Magnetic-Moment Fragmentation and Monopole Crystallization

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

Out-of-equilibrium dynamics and extended textures of topological defects in spin ice

“…The FCSL has been predicted theoretically 40,41 and observed experimentally [42][43][44] in nano-lithographic artificial kagome ice whose geometry prevents the existence of a charge-free Coulomb phase 45 . But in three dimensions, it has so far only been partially stabilized at equilibrium in the spin-ice model with dipolar interactions 46 , or requires four-body interactions 39 or the suppression of double charges 37,38 . The nonequilibrium magnetic-field quench proposed here provides a promising tool to realize this state and demonstrates the possibility of engineering a macroscopic state via nonequilibrium techniques 47 .…”

“…As a result, charge relaxation becomes extremely slow, while the magnetization decays rather quickly. The system finally forms the co-called fragmented Coulomb spin liquid (FCSL) [37][38][39] . In the FCSL charges are long-range ordered but the spin texture remains disordered, extensively degenerate and described by a Coulomb gauge theory.…”

“…These values correspond to the plateaux observed in Fig. 12 With the peculiar exceptions of models where double charges were explicitly forbidden 37,38 , and 2D artificial kagome ice systems [40][41][42][43][44] , the FCSL has been noticeably elusive at equilibrium in 3D spin-ice models 46 , despite its deceptive simplicity.…”

“…But as opposed to the all-in / all-out order, the magnetic "crystal" of the FCSL phase is made of single charges, which retains an extensive spin degeneracy. It also supports an emergent Coulomb gauge field due to a local divergencefree constraint on the magnetization flux 38 . When averaged over its Gibbs ensemble, the FCSL phase does not bear any finite total magnetization.…”

“…14]. This corresponds to the fragmented Coulomb spin liquid 37,38 where long-range charge order co-exists with extensive spin degeneracy.…”