Electro-nuclear quantum phase transition in TmVO$_4$

Hyperfine interactions couple nuclear and electronic degrees of freedom. The present work explores how hyperfine coupling within the Tm ions in TmVO$_4$ single crystals affects an electronic ferroquadrupole ordered ground state and its associated field-tuned quantum phase transition. For temperatures below the hyperfine energy scale, the nuclear moments reduce the critical field for the electronic order, resulting in a dramatic back-bending of the phase boundary delineating the ferroquadrupole order. This behavior is well described by a single-ion semiclassical mean-field model. Moreover, analysis of the effective Hamiltonian leads to a prediction of spontaneous nuclear magnetic order mediated by 4$f$ electrons, which in principle persists with the application of orthogonal antisymmetric strain, yielding a proposed electro-nuclear tetracritical point.