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The suppression of ferroquadrupolar order in TmVO₄in a magnetic field is well-described by the transverse field Ising model, enabling detailed studies of critical dynamics near the quantum phase transition. We describe nuclear magnetic resonance measurements in pure and Y-doped single crystals. The non-Kramers nature of the ground state doublet leads to a unique form of the hyperfine coupling that exclusively probes the transverse field susceptibility. Our results show that this quantity diverges at the critical field, in contrast to the mean-field prediction. Furthermore, we find evidence for quantum critical fluctuations present near Tm-rich regions in Y-doped crystals at levels beyond which long-range order is suppressed, suggesting the presence of quantum Griffiths phases. 

Abstract TmVO₄exhibits ferroquadrupolar order of the Tm 4f electronic orbitals at low temperatures, and is a model system for Ising nematicity. A magnetic field oriented along thec-axis constitutes a transverse effective field for the quadrupolar order parameter, continuously tuning the system to a quantum phase transition as the field is increased from zero. In contrast, in-plane magnetic fields couple to the order parameter only at second order, such that orienting along the primary axes of the quadrupole order results in an effective longitudinal field, whereas orienting at 45 degrees results in a second effective transverse field. Not only do in-plane fields engender a marked in-plane anisotropy of the critical magnetic and quadrupole fluctuations above the ferroquadrupolar ordering temperature, but in-plane transverse fields initially enhance the ferroquadrupolar order, before eventually suppressing it, an effect that we attribute to admixing of the higher crystalline electric field levels. 

Abstract We present Raman-scattering results for three materials, CeB 6 , TbInO 3 , and YbRu 2 Ge 2 , to illustrate the essential aspects of crystal-field (CF) excitations and quadrupolar fluctuations of 4 f -electron systems. For CF excitations, we illustrate how the 4 f orbits are split by spin-orbit coupling and CF potential by presenting spectra for inter- and intra-multiplet excitations over a large energy range. We discuss identification of the CF ground state and establishment of low-energy CF level scheme from the symmetry and energy of measured CF excitations. In addition, we demonstrate that the CF linewidth is a sensitive probe of electron correlation by virtue of self-energy effect. For quadrupolar fluctuations, we discuss both ferroquadrupolar (FQ) and antiferroquadrupolar (AFQ) cases. Long-wavelength quadrupolar fluctuations of the same symmetry as the FQ order parameter persists well above the transition temperature, from which the strength of electronic intersite quadrupolar interaction can be evaluated. The tendency towards AFQ ordering induces ferromagnetic correlation between neighboring 4 f -ion sites, leading to long-wavelength magnetic fluctuations.