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Magnetisation process in Er2Ti2O7 and Tb2Ti2O7 at very low temperature

We present a model which accounts for the high field magnetisation at very low temperature in two pyrochlore frustrated systems, Er2Ti2O7 and Tb2Ti2O7. The two compounds present very different ground states: Er2Ti2O7, which has a planar crystal field anisotropy, is an antiferromagnet with T_N=1.2K, whereas Tb2Ti2O7 is expected to have Ising character and shows no magnetic ordering down to 0.05K, being thus labelled a ``spin liquid''. Our model is a mean field self-consistent calculation involving the 4 rare earth sites of a tetrahedron, the building unit of the pyrochlore lattice. It includes the full crystal field hamiltonian, the infinite range dipolar interaction and anisotropic nearest neighbour exchange described by a 3-component tensor. For Er2Ti2O7, we discuss the equivalence of our treatment of the exchange tensor, taken to be diagonal in a frame linked to a rare earth - rare earth bond, with the pseudo-spin hamiltonian recently developped for Kramers doublets in a pyrochlore lattice. In Tb2Ti2O7, an essential ingredient of our model is a symmetry breaking developping at very low temperature. We compare its prediction for the isothermal magnetisation with that of ``the quantum spin ice'' model.