Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

In rare-earth borides we encounter many cases where not only the magnetic dipole but also the electric quadrupole and magnetic octupole are involved in the ordering phenomena, as summarized in several recent review articles [1–4]. A typical case is the rich magnetic phase diagram of cubic Ce _x La_1−x B₆ with the Г₈ quartet crystal-field ground state. Among many experimental and theoretical efforts dedicated to observe and establish the quadrupole and octupole ordered states, in this chapter we focus on the role of resonant x-ray diffraction. Another topic in this chapter is a competition of order parameters and ordered structures in tetragonal rare-earth tetraborides, RB₄. With the so-called Shastry–Sutherland lattice, a network composed of orthogonal dimer bonds, the magnetic structures of RB₄ result in a macroscopic degeneracy due to geometrically frustrated exchange interaction. In this series of compounds, interesting successive ordering phenomena are observed, which is possibly associated with a competition of magnetic and quadrupolar order parameters. Another competition among the geometrically frustrated short-ranged interactions, longranged RKKY interaction, and the Zeeman energy in magnetic fields, leads to an emergence of interesting and unpredictable arrangements of magnetic moments, manifested in the appearance of fractional magnetization plateaus.