Page 71 - ILL Annual Report 2019

SCIENTIFIC HIGHLIGHTS
68-69
2
Figure 2. (a-c) Choice of all 3 inequivalent low-energy configurations of a 2-tetrahedron system hosting only one monopole
a) b) c)
Figure 2
(red sphere). Both dipolar and exchange fields on the central site 0 due to its j = 1,2,...,6 n.n. are purely transverse to The three inequivalent, low-energy configurations of a 2-tetrahedron
z0 / h111i in 2/3 of the cases (a,b – vectors in green), and are identically null in the remaining 1/3 (c). (d) Histogram of the system hosting only one monopole (red sphere). Effective fields on the
dipolar fields resulting on 0 from its 6 n.n. spins (left panel), and from the inclusion of further 18 spins in the n.n.n tetrahedra central spin due to its nearest neighbours spins are purely transverse
(right panel). This verifies that the bimodal distribution (2:1) is largely unaffected when we consider an 8-tetrahedron system
(green arrows) to the central axis of anisotropy in (a) and (b), and (24 surrounding spins).
identically null in (c).
literature [7, 8] as foundation. We study the local dy-
in these systems of coherent and incoherent processes. Model — Spin ices are magnets where anisotropic
namics of emergent monopole excitations, which has a
3-in/1-out or 3-out/1-in spin configurations (a few 3-in/1-
As transverse fields induce spin-tunnelling, these results
quantum mechanical (tunnelling) origin, rooted in the
Ising-like spins reside on a pyrochlore lattice of corner-
out examples are shown in figure 2). These defects are transverse terms of the dipolar and exchange interac-
imply that for a ‘flippable’ spin next to a monopole two
called magnetic monopoles, because they act as sources
very distinct rates,τfast and τslow, appear with a 2:1 ratio. tions between the spins are largely frustrated, and at low
Tthemepspeirnatduyrneasmtihcesgisrothuenredfosrteatremisadrkeasbcrlyibceodrreblyataendewxithen-
tshievelloycadlegeennvierroantmeemntaannifdoltdheofprcopnafigautrioatnionf smobneoypinoglesthe
tions between rare-earth (RE) ions [18, 19]. We focus
oannedlassintkics opfromcaegssneestis(amtionoflpuoxleshinopthpeinogt)h,esrwinicse ‘ivnaeclausutmic’
oonfesth(em2o-ino/p2o-oleutcrspeaintibona/cakgnrnoiuhnildat[i2o]n.) are suppressed at
low temperatures. The key question is: how do the pre-
so called ‘ice rules’ (in each tetrahedron, 2 spins point
The success of this formulation aside, magneto-dynamics
is intrinsically stochastic. Contrary to conventional
dominantly off-diagonal terms (‘transverse fields’) nec-
‘in’, towards its centre, and 2 point ‘out’) [7]. The lowest modelling, which assumes a uniform distribution of
are nevertheless poorly understood [3]. A comprehensive
essary to induce monopole hopping arise in a material
excitations above such ground state are effective mag-
understanding of experimental relaxation time scales is still
hopping amplitudes, this opens up the study of monopole
whose statistics are excellently described by a classical
netic monopoles with Coulomb interactions [8]. dynamics accounting for correlated statistics as well as
lacking and the detection of a temperature-independent
Ising model? Our central result is that there is a fun-
The local Ising anisotropies originate from the strong
draemgiemnetails tfheedobnalyckcomemchoanndisemnombeintwateoernamspoingdayllnamics,
mporonboepsol[e4]q.uAasitphaisrtiiscalesc,riabnedtthoeqluoacanltusmpisnpein-vtuinrnoenlmlinegnt.
Wduherteoams tohneopvoaslet hmoapjpoirnitgy, owfespbianseidnotuhrethseaomryploenexspienri-
percolation effects which may also play a prominent role
ence longitudinal fields, which justify a classical descrip-
Tthael atinmgeulsacramleosmobentatiunmedqfuoarnmtounmopnoulmebmeortios,nraesrepectively, J = 8,15/2 and 4) [19, 20]. Low energy dynamics be-
fluctuations originating from localised, isolated monopoles.
found to be slower than expected decoherence times.
tion, some of the spins adjacent to a monopole experience
tween the single-ion states of the ground-state doublet,
Our model is material-dependent, with the crystal-field
This reconciles the view of spin-ice monopoles as
predominantly transverse fields. As illustrated in Fig. 2,
|−i and |+i (labelled by S = −1,1), necessarily in- i
Hamiltonian being the ‘foundation’ for setting up the Hilbert
clasisically diffiusing quasi-particles with the fact that
a monopole has 3 available lattice bonds to hop across,
volve transitions via the CEF excited states, with ener-
space, and with the dipolar and exchange energy-terms as
their motion requires quantum fluctuations. We propose
and, statistically, we find a bimodal distribution of trans-
gies E & 102 K [21, 22]. In the temperature range a model based on the quantum Zeno effect, where the where the monopole description is valid (T . 1 K), ther- hopping is effected by decoherence.
the ‘building blocks’ of interactions for creating dynamics
verse fields, and thence of quasiparticle hopping rates, in
and realistic estimation of timescales thereof.fast slow ratio 2:1 (fast:slow). We posit that these ⌧ and ⌧
mal activation of CEF excited states is negligible so that quantum tunnelling must underpin the spin dynamics [9].
are the fundamental (tunnelling) timescales underlying
Our principal finding is that there is a fundamental
The quantitative benefit of this theory is intimately
This provides a mechanism for the flipping of the minor-
a broad range of dynamic phenomena in spin ices [9–16] correlation between spin dynamics, monopole excitations
related to neutron scattering in condensed matter. The
and find they are consistent with experimental timescales
ity of spins that are not frozen by a local (longitudinal)
and their local environment. Indeed, in the case of low
best determinations of the crystal field Hamiltonian
(Table II). We extend our calculations to ‘quantum spin
combined dipolar and effective exchange field. These are
densities of monopoles, whereas the vast majority of spins
are obtained thanks to the inelastic interaction of
ices’ Pr2 Sn2 O7 (PSO) and Pr2 Zr2 O7 (PZO) and find, as in the sample experience net longitudinal fields, those
of course the flippable spins next to a monopole.
expected, much faster timescales and also, more surpris-
the spin-1⁄2 magnetic moment of the neutron with the
adjacent to a monopole have such longitudinfasltcomponsleonwts ingly, much greater separation between ⌧ and ⌧ .
We focus on a given spin, say at i = 0, to study the magnetic moments of a given compound. Our approach single spin-flip dynamics which amounts to the hopping
suppressed. Moreover, as illustrated in figure 2, the same Finally we argue that decoherence may play an essential
demonstrates how theoretical analyses can tailor the
monopole induces, statistically, a bimodal distribution of role in the emergence of slow, classical spin flips out of
of a monopole. Our Hamiltonian,
fatrsatn, sqvuerasnetufiemldtsu: nspnelclifincgaallyn,dwwiteh ptwrov-tihdierdas psirmobpalebimlityodtheel bcaesnetdraol nsptihneexZpeenroieenffcecsta. Ifintiteretrsatninsvgelyrsethfieeldarogfesstrepnagrtah- ti~on0.o3f5timteselsacsa(lfeisgiunrPeSsO2anadnPdZ2Ob)im,wplhiielesawcitoheoxnseit-ethnircde
probability the field vanishes (figure 2c).
ˆˆˆˆ
conventionaHl (e0x)p=ecHtatCioEnFs.+ Hdip(0) + Hexc(0), (1)
describes a RE-ion at site 0 of an N-site pyrochlore sys- tem. Hˆ acts on the Hilbert space of the RE3+ of inter-
sharing tetrahedra. The exchange and dipolar interac-
crystalline-electric-fields (CEF) acting on the J-manifold in understanding the so-called quantum spin ices.
of the RE3+ ions (for Ho3+, Dy3+ and Pr3+ ions, the to-
quantitative prediction of neutron studies well beyond
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