Geometrically Frustrated Matter—Magnets to Molecules

Abstract

AbstractGeometric frustration occurs when a set of degrees of freedom is incompatible with the space it occupies. A purely geometric example is the impossibility of close-packing pentagons in two dimensions. Another simple example is atomic magnetic moments with antiferromagnetic interactions. These moments lower their interaction energy by pointing antiparallel to their neighbors, an arrangement incompatible with the occupation of a crystal lattice of triangular symmetry. Other manifestations of frustration occur in ice, glass, liquid crystals, and correlated metals. Because frustration governs the rules of packing, examples are also found in biological materials, as in the self-assembly of liposomes that form nanotubules. Geometric frustration is essentially “many-body” in nature: the basic concept is trivial on the scale of three particles, but complex and anharmonic for an Avogadro's number of particles. In fact, geometrically frustrated systems are so anharmonic that no general theoretical framework exists to explain their collective behavior. This article will explore the basic concepts of geometric frustration and illustrate these concepts with examples from magnetism, crystal structures, and molecular systems.