Figure 1

Some examples of lattices for which the exact entropy is calculated in this Letter. (a) The kagome lattice. (b) A three-dimensional lattice with corner-sharing triangles. (c) The unit cell of the ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ lattice. There are 12 iridium atoms per unit cell (shown as red cubes or blue spheres). Sodium and oxygen atoms are not shown. The (red) cubes mark the articulation points of the lattice and the spheres mark the rest. The lattices in (b) and (c) can be shown to be equivalent. (d) Unit cell of a lattice obtained by the decoration procedure discussed in text.Reuse & Permissions

Figure 2

On deleting two edges from each (red) square, the kagome lattice breaks into disconnected chains. The values of the $\sigma$ variables at some of the (red) squares are indicated by the numbers in parentheses.Reuse & Permissions