Composite frustration

Figure 1 The organization of spins in antiferromagnetic materials. When located at the vertices of a square, the alignment of neighbouring antiparallel spins is straightforward (a). But when located at the vertices of a tetrahedron, such alignment is impossible, which results in a frustrated system (b). In ZnCr₂O₄, ordering in antiparallel hexagonal loops develops, and composite degrees of freedom occur in the form of 'spin directors' (c). The coloured arrows indicate the alignment of the directors.

Frustrated magnetic materials — in which there is little or no long-range ordering between magnetic spins in the ground state — represent a subset of frustrated systems that occur throughout nature. Despite their ubiquity, however, their behaviour is poorly understood. Writing in this week's Nature, Seung-Hun Lee and colleagues report neutron-scattering measurements of the frustrated magnetic spinel ZnCr₂O₄, which demonstrate the self-organisation of hexagonal spin clusters to form composite degrees of freedom.

In most magnetic materials, the lowest-energy ordering of individual magnetic moments, or spins, is unique. In a ferromagnetic material, this ordering is characterized by having all spins pointing in the same direction. Conversely, in an antiferromagnetic material, the ordering is characterized by having all neighbouring spins pointing in opposite directions. The uniqueness of these spin arrangements means that the ground state in both cases is non-degenerate. And when such systems are cooled, long-range order arises as the orientation of their individual spins converge on the ground state.

But there are magnetic materials whose spins cannot be arranged in such an orderly or thermodynamically unique way. Consider an antiferromagnetic material whose lowest energy constraints are that neighbouring spins are antiparallel, and that their net spin must be zero. If these spins are located at the vertices of a square array, the relative antiparallel orientations of neighbouring spins is clearly defined (see Fig. 1a). But if they are located at the vertices of a tetrahedron, the antiparallel arrangement of all four neighbouring spins is geometrically impossible (see Fig. 1b) — a situation that is known as 'geometrical frustration'.

Because of the unlimited number of arrangements that satisfy a net spin of zero — and none that allow all neighbouring spins to be antiparallel — the ground state of a frustrated magnet is strongly degenerate. As a result of this, and the quasi-random orientation of spins over scales greater than nearest neighbour clusters, the emergence of medium- and long-range order in such systems is not considered possible, irrespective of temperature.

By carrying out neutron-diffraction measurements on ZnCr₂O₄, Lee et al. have found that ordering in this frustrated magnetic system is not limited to the relative orientation of nearest neighbour spins centred around individual tetrahedra, but also occurs within hexagonal rings defined by six different tetrahedra. As the material is cooled, the spins within these rings self-organize into loops in which all six spins are oriented antiparallel to neighbouring members of the loop.

The organization of these spins into antiferromagnetic loops with well-defined orientation directions results in an effective reduction of the number degrees of freedom of the ZnCr₂O₄ system, with each loop representing a composite degree of freedom that the authors refer to as a 'spin director'. By leading to an understanding of frustrated magnetic systems in terms of the interaction between these spin directors instead of between individual spins, the discovery of such composite degrees of freedom could help explain a number of anomalous aspects of their behaviour.

letters to nature
Emergent excitations in a geometrically frustrated magnet
S.-H. LEE, C. BROHOLM, W. RATCLIFF, G. GASPAROVIC, Q. HUANG T. H. KIM & S.-W. CHEONG
Frustrated systems are ubiquitous, and they are interesting because their behaviour is difficult to predict; frustration can lead to macroscopic degeneracies and qualitatively new states of matter. Magnetic systems offer good examples in the form of spin lattices, where all interactions between spins cannot be simultaneously satisfied. Here we report how unusual composite spin degrees of freedom can emerge from frustrated magnetic interactions in the cubic spinel ZnCr₂O₄. Upon cooling, groups of six spins self-organize into weakly interacting antiferromagnetic loops, whose directors—the unique direction along which the spins are aligned, parallel or antiparallel—govern all low-temperature dynamics. The experimental evidence comes from a measurement of the magnetic form factor by inelastic neutron scattering; the data show that neutrons scatter from hexagonal spin clusters rather than individual spins. The hexagon directors are, to a first approximation, decoupled from each other, and hence their reorientations embody the long-sought local zero energy modes for the pyrochlore lattice.
Nature 418, 856–858 (22 August 2002)
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