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We have measured neutron diffraction patterns in a single crystal sample ofthe pyrochlore compound Er$_2$Ti$_2$O$_7$ in the antiferromagnetic phase(T=0.3\,K), as a function of the magnetic field, up to 6\,T, applied along the[110] direction. We determine all the characteristics of the magnetic structurethroughout the quantum critical point at $H_c$=2\,T. As a main result, all Ermoments align along the field at $H_c$ and their values reach a minimum. Usinga four-sublattice self-consistent calculation, we show that the evolution ofthe magnetic structure and the value of the critical field are rather wellreproduced using the same anisotropic exchange tensor as that accounting forthe local paramagnetic susceptibility. In contrast, an isotropic exchangetensor does not match the moment variations through the critical point. Themodel also accounts semi-quantitatively for other experimental data previouslymeasured, such as the field dependence of the heat capacity, energy of thedispersionless inelastic modes and transition temperature.

Field evolution of the magnetic structures in Er$_2$Ti$_2$O$_7$ through the critical point