Anisotropic magneto-elastic coupling in the quasi-2D Ising magnet Li$_2$FeSiO$_4$ and thermodynamic signature of low-energy orbital excitations

The relevance of spin, lattice and orbital degrees of freedom in the quasi-2D Ising-like magnet Li$_2$FeSiO$_4$ is demonstrated by thermal expansion and specific heat measurements supported by numerical studies. By means of thermodynamic and Grüneisen analysis, we separate the orbital signatures and quantify the energy gap and its pressure dependence from the experimental data as $Δ\sim 300 - 350$ K and $\partial \ln Δ/\partial p_{\rm c} \approx -4.6$ %/GPa). Both values excellently agree with the theoretical DFT predictions. The thermal expansion anomalies associated with the onset of long-range magnetic order imply significant magneto-elastic coupling and are used to determine the anisotropic uniaxial strain dependencies of $T_{\rm N}$. They are dominated by the in-plane (i.e., $c$ axis) response $(\partial \ln{ T_{\rm N}} )/\partial p_{\rm c} \simeq 15$ %/GPa while the (negative) $a$ axis and (positive) out-of-plane ($b$ axis) pressure dependencies are much smaller. Short-range magnetic order extends well above $T_{\rm N}$ but is not driven by the same energy scale as $T_{\rm N}$ as evidenced by the observation of non-monotonic in-plane distortion, sign-change of the Grüneisen parameter, and failure of Grüneisen scaling.