Geometrical Frustration

© 2006 American Institute of Physics, S-0031-9228-0602-010-2 T ancient Greeks were aware of the phenomenon of magnetic order in lodestone, a type of rock containing the ferromagnet magnetite Fe3O4. Magnetic moments in a ferromagnet tend to align and thereby sum to an easily observed macroscopic magnetic moment. The absence of such a moment even in ordered antiferromagnets is the reason their discovery is comparatively recent. It had to await the development of Louis Néel’s microscopic theory of spin interactions in the 1930s and the neutron diffraction measurement of MnO in 1949 by Clifford Shull and Stuart Smart. There are magnets, however, that today present greater experimental and theoretical challenges than those posed by simple antiferromagnets in the 1930s.1 The origin of their complex and varied behavior is remarkably simple and can be illustrated by as few as three spins on a triangular lattice. Once two of the spins on an elementary triangle are antialigned to satisfy their antiferromagnetic interaction, the third one can no longer point in a direction opposite to both other spins (see figure 1). Thus, not all interactions can be minimized simultaneously—that is, exist in their lowest energy state. In other words, antiferromagnetic interactions are incompatible with triangular lattice symmetry, a situation known as geometrical frustration. The antiferromagnetic triangle is the simplest case in which a conflict arises between the geometry of the space inhabited by a set of degrees of freedom and the local correlations favored by their interactions. This phenomenon is one aspect of a powerful paradigm for discovery over the past few decades—namely, our ability to experimentally manipulate the space in which magnetic, charge, or vibrational degrees of freedom interact. Two other particularly well-studied aspects are low effective dimensionality for electronic systems and tunable optical lattices for systems of cold atoms. The study of geometrically frustrated magnets is concerned with what happens when lattice geometry inhibits the formation of a simple, ordered, low-temperature spin configuration. Typically, geometrical frustration gives rise to a degenerate manifold of ground states rather than a single stable ground-state configuration, leading to magnetic analogues of liquids and ice. Not surprisingly, even slight perturbations induce instabilities in such systems and prompt the emergence of further unusual phenomena, even including an incarnation of artificial electrodynamics in which the frustrated magnet acts as an “ether” for novel magnetic excitations. Frustrated magnets thus lie at the crossroads of two fundamental enterprises in condensed matter physics. On the applied side, the instabilities exhibited by frustrated magnets open a window on the richness of nature realized in different materials. On the fundamental side is the search for principles that help organize the variety of behavior we observe around us. This article addresses two possible principles: Underconstraint, which here arises for spins residing on a weakly connected lattice whose geometry frustrates their mutual interactions, and emergence, the dynamical generation of new types of degrees of freedom.