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Register a new userTwo-stage multipolar ordering in Pr(TM)$_2$Al$_{20}$ Kondo materials
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Among heavy fermion materials, there is a set of rare-earth intermetallics with non-Kramers Pr$^{3+}$ $4f^2$ moments which exhibit a rich phase diagram with intertwined quadrupolar orders, superconductivity, and non-Fermi liquid behavior. However, more subtle broken symmetries such as multipolar orders in these Kondo materials remain poorly studied. Here, we argue that multi-spin interactions between local moments beyond the conventional two-spin exchange must play an important role in Kondo materials near the ordered to heavy Fermi liquid transition. We show that this drives a plethora of phases with coexisting multipolar orders and multiple thermal phase transitions, providing a natural framework for interpreting experiments on the Pr(TM)$_2$Al$_{20}$ class of compounds.
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A series of Pr(TM)$_2$X$_{20}$ (with TM=Ti,V,Rh,Ir and X=Al,Zn) Kondo materials, containing non-Kramers Pr$^{3+}$ $4f^2$ moments on a diamond lattice, have been shown to exhibit intertwined orders such as quadrupolar order and superconductivity. Motivated by these experiments, we propose and study a Landau theory of multipolar order to capture the phase diagram and its field dependence. In zero magnetic field, we show that different quadrupolar states, or the coexistence of quadrupolar and octupolar orderings, may lead to ground states with multiple broken symmetries. Upon heating, such states may undergo two-step thermal transitions into the symmetric paramagnetic phase, with partial restoration of broken symmetries in the intervening phase. For nonzero magnetic field, we show the evolution of these thermal phase transitions strongly depends on the field direction, due to clock anisotropy terms in the free energy. Our findings shed substantial light on experimental results in the Pr(TM)$_2$Al$_{20}$ materials. We propose further experimental tests to distinguish purely quadrupolar orders from coexisting quadrupolar-octupolar orders.
The nature of multipolar order and hyperfine-enhanced (HE) $^{141}$Pr nuclear spin dynamics in PrV$_2$Al$_{20}$ was investigated using the muon spin relaxation technique. No explicit sign of time-reversal symmetry breaking was found below the multipolar order temperature $T_Qsim 0.6$ K in a zero applied field as anticipated on the basis of the antiferroquadrupolar (AFQ) order picture proposed by Sakai and Nakatsuji [J. Phys. Soc. Jpn. 80, 063701 (2011)]. Further evidence of the nonmagnetic ground state was obtained from the observation of HE $^{141}$Pr nuclear spin fluctuations in the MHz scale. A marked increase in the muon spin-lattice relaxation rate (1/$T_{rm 1,mu}$) was observed below 1 K with decreasing temperature, which was attributed to the perturbation on the HE $^{141}$Pr nuclear spin dynamics associated with the development of AFQ correlations. The longitudinal field dependence of 1/$T_{rm 1,mu}$ revealed that the enhanced $^{141}$Pr nuclear spin accidentally has an effective gyromagnetic ratio close to that of the muon.
The intermetallic compound PFA that possesses a three-dimensional network structure of Al polyhedra centered at the transition metal element Fe and the rare earth Pr is investigated through neutron powder diffraction and inelastic neutron scattering in order to elucidate the magnetic ground state of Pr and Fe and the crystal field effects of Pr. Our neutron diffraction study confirms long-range magnetic order of Pr below $T_N $ = 4.5~K in this compound. Subsequent magnetic structure estimation reveals a magnetic propagation vector $k$ = ($frac{1}{2}$~0~$frac{1}{2}$) with a magnetic moment value of 2.5~$mu_mathrm{B}$/Pr along the orthorhombic $c$-axis and evidence the lack of ordering in the Fe sublattice. The inelastic neutron scattering study reveals one crystalline electric field excitation near 19~meV at 5~K in PFA. The energy-integrated intensity of the 19~meV excitation as a function of |$Q$| $(A^{-1})$ follows the square of the magnetic form factor of Pr$^{3+}$ thereby confirming that the inelastic excitation belongs to the Pr sublattice. The second sum rule applied to the dynamic structure factor indicates only 1.6(2)~$mu_mathrm{B}$ evolving at the 19~meV peak compared to the 3.58~$mu_mathrm{B}$ for free Pr$^{3+}$, indicating that the crystal field ground state is magnetic and the missing moment is associated with the resolution limited quasi-elastic line. The magnetic order occurring in Pr in PFA is counter-intuitive to the symmetry-allowed crystal field level scheme, hence, is suggestive of exchange-mediated mechanisms of ordering stemming from the magnetic ground state of the crystal field levels.
Acoustic signatures of the single-site quadrupolar Kondo effect in Y$_{0.966}$Pr$_{0.034}$Ir$_2$Zn$_{20}$ are presented. The elastic constant ($C_{11}-C_{12}$)/2, corresponding to the $Gamma_3$(E)-symmetry electric-quadrupolar response, reveals a logarithmic temperature dependence of the quadrupolar susceptibility in the low-magnetic-field region below $sim$0.3 K. Furthermore, the Curie-type divergence of the elastic constant down to $sim$1 K indicates that the Pr ions in this diluted system have a non-Kramers ground-state doublet. These observations evidence the single-site quadrupolar Kondo effect, as previously suggested based on specific-heat and electrical resistivity data.
The ground-state phase diagrams of the three-orbital t2g Hubbard model are studied using a Hartree-Fock approximation. First, a complete set of multipolar order parameters for t2g models defined in terms of the effective total angular momentum jeff are theoretically derived. These order parameters can classify off-diagonal orders between jeff = 1/2 and jeff = 3/2 manifolds. Second, through extensive Hartree-Fock calculations, the ground-state phase diagrams in the space of (1) the onsite Coulomb repulsion U, (2) the spin-orbit coupling (SOC), and (3) the number of electrons are mapped out. A variety of nontrivial quantum phases with jeff-diagonal and jeff-off-diagonal multipole orders are found. Finally, future studies using more numerically expensive methods, such as dynamical mean-field theory are discussed.
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