O 7 . Magnetic ions in these materials are located at the vertices of regular tetrahedra linked into a three dimensional pyrochlore lattice (Fig. 1). Harris et al. [1] showed that such a system of spins with the initial Heisenberg fer romagnetic exchange interaction is equivalent to the Ising antiferromagnetic model on a geometrically frustrated lattice. The reason is strong magnetic anisotropy owing to which the spins of magnetic ions can be directed either only toward or only from the center of each tetrahedron. The ground state of this magnetic system is characterized by the ice rule: two spins of each tetrahedron are directed toward its cen ter, and two other spins, from the center. In view of the close analogy with the distribution of protons on hydrogen bonds in normal ice, these materials are referred to as spin ice [1,2]. Den Hertog and Gingras [3,4] proposed an alternative model of spin ice based on the magnetic dipole-dipole interaction that leads to a similar structure of the ground state. The close model of a proton sublattice in water ice based on the dipole-dipole interaction between dipoles of hydro gen bonds was proposed in [5].The ground state of spin ice is conveniently described by a divergence free vector field [6-8], which can be defined in terms of the spin variables asHere, summation is performed over all spins inside a physically infinitesimal volume V near the point r, σ i = ±1 are the spin variables, and e i are the unit vectors directed from the centers of tetrahedra (one of two sublattices taken as the basis one) toward their vertices (see [9] and Fig. 1). In this case, the ice rule is transThe temperature dependence of the concentration of magnetic monopoles in spin ice has been analyzed. It has been shown that magnetic Coulomb interaction between monopoles can lead to a temperature induced stepwise increase in their concentration by several orders of magnitude. This first order phase transition is also accompanied by a decrease in the relaxation time by several orders of magnitude and can be considered as the melting of the Coulomb phase of spin ice. The phase diagram, as well as the critical point, has been determined. Fig. 1. Pyrochlore lattice. Magnetic ions are at the vertices of the regular tetrahedra. The centers of the tetrahedra form a diamond type lattice consisting of two face cen tered sublattices (one of them is taken as the basis sublat tice). The magnetic moments directed toward (from) the tetrahedron centers are shown by closed (open) circles.

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