between all pairs of nearest neighbours $\langle
i,j \rangle$ on the lattice. The coordination number of the lattice is $c$, and $\Delta$ is the energy required to create a vacancy. It functions as minus the chemical potential for the vacancies. The last term represents the vacancy-vacancy repulsion. It could be omitted in favour of a change to on-site Fermi commutation relations for the $b_i^{(\dagger)}$, but we prefer to work with standard commutation relations and a potential $U$ which penalizes the simultaneous occurrence of two or more vacancies at the same site. We will let $U\rightarrow \infty$, or $U$ is much larger than any other energy in the problem.

Without the 