Non-Collinear Magnetism due to Orbital Degeneracy and Multipolar Interactions

Abstract

The origin of non-collinear magnetism under quadrupolar ordering is investigated with CeB6 taken as a target system. The mode-mixing effect among 15 multipoles is analyzed based on the Ginzburg-Landau free energy. Then the lower magnetic transition temperature and the order parameters are derived within the mean-field approximation. In the presence of pseudodipole-type interactions for the next-nearest neighbors, the observed pattern of non-collinear ordering is indeed stabilized for certain set of interaction parameters. The stability of the phase III ′ in the magnetic field is also explained, which points to the importance of the next-nearestneighbor octupole-octupole interaction. Concerning the phase IV in CexLa1−xB6 with x ∼ 0.75, a possibility of pure octupole ordering is discussed based on slight modifications of the strength of interactions.KEYWORDS: orbital degeneracy, multipole moment, non-collinear magnetism, CeB 6 , CexLa 1−x B 6 , GinzburgLandau free energy, mode mixing §1. IntroductionFor the last decade, CeB 6 and Ce x La 1−x B 6 have attracted great interest because of their exotic properties originating from orbital degeneracy. Due to strong spin-orbit interaction of f electrons, the entangled spin and orbital degrees of freedom show unusual responses against various external perturbations.One of the mysteries was concerned with so-called phase II: (i) the inconsistency between NMR 1) and neutron scattering 2, 3) measurements on the pattern of the magnetic field induced antiferro (AF) magnetic moment and (ii) the increase of the transition temperature with increase of the magnetic field.1, 2, 4) The latter has been interpreted theoretically in terms of the intersite interaction between the field-induced Γ 2 -type octupole, i.e., T xyz , 5) and the quadrupolar fluctuations suppressed by the magnetic field.6, 7) The former mystery has been resolved by reconsidering the NMR analysis, where the coupling to the induced octupole in the hyperfine interaction is taken into account.8) Through these studies, the AF-quadrupolar ordering has been identified as of Γ 5 type (O yz , O zx , O xy ).In spite of these successful studies, understanding of phase III is still obscure. A complicated magnetic structure with double k, i.e., k 1 = (1/4, 1/4, 1/2) and k 2 = (1/4, −1/4, 1/2) in units of 2π/a with a being the lattice constant, has been established by neutron scattering.2, 3) The non-collinear spin orientation has no gain of energy from the nearest neighbor exchange interactions. Besides, the next-nearest neighbors give the same number of parallel and anti-parallel spins. Thus, the double k structure is hardly stabilized by simple exchange interactions. * E-mail: kusu@cmpt.phys.tohoku.ac.jp Furthermore, a new phase called IV was discovered recently in Ce x La 1−x B 6 with x ∼ 0.75, whose order parameter has not been identified yet. The phase IV has prominent features: (i) a drastic softening of the transverse elastic constant C 44 (∼ 30%), 9) (ii) almost no magneto resistance in contrast with other phases, 10...

Group theory and octupolar order inURu2Si2

“…This would make URu 2 Si 2 the third well-argued case of primary octupolar order in an f -electron system (the first two cases being NpO 2 [21] and Ce 1−x La x B 6 [22]). …”

Theory of Coupled Multipole Moments Probed by X-ray Scattering in CeB6

“…10 This interaction stabilizes phase III with the peculiar pattern of dipole moments. For simplicity, we do not consider such part of dipole interactions that comes from σ i in eq.…”

“…For simplicity, we do not consider such part of dipole interactions that comes from σ i in eq. (3), nor the Γ 5u type pseudo-dipole interaction, which is known to be important for realizing phase III' in magnetic L Author Name field larger than 1 T. 10 Hence discussion of phase III' is out of the scope of this paper. We assume that D 5g is positive and is the largest among all interactions.…”

“…We work with a RKKY type multipole Hamiltonian [8][9][10] which reproduces phases II and III with minimum number of interactions. The model under magnetic field H is given by…”

“…In other words, dipole and octupole moments combine to give η under the cubic symmetry. 9,10 In phase III, on the other hand, simultaneous presence of dipole and octupole orders restrict the direction of µ even without magnetic field. In real CeB 6 , there should be various symmetry breaking perturbations to fix the direction of µ even in phase II.…”