content_image/421816.jpg content_image/421827.jpg content_image/421828.jpg
<img src='content_image/421816.jpg'>

“turned on”, focusing on the half–magnetization plateau fo r h ≈ 4 . For small | J$_{3}$ | , the system now exhibits two charac- teristic temperature scales — an upper temperature T ∗ ≈ b at which the gap protecting the magnetization plateau opens, and a lower temperature T$_{N}$ ≈ O ( | J$_{3}$ | ) at which the system exhibits long–range magnetic order. This is contrasted wit h the situation for h = 0 , where the system also exhibits two characteristic temperature scales, but these correspond to suc- cessive phase transitions : a nematic transition at T$_{Q}$ ∼ b and a N´eel ordering at T$_{N}$ ∼ O ( | J$_{3}$ | ) . We discuss the nature of these transitions for J$_{3}$ → 0 , identifying a line of first–order multifurcative points at J$_{3}$ = 0 . And, for J$_{3}$ = 0 , we identify an unusual continuous transition from the coulombic plateau liquid to the vector-multipole phase. At low temperatures this transition appears to have mean field character. 

Finally, in Sec. V we conclude with a discussion of the broader implications of these results. 

## II. DEGENERACIES IN FINITE MAGNETIC FIELD 

The pyrochlore lattice (Fig. 1) is the simplest example of a three–dimensional (3D) network of corner sharing complete graphs. Its elementary building block is the tetrahedron, in which every site is connected to every other site, i.e. the tetra- hedron is a complete graph of order four. Tetrahedra in the py- rochlore lattice can be divided into A and B sublattices, with each lattice site shared between an A- and a B-sublattice tetra- hedron. The centres of the two types of tetrahedra together form a (bipartite) diamond lattice. 63 The overall symmetry of the lattice is cubic. 

## A. Geometrical arguments 

As such, the pyrochlore lattice is a natural 3D analogue of the 2D kagome lattice, a corner sharing network of triangles (complete graphs of order three). In fact the [111] planes of the pyrochlore lattice are alternate kagome and triangu- lar lattices, composed of the triangular “bases” of tetrahe dra and their “points”, respectively. Much of the unusual physi cs of the kagome lattice also extends to its higher dimensional cousin. 

Lattices composed of complete graphs have the special property that bilinear quantities on nearest neighbor bonds can be recast as a sum of squares. Thus for b = 0 the Hamiltonian Eq. (1) can be written 

where the sum runs over tetrahedra, and 

is the magnetization (per site) of a given tetrahedron. For h = 0 , a simple classical counting argument shows that two of the eight angles needed to determine the orientation of the four spins in any given tetrahedron remain undetermined. Nearest neighbor interactions do not select one unique ground state on the pyrochlore lattice but rather the entire manifold of states for which | M | = 0 in each tetrahedron. Thus at T = 0 , the system is disordered. For fields h < h$_{sat}$ = 8 this conclu- sion is unaltered by the presence of magnetic field. In this case the manifold of ground states is determined by the con- dition M = h / 8 in each tetrahedron, and the magnetization is linear in h up to the saturation field h$_{sat}$ = 8 . (We recall that magnetic field is measured in units of J$_{1}$ , so that in fact h$_{sat}$ = 8 J$_{1}$ .) 

<img src='content_image/421827.jpg'>

<img src='content_image/421828.jpg'>

