Inducing Ferroquadrupolar Order with Applied Magnetic Field in $TbPO_4$

In this work, we demonstrate a remarkable field-induced ferroquadrupolar phase emanating from the magnetoelastic coupling. Using synchrotron x-ray and neutron diffraction, heat capacity measurements, as well as mean-field modeling, we show that above the critical magnetic field, ferroquadrupolar $O$$_2$$^{-2}$order is stabilized as the main order parameter, in competition with the field-induced magnetic polarization. This is revealed by a severe lattice distortion and the suppression of antiferromagnetic ordering of the dipole moments. Even in zero magnetic field, the previously reported antiferromagnetic order is a mixed quadrupolar-dipolar phase, further highlighting the crucial role of spin-lattice interactions. This complete understanding of the magnetoelastic phase diagram establishes ${TbPO}$$_4$as a model system for studying nematic-antiferromagnetic transitions and provides key Hamiltonian parameters for its description.