Abstract
The Coulomb phase, with its dipolar correlations and pinch-point–scattering patterns, is central to discussions of geometrically frustrated systems, from water ice to binary and mixed-valence alloys, as well as numerous examples of frustrated magnets. The emergent Coulomb phase of lattice-based systems has been associated with divergence-free fields and the absence of long-range order. Here, we go beyond this paradigm, demonstrating that a Coulomb phase can emerge naturally as a persistent fluctuating background in an otherwise ordered system. To explain this behavior, we introduce the concept of the fragmentation of the field of magnetic moments into two parts, one giving rise to a magnetic monopole crystal, the other a magnetic fluid with all the characteristics of an emergent Coulomb phase. Our theory is backed up by numerical simulations, and we discuss its importance with regard to the interpretation of a number of experimental results.
- Received 3 July 2013
DOI:https://doi.org/10.1103/PhysRevX.4.011007
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Published by the American Physical Society
Popular Summary
Isolating the north or south pole of a fridge magnet is a hopeless task: Cutting such a magnet in half simply yields two scaled-down replicas, each with their own connected north and south poles. Indeed, fundamental laws of electromagnetism strongly suggest that the separation of these poles is impossible. However, recent advances in materials science have yielded surprises on a microscopic scale. In materials known as spin ice, in which atomic-scale magnets sit on particular types of lattice structures, Mother Nature seems to have played a sleight of hand: The tiny magnets can orient themselves collectively into such patterns that the magnetic field they generate looks like that generated by freely moving and independent north and south poles. These effective magnetic “monopoles” have been the subject of intense theoretical and experimental study over the last five years, particularly under conditions when there are so few of them that they behave like a very dilute gas of charged particles. Very little is known, however, about the behavior of a dense gas of such magnetic monopoles.
In this theoretical paper, we demonstrate how a dense gas of equal numbers of north and south monopoles can solidify at low temperature into a monopole crystal in much the same way as electrical charges form ionic solids. Our finding, and understanding, of this monopole crystal leads to a new conceptual picture: magnetic-moment fragmentation. In this picture, the atomic moments appear to effectively “fragment” into two parts. The first part gives the effective monopoles, and the second part is a “leftover” fragment that is free to randomly fluctuate, subject to the overall constraints imposed by Maxwell’s equations of electromagnetism. Magnetic-moment fragmentation has a particularly spectacular manifestation in the monopole crystal we have observed: The set of first fragments collectively acquires the frozen order of a monopole crystal, while the set of second fragments forms a fluctuating and disordered magnetic background reminiscent of a magnetic liquid.
We do not expect magnetic-moment fragmentation to be limited only to monopole crystals. Indeed, as partial ordering in the presence of background magnetic liquid fluctuations appears to be a characteristic feature of many experimental systems, we believe this new conceptual picture will be of broad relevance.
