Fig. 1 | Nature Communications

Fig. 1

From: Fragmentation in spin ice from magnetic charge injection

Fig. 1

Staggered field in a pyrochlore lattice and magnetic fragmentation. a Pyrochlore lattice submitted to 〈111〉 staggered field represented by the arrows on each site, with in and out tetrahedra when this field points either inward or outward of the tetrahedra. b Structure of the pyrochlore iridate Ho2Ir2O7 with the Ho3+ ions (red) and the Ir4+ ions (blue), both occupying a pyrochlore lattice. Each rare-earth is surrounded by a hexagon of six Ir nearest neighbors. When the iridium lattice orders magnetically in the AIAO phase, as shown on the blue lattice, a local magnetic field, perpendicular to the hexagon plane, hence aligned along the local 〈111〉 directions, is felt by the central rare-earth ion, as represented in a. c Phase diagram as a function of reduced temperature \(T{\rm{/}}{{\cal J}_{{\rm{eff}}}}\) and local field \({h_{{\rm{loc}}}}{\rm{/}}{{\cal J}_{{\rm{eff}}}}\), defined through the charge order parameter \({M_q} = \left\langle {\left| {\frac{1}{N}\mathop {\sum}\nolimits_{\alpha = 1}^N {{{\it{\Delta }}_\alpha }{q_\alpha }} } \right|} \right\rangle \)where N is the number of tetrahedra and Δ α  = +1 (−1) for in (out) tetrahedra carrying a charge q α . As \({h_{{\rm{loc}}}}{\rm{/}}{{\cal J}_{{\rm{eff}}}}\) increases, the ground state changes from the spin ice (M q  = 0, blue), to the AIAO state (M q  = 2, yellow), going through the charge crystal intermediate phase which fragments (M q  = 1, orange). The fragmentation can be described by the splitting of the pseudo-spin σ = ±1: the divergence free fragment on a tetrahedron can then be written as a state with four spins such that \({\sigma _{\left\langle {\left| q \right|} \right\rangle = 0}}\) = (±1/2, ±1/2, ±1/2, and 3/2), while the divergence full fragment that carries the magnetic charge corresponds to an AIAO configuration with half of the moment so that \({\sigma _{\left\langle {\left| q \right|} \right\rangle = 1}}\) = (±1/2, ±1/2, ±1/2, ±1/2)8. All the spin and charge configurations are shown in the left panel of c at T = 0. d Magnetic scattering function S(Q), with Q the scattering vector, calculated in the fragmented phase. It shows a diffuse pattern with pinch points (evidenced by arrows) together with the magnetic Bragg peaks (highlighted by circles) characteristic of the AIAO ordered state, but whose intensity (proportional to the magnetic moment squared) is a quarter of that expected when the whole magnetic moment is ordered

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