sufficiently negative $\zeta$ one has $\Phi =-\pi$ while for $\zeta$ sufficiently positive $\Phi=0$. In the zone in between one has for $d>1$ in general a continuous transition from $-\pi$ to 0, with eventually also zones with $\Phi=-\pi/2$.

The two- and three-dimensional lattices
=======================================

The calculation of $b_2$ for hypercubic lattices in higher dimensions runs along the same lines as what was done in section \[oned\] for one dimension. The band structure is now given by (again $a=1$) $$\varepsilon (\vec{k}) =\left\{
\begin{array}{l}
2t(2-\cos k_x - \cos k_y ) ~~~~~~~~~~~~~~~~(d=2)\\
2t(3-\cos k_x - \cos k_y -\cos k_z) ~~~~~(d=3).
\end{array}
\right.
\label{hdeps}$$ In $d$ dimensions, $b_1(\beta)$ is simply given by $$\begin{aligned}
b_1(\beta) &=& \frac{1}{(2\pi)^d} \left[ \int_{-\pi}^{\pi}
dk\, e^{-2\beta t(1-\cos k)}\right]^d\nonumber\\
&=&e^{-2d\beta t}\left[ I_0
(2\beta t) \right]^d.
\label{mdb1}\end{aligned}$$

As before, we rearrange the Brillouin zone such that its boundaries in $\vec{K},\vec{k}$ space are convenient. The different components of $\vec{K},\vec{k}$ are independent, and we have $0\leq K_i
\leq 2\pi\;,
\; 0 \leq k_x \leq \pi \; , \; -\pi \leq k_y ,
k_z \leq \pi$. The $k_x$-interval is halved to avoid counting the same (symmetric) state twice. The two-vacancy energy bands are given by $$\begin{aligned}
E_0(\vec{K},\vec{k})&=&
4t(2-\cos\displaystyle{\frac{K_x}{2}}\cos k_x -\cos
\displaystyle{\frac{K_y}{2}}\cos k_y )\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(d=2),\nonumber\\
E_0(\vec{K},\vec{k})&=&
4t(3-\cos\displaystyle{\frac{K_x}{2}}\cos 