Magnetic clustering, half-moons, and shadow pinch points as signals of a proximate Coulomb phase in frustrated Heisenberg magnets

TL;DR

This paper investigates magnetic clustering phenomena in frustrated Heisenberg magnets near Coulomb phases, revealing characteristic half-moon and star patterns in structure factors that can be observed via neutron scattering.

Contribution

It demonstrates how farther-neighbor couplings induce magnetic clusters and characteristic scattering patterns, connecting these features to Coulomb phase proximity in classical Heisenberg models.

Findings

Half-moon patterns in structure factors develop with increasing coupling.

Connected star structures emerge, spanning multiple Brillouin zones.

Pinch point shadows persist at finite energy, observable in experiments.

Abstract

We study the formation of magnetic clusters in frustrated magnets in their cooperative paramagnetic regime. For this purpose, we consider the $J_1$-$J_2$-$J_3$ classical Heisenberg model on kagome and pyrochlore lattices with $J_2 = J_3=J$. In the absence of farther-neighbor couplings, $J=0$, the system is in the Coulomb phase with magnetic correlations well characterized by pinch-point singularities. Farther-neighbor couplings lead to the formation of magnetic clusters, which can be interpreted as a counterpart of topological-charge clusters in Ising frustrated magnets [T. Mizoguchi, L. D. C. Jaubert and M. Udagawa, Phys. Rev. Lett. {\bf 119}, 077207 (2017)]. The concomitant static and dynamical magnetic structure factors, respectively $\mathcal{S}({\bm{q}})$ and $\mathcal{S}({\bm{q}},\omega)$, develop half-moon patterns. As $J$ increases, the continuous nature of the Heisenberg spins enables the half-moons to coalesce into connected `star' structures spreading across multiple Brillouin zones. These characteristic patterns are a dispersive complement of the pinch point singularities, and signal the proximity to a Coulomb phase. Shadows of the pinch points remain visible at finite energy, $\omega$. This opens the way to observe these clusters through (in)elastic neutron scattering experiments. The origin of these features are clarified by complementary methods: large-$N$ calculations, semi-classical dynamics of the Landau-Lifshitz equation, and Monte Carlo simulations. As promising candidates to observe the clustering states, we revisit the origin of "spin molecules" observed in a family of spinel oxides $AB_2$O$_4$ ($A=$ Zn, Hg, Mg, $B=$ Cr, Fe).