We study an infinite-range coupled electronic-quadrupole and nuclear-spin model for ferro-quadrupolar and nuclear-spin ordering in TmVO4 in external magnetic and strain fields. This material is an experimental realization of a Transverse-Field Ising Model, where the Ising degree of freedom is quadrupolar and non-magnetic, but a transverse component is magnetic and couples both to external magnetic fields and to the nuclear spins via a hyperfine coupling. In zero external magnetic-field, there is a well-separated two-step order of the electronic and nuclear degrees of freedom and the release of their respective entropies. A transverse magnetic-field polarizes the electronic orbital moments and also the nuclear spins via the hyperfine coupling. The quadrupolar ordering temperature is gradually reduced to zero. But, there is no longer a nuclear transition in non-zero fields. Quantum fluctuations are magnified near the phase transitions and lead to peaks in the magnetic susceptibility. The spectral functions reveal a softening of a low-energy mode near the quantum critical point, consistent with the closing of the excitation gap and its reopening in the disordered phase, providing direct dynamical signatures of the field-driven quantum critical phenomena.
@article{arxiv.2507.16619,
title = {A long-range model for the electron-nuclear coupling and two-stage order in TmVO$_4$},
author = {Sayan Ghosh and Anirudha Menon and Manoranjan Kumar and Rajiv R. P. Singh},
journal= {arXiv preprint arXiv:2507.16619},
year = {2026}
}