High-magnetic field phase diagram and failure of magnetic Gr\"uneisen scaling in LiFePO${}_4$

We report the magnetic phase diagram of single-crystalline LiFePO${}_4$ in magnetic fields up to 58~T and present a detailed study of magneto-elastic coupling by means of high-resolution capacitance dilatometry. Large anomalies at \tn\ in the thermal expansion coefficient $\alpha$ imply pronounced magneto-elastic coupling. Quantitative analysis yields the magnetic Gr"uneisen parameter ${\gamma }_{\mathrm{m}\mathrm{a}\mathrm{g}}=6.7(5)\cdot {10}^{-7}$~mol/J. The positive hydrostatic pressure dependence $d{T}_{\mathrm{N}}\mathrm{/}dp=1.46(11)$~K/GPa is dominated by uniaxial effects along the $a$-axis. Failure of Gr"uneisen scaling below $\approx 40$~K, i.e., below the peak temperature in the magneto-electric coupling coefficient [\onlinecite{toft2015anomalous}], implies several competing degrees of freedom and indicates relevance of recently observed hybrid excitations~[\onlinecite{yiu2017hybrid}]. A broad and strongly magnetic-field-dependent anomaly in $\alpha$ in this temperature regime highlight the relevance of structure changes. Upon application of magnetic fields $B\mathrm{\mid }\mathrm{\mid }b$-axis, a pronounced jump in the magnetisation implies spin-reorientation at $B_{\mathrm{S}\mathrm{F}}=32$~T as well as a precursing phase at 29~T and $T=1.5$~K. In a two-sublattice mean-field model, the saturation field $B_{\mathrm{s}\mathrm{a}\mathrm{t},\mathrm{b}}=64(2)$~T enables the determination of the effective antiferromagnetic exchange interaction $J_{\mathrm{a}\mathrm{f}}=2.68(5)$~meV as well as the anisotropies $D_{\mathrm{b}}=-0.53(4)$~meV and $D_{\mathrm{c}}=0.44(8)$~meV.