Observation of magnetic fragmentation in spin ice

Abstract

The physics of spin ice materials is intimately connected with the pyrochlore lattice, composed of corner-sharing tetrahedra. On the corners of these tetrahedra reside rare-earth magnetic moments J i , which, as a consequence of the strong crystal electric field, are constrained to point along their local trigonal axes z i , and behave like Ising spins. The magnetic interactions are composed of nearestneighbour exchange J and dipolar interactions between spins i and j separated by a distance r ij (ref. 7):wherenn ), µ 0 is the permeability of free space, g J is the Landé factor of the magnetic moment, µ B is the Bohr magneton and r nn is the nearest-neighbour distance between rareearth ions. The nearest-neighbour spin ice Hamiltonian is obtained by truncating the Hamiltonian (1), yielding:When the effective interaction J eff = (−J + 5D)/3 is positivethat is, when the dipolar term overcomes the antiferromagnetic exchange-a very unusual magnetic state develops, known as the spin ice state. The system remains in a highly correlated but disordered ground state where the local magnetization fulfils the so-called 'ice rule': each tetrahedron has two spins pointing in and two spins pointing out (see Fig. 1a), in close analogy with the rule which controls the hydrogen position in water ice 8 . The extensive degeneracy of this ground state results in a residual entropy at low temperature which is well approximated by the Pauling entropy for water ice 9 . Such highly degenerate states, where the organizing principle is dictated by a local constraint, belong to the class of Coulomb phases 5,10,11 : the constraint (the ice rule for spin ice) can be interpreted as a divergence-free condition of an emergent gauge field. This field has correlations that fall off with distance like the dipolar interaction 12,13 . In reciprocal space, this power-law character leads to bow-tie singularities, called pinch points, in the magnetic structure factor. They form a key experimental signature of the Coulomb phase physics. They have been observed by neutron diffraction in the spin ice materials Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7 , in excellent agreement with theoretical predictions 14,15 . Classical excitations above the spin ice manifold are defects that locally violate the ice rule and so the divergence-free condition: by reversing the orientation of a moment, 'three in-one out' and 'one in-three out' configurations are created (see Fig. 1b). Considering the Ising spins as dumbbells with two opposite magnetic charges at their extremities, such defects result in a magnetic charge in the centre of the tetrahedron, called a magnetic monopole, that give rise to a non-zero divergence of the local magnetization 4 . Recently, theoreticians have introduced the concept of magnetic moment fragmentation 6 , whereby the local magnetic moment field fragments into the sum of two parts, a divergence-full and a divergence-free part (see Fig. 1c): for example, a monopole in the spin configuration m = {1, 1, 1, −1} on a tetrahedron can be written m = 1/2{1, ...

Antiferroquadrupolar correlations in the quantum spin ice candidatePr2Zr2O7

“…It can be described by a flat mode, whose width might be induced by inhomogeneities in the sample. This observation reminds the case of the kagome antiferromagnet KFe 3 (OH) 6 (SO 4 ) 2 [53], and more recently the pyrochlore system Nd 2 Zr 2 O 7 [54], in which an inelastic flat mode was interpreted as a zeroenergy mode (the kagome weather vane mode and the spin ice pattern, respectively) lifted up to finite energy by an additional term in the Hamiltonian (a Dzyaloshinskii-Moriya term and an octopolar term, respectively).…”

Spin correlations in the dipolar pyrochlore antiferromagnet Gd2Sn2O7

“…The diversity of magnetic states observed in this class of materials includes spin ices [1,2], spin liquids [3,4], fractionalised order [5], and complex long-range order [6,7]. In general terms, the common origin of these states is the low connectivity of the tetrahedral building-blocks of the pyrochlore lattice [ Fig.…”

Clustering of Topological Charges in a Kagome Classical Spin Liquid

“…Below T c , where the kagome ice correlation is well developed, pinchpoint singularities are observed [ Fig. 3(b)], coexisting with the Bragg peaks of ferrimagnetic order [36,37,[44][45][46].…”