Second order Zeeman interaction and ferroquadrupolar order in TmVO4

Abstract

TmVO4 exhibits ferroquadrupolar order of the Tm 4f electronic orbitals at low temperatures, and is a model system for Ising nematicity. A magnetic field oriented along the c-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.

Possible realization of three-dimensional quantum spin liquid behavior in HoVO4

“…Recently, this series is in focus, as a few compounds show exotic properties. In TmVO 4 , magnetic field tuned ferroquadrupolar quantum phase transition is reported [29,30] and anisotropic nematic fluctuations are found to exist above this transition [31]. Two other members of the family reports the presence of field tuned quantum criticality in DyVO 4 [32] and signatures of effective spin-1 /2 ground state in CeVO 4 [33].…”

Evolution of field induced magnetic phase attributed to higher order magnetic moments in TbVO4

“…The origin of this is believed to be unusually large magnetoelastic coupling along the easy c-axis of magnetization which is in contradiction to other members of the series. Along with this, TmVO 4 reveals magnetic field tuned ferroquadrupolar quantum phase transition 12 , 13 and anisotropic nematic fluctuations above this ferroquadruople transition 14 . Investigations on DyVO 4 divulge the presence of field tuned quantum criticality 15 .…”

Nuclear magnetic resonance studies in a model transverse field Ising system

“…For sufficiently low perpendicular fields, H x,y ≤ 3 T, the second order Zeeman interaction in the perpendicular direction will be less than 0.1k B T Q , and can be safely ignored. At higher fields, H x and H y can also act as either longitudinal or transverse fields for the Ising order, and can in fact be used to detwin the ferroquadrupolar order [24].…”

“…Mean field theory predicts a QCP for a c-axis field of T Q /g c μ B ≈ 0.3 T, which is close to the experimental value of H c * 0.5 T. Note that if there is a perpendicular field oriented such that H x or H y is zero, the system can still be described by the TFIM, because H Z does not couple to the longitudinal order in pseudospin space (σ y ). Rather, there is an effective transverse field in the x-z plane of pseudospin space leading to a different value of the critical field [24].…”