SWISS NEUTRON NEWS Number 56 | November 2020 - Swiss Neutron Scattering Society SGN ...

SWISS
 Number 56 | November 2020

NEUTRON
NEWS

 SSSN

 Schweizerische Gesellschaft für Neutronenforschung
 Société Suisse de la Science Neutronique
SGN Swiss Neutron Science Society

2

On the cover
Artist’s view of a liquid of octupoles, see the related article "Neutron scattering by
magnetic octupoles of a quantum liquid" by N. Gauthier, V. Porée, S. Petit,
V. Pomjakushin, E. Lhotel, T. Fennel, and R. Sibille.

3

Contents

 4 The President‘s Page

 6 Neutron scattering by magnetic octupoles of a quantum liquid

20 Expansion of experimental infrastructure at HRPT:
 Stroboscopic neutron diffraction

38 Announcements

40 Winners of the Young Scientist Prize 2020 of the Swiss Neutron
 Science Society, sponsored by SwissNeutronics

 41 Young Scientist Prize 2021 of the Swiss Neutron Science
 Society, sponsored by SwissNeutronics

42 Conferences and Workshops

47 Editorial

4

The President’s Page

Dear fellow neutron scientists, recently engaged with the EU funded Bright-
 nESS2 project to analyze the European neu-
Welcome to this issue of Swiss Neutron News. tron science community with the aim of ex-
Let me start by thanking those who joined our trapolating trends and needs into the future.
recent general assembly. Being forced to hold As part of this, we used natural language
it online prevented us from enjoying the com- processing and machine learning to analyze
pany of colleagues and friends around an neutron publications to identify domains and
apero, but had the advantage that everyone trends within the neutron science community.
could join without need to travel. We had two We coupled this to a survey that all of you
exciting presentations from the 2020 Young received, and I thank those of you who took
Scientist Prize recipients Muriel Siegwart and the time to answer. Within the coming year
Jiri Ulrich, who gave very exciting presenta- receive a second survey focused specifically
tions at our recent general assembly. I take on the Swiss neutron science community and
the opportunity to congratulate them once impact. We are currently in an exciting time
more on their achievements. Muriel reported where SINQ has just been upgraded and ESS
sophisticated neutron imaging techniques for will come on-line, but where access to ILL will
studying fuel cells and other energy solution remain vitally important to many Swiss neu-
materials. Jiri's work on high precision nuclear tron scientists. I therefore kindly ask you to
data for astrophysics and geosciences repre- help in these efforts to map and shape the
sent the fields of fundamental physics using future of our scientific possibilities.
neutrons, which we wanted to encompass On the topic of SINQ's upgrade, it is my
with the name change from Swiss Neutron understanding it has been extremely success-
Scattering Society to Swiss Neutron Science ful with flux gains of 2 or better on most in-
Society. In short, the scope of our interest struments. This is a great achievement, which
sphere is science that relies on neutron everybody involved should be very proud of.
sources. Another point where we as user community
 On this topic, our umbrella organization can be grateful not just to PSI but to many of
ENSA, which I chair since beginning of 2020, the neutron facilities is their efforts to enable

5

experiments during this unprecedented pan- we gratefully thank them for taking on this
demic. The facilities are offering mail-in and extra burden, but in the longer term this
remote experiments, which is extremely val- means we must make sure that the operation
uable to the scientific community, especially budgets of facilities are adequate to optimize
PhD students and postdocs who need results the total science impact.
for their next career steps. I believe such
neutron access solutions can be very benefi- I wish you all a healthy autumn and hope
cial for the total science output and for the we can soon meet again in experimental
traveling footprint of science. However, it halls and conference theaters,
places increased work load on the instrument
scientists and facility staff. In the short term Henrik M. Rønnow

6

Neutron scattering by magnetic
octupoles of a quantum liquid

Nicolas Gauthier1,2, Victor Porée1, After R. Sibille et al. Nature Physics 16, 546-552
Sylvain Petit3,*, Vladimir Pomjakushin1,*, (2020)
Elsa Lhotel4, Tom Fennell1 &
Romain Sibille1,*
 Neutron scattering is a powerful tool to study
1 Laboratory for Neutron Scattering and Ima- magnetic structures and dynamics, benefiting
ging, Paul Scherrer Institut, 5232 Villigen from a precisely established theoretical
PSI, Switzerland. framework. The neutron dipole moment in-
2 Stanford Institute for Materials and Energy teracts with electrons in materials via their
Science, SLAC National Accelerator Labora- magnetic field, which can have spin and or-
tory and Stanford University, Menlo Park, bital origins. Yet in most experimentally stud-
California 94025, USA. ied cases the individual degrees of freedom
3 LLB,
 CEA, CNRS, Université Paris-Saclay, are well described within the dipole approx-
CEA Saclay, 91191 Gif-sur-Yvette, France. imation, sometimes accompanied by further
4 InstitutNéel, CNRS–Université Joseph terms of a multipolar expansion that usually
Fourier, 38042 Grenoble, France. act as minor corrections to the dipole form
* email:
 romain.sibille@psi.ch ; sylvain.pe-
 factor. Here we report a unique example of
tit@cea.fr ; vladimir.pomjakushin@psi.ch neutrons diffracted mainly by magnetic octu-
 poles. This unusual situation arises in a quan-
 tum spin ice where the electronic wavefunc-
 tion becomes essentially octupolar under the
 effect of correlations. The discovery of such
 a new type of quantum spin liquid that comes
 with a specific experimental signature in
 neutron scattering is remarkable, because
 these topical states of matter are notoriously
 difficult to detect.

7

ON MAGNETIC NEUTRON pole moments. The calculations of these
SCATTERING higher-order contributions to the scattered
 neutron intensity is mathematically quite in-
Neutrons, thanks to their spin, are employed volved and requires the use of spherical ten-
to discern materials’ magnetic properties. sors and Racah tensor algebra with the de-
They are sensitive to the magnetic field pro- tailed procedures given in [1]. Despite this
duced by unpaired electrons, which can have complexity, we can identify two main charac-
various symmetries and properties depending teristics expected from the conventional mag-
on the particular atom and its crystal field netic multipoles based on the neutron-elec-
environment. The magnetization density orig- tron interaction. First, conventional magnetic
inates from both the spin and orbital distri- multipoles give significantly smaller contribu-
butions of open shell electrons and can be tion to neutron scattering than the dipole one.
expanded in multipoles with the use of spher- Second, their form-factor is zero at q=0 with
ical harmonics. In a vast number of cases, a maximum at relatively high q, and is also
neutron scattering results are well accounted anisotropic. The expected signatures of mag-
for by considering the magnetic dipole mo- netic multipoles are therefore a weak aniso-
ment of the atom – a parity-even tensor of tropic signal at high q, making their experi-
rank 1 (axial vector). The tensors of higher mental detection a real challenge.
odd-rank K are conventional magnetic Multipoles that are observable in neutron
multipoles such as the octupole (K=3) and scattering must be odd under time-reversal
the triakontadipole (K=5). However, their con- symmetry. This includes the conventional
tribution to the neutron scattering form factor magnetic multipoles, which are parity-even
is usually marginal in comparison to the di- multipoles and are the main topic of this ar-
pole moment. ticle. We note, however, that parity-odd
 The interaction of the magnetic multipole multipoles can also exist and be observed if
degrees of freedom with the neutron spin are the atomic wavefunction does not have a
described by the neutron-electron interaction well-defined parity [2-4]. These parity-odd
operator Q=exp(iqr)(s–ih/q2 [q×p]) [1]. This multipoles are fundamentally different from
operator can be expressed using spherical the conventional (parity-even) magnetic
Bessel functions jn (q) in powers of (qr)m multipoles. For example, the first order pari-
(m=0,1,2…), where q is the neutron scattering ty-odd multipoles are called anapoles, or
wavevector, and s, r and p are the spin, posi- toroidal moments, which are the cross prod-
tion and momentum of the electron. The first ucts of spin s or orbital l angular momentum
two leading terms in this expansion give the with the electron position r.
so-called ‘dipole approximation’. The domi-
nating contribution to the neutron scattering MULTIPOLES IN CONDENSED
is given by the conventional radial integral MATTER RESEARCH
‹ ›
 j0 (q) , which has maximum at q=0. Higher
terms in the expansion of Q contain the con- Although a vast majority of materials with
tribution of the conventional magnetic multi- strong electronic correlations can be well

8

understood based on individual degrees of poles was established, indirectly, using res-
freedom described using the first term of the onant X-ray diffraction through the measure-
multipolar expansion, further terms are re- ment of a parasitic order of electric
quired to explain an increasing number of quadrupoles having the same structure
novel phenomena. Such multipole moments [14,15]. In cerium hexaboride, the first corre-
can in principle lead to the emergence of lated phase entered upon cooling in zero field
macroscopic orders that are sometimes called below TQ = 3.4 K is an antiferroquadrupolar
‘hidden’ due to the challenge of determining order, which is followed by an antiferromag-
their order parameter [5-6]. Multipoles in netic order of dipoles at TN = 2.3 K [16,17]. The
condensed matter correspond to anisotropic dipole-dipole nature of the intersite magnetic
distributions of electric and magnetic charges interactions make the associated correlations
around given points of the crystal structure more resistant against disorder than for the
– a situation that can arise at the atomic scale electric quadrupoles, so that TQ decreases
from spin-orbit coupling, such as for the faster than TN upon diluting the cerium lattice
multipoles proposed to explain a famously with lanthanum [18]. At some doping level,
mysterious phase in the heavy-fermion mate- these phase transitions intersect and a new
rial URu2Si2 [7], or at the scale of atomic phase pocket appears, characterized by an
clusters where the established correlations antiferromagnetic ordering of octupoles that
lead to the emergence of novel degrees of was measured directly using resonant X-ray
freedom, such as in the spin-liquid regime of diffraction [19].
Gd3Ga5O12 [8]. The contribution of parasitic magnetic
 As already noted, neutrons are also sen- octupoles to the total magnetic scattering
sitive to odd-parity multipoles, and a number intensity – of mainly dipole origin, is well
of studies have pointed to their role to explain known in materials such as elemental hol-
phase transitions that break both space in- mium for instance [20]. However, the exper-
version and time reversal. This has been es- imental results for Ce0.7La0.3B6 presented
pecially discussed in the context of magne- in ref. [21], where magnetic multipoles are
toelectric insulators [9-10], and in high-TC the primary order parameter, is to the best
superconductors where magneto-electric of our knowledge a unique example in terms
quadrupoles were proposed as the order of ordered magnetic multipoles scattered
parameter of the transition appearing in the by neutrons. Only three independent mag-
pseudogap region [11-12]. netic Bragg peaks were detected, but the
 Well characterized examples of ‘hidden’ fact that the intensity at q=6 Å-1 is larger than
orders of (conventional) multipoles exist, such at q=1.3 Å-1 has led the authors to argue
as in NpO2 or CeB6 and its substitutional alloys that these have octupolar origin. A later
Ce1−xLaxB6 [5-6]. In neptunium dioxide [13], the theoretical study [22] qualitatively confirms
primary order parameter is associated with that the q-dependence of the observed
magnetic octupoles that order around 25 K in peaks agrees with the calculated octupolar
a longitudinal structure defined by three neutron magnetic form factor of cerium in
wavevectors. This structure of ordered octu- this material.

a bb c 9
 150 meV
 4

 Intensity (arb. u.)
 a bb c
 2
 150 meV 450 meV Figure 1
 4
 2 Inelastic neutron scattering (INS) data
 2
Intensity (arb. u.)

 0
 of Ce2Sn2O7 probing the crystal-elec-
 40 60 80 100 120 tric field levels within the ground mul-
 EE (meV)
 (meV) E (meV)
 e tiplet 2F5/2 of Ce3+ (a) [29]. The bulk
 2 magnetic susceptibility χ [28], shown
 Figure 11Inelastic neutron scattering (INS) 1 data of Ce2Sn2O7 probing the crystal-electric field
 levels within the ground multiplet F5/2 of Cein
 2 3+ blue as the effective magnetic mo-
 (a) [29]. The bulk magnetic susceptibility [28],
 shown in blue as the effective magnetic moment ment QRR ∝ T as asaafunction
 functionofof
 temperature
 reveals three regimes. At high (T > 100 K) temperature QRR decreases due to crystal-field
 temperature (b) reveals three re-
 effects – a regime that is well reproduced by the calculation from the fit of the INS data [29].
 0
 At moderate (1 K < T < 10 K) temperatures,gimes. At high
 a plateau (T > 100
 of ~1.2 K) temperature
 X corresponds to the dipole
 0 0
 moment of the ground state ‘dipole-octupole’ μeff decreases
 doublet. At due
 low (Tto

Hamiltonian, and these can reproduce the data at higher energy transfers as well as the bulk
 10
 present work on Ce2Sn2O7 isHamiltonian,
 that the further
 susceptibility atdecrease
 and these
 high of
 candipole
 temperature. moment,
 reproduce observed
 The the data
 latter atinhigher
 QRR when
 is represented energy transfers
 in Figure as wellthe
 1b, where as effective
 the bulk
 cooling down in the correlated regime below
 susceptibility at high1 temperature.
 K, is due to dominant
 The latteroctupole–octupole
 is represented in Figure 1b, where the effective
 magnetic dipole moment QRR ∝ T is plotted as a function of temperature. This quantity
 couplings, causing the octupole moment to strengthen at the expense of the dipole one. In other
 magnetic dipole
 decreases moment QRRexcited
 upon depopulating ∝ T is plottedlevels
 crystal-field as a function of temperature.
 when cooling, to reach an This quantity
 approximate
 words, dominant octupole–octupole interactions mix the otherwise degenerate ` ab = ± 3⁄2c
 decreases
 plateau of upon
 Hamiltonian, depopulating
 ~1.2and these
 X in the can excited
 from crystal-field
 reproduce
 range 1 tothe
 10 data levels
 K. This when
 at value
 higher cooling,
 energy toto
 reach
 theasan
 transfers
 corresponds approximate
 well
 dipole as the bulk
 moment
 states to form new split eigenstates – the driving force being to minimize the energy of the
 plateau of Hamiltonian,
 susceptibility
 ~1.2 at
 in
 highthe and
 range these
 from
 temperature. can
 1
 Theto reproduce
 10 K.
 latter This
 is the data
 value at higher
 corresponds
 represented in Figure energy
 to the
 1b, transfers
 dipole
 where the as well
 moment
 effective
 calculated from the wavefunction of the ground doublet only.
 X
 system due to different magnetic dipole and octupole moment sizes.
 calculated
 magneticfromsusceptibility
 themoment
 dipole at
 wavefunction high
 QRR ∝oftemperature.
 the ground
 T The
 aslatter
 doublet
 is plotted is represented
 a only.
 function in FigureThis
 of temperature. 1b,quantity
 where t
 magnetic
 decreases upon dipole
 a
 depopulatingmoment QRR ∝ bT 
 b
 excited crystal-field is plotted
 levels as a function
 c
 when cooling, of an
 to reach temperature.
 approximateT
 a 150 meV bb c
 decreases upon4 depopulating excited crystal-field levels when cooling, to reach an a
 plateau of ~1.2 X in the range from 1 to 10 K. This value corresponds to the dipole moment
 150 meV
 4

 u.) u.)
 plateau
 calculated from of ~1.2 X in of
 the wavefunction thethe
 range from
 ground 1 to 10only.
 doublet K. This value corresponds to the dipo

 (arb.(arb.
 calculated from the wavefunction of the ground doublet only.

 Intensity
 a2 bb c

 Intensity
 2 150
 a meV bb c
 4
 Figure 2 150 meV
 4

 Intensity (arb. u.)
 Magnetic chargecharge
 density calculated from
 fromthethetype
 0 of of ground statewavefunction
 wavefunction of of Ce deter-
 3+

 Intensity (arb. u.)
 Figure 2 Magnetic density calculated type ground state
 40 60 80 100 120
 mined
 Ce 3+
 from the
 determined fit the
 from of the
 fit ofneutron data
 the neutron in in
 data Figure
 Figure1,
 01,i.e.
 i.e. |±⟩ = |± 3⁄2⟩ ± |∓ 3⁄2⟩.. The values of Jz
 E E (meV)
 (meV) E (meV)
 The values of e correspond to different values of the e and240 coefficients.
 60 80 100 120
 correspond to different values of the A and B coefficients. EE (meV)
 (meV) E (meV) 2
 e
 SIGNATURES OF CORRELATIONS IN MACROSCOPIC MEASUREMENTS
 Figure 1 Inelastic neutron scattering (INS) data of Ce2Sn2O7 probing the crystal-electric field
 Our investigations of Ce2Sn2levels
 O within
 7 started
 Figure withthe
 1 Inelastic ground
 measurements
 neutron multiplet
 of the2Fbulk
 scattering ofmagnetization
 5/2(INS) Cedata
 3+
 (a)of[29].Ceand TheObulk
 2Sn2heat
 magnetic susceptibility [28],
 7 probing the crystal-electric field
 0
incapacity
 the material of interest,shown
 levelsCe inSn
 within
 2 blue
 2 Otheas
 7 [28-29].
 the
 ground
 down to very low temperature [28]. As already exemplified effective proximate
 multiplet magnetic
 2
 40 60 F with
 5/2 of plateau
 moment
 Ce 3+
 the 120
 80 100 (a)of
 plot of
 [29]. ~1.2
 The∝ T 
 QRR shown
 QRR μB in
 bulk asthe
 magnetic range
 a function from
 of temperature
 susceptibility [28],
 reveals three 0 temperature E (meV)
Trivalent cerium has one f
 shownelectron,
 in blueregimes.
 with
 as thespin At high
 effective 1 to(T 10
 >EE100
 magnetic (meV)
 K.
 (meV) K)
 This
 moment
 e data below about401 Kelvin. value 
 60 80 100 corresponds
 QRR ∝ 120
 QRR decreases
 T as a to the
 function due totemperature
 dipole
 of crystal-field
 in Figure 1b, signatures of a correlated state appear in this At these
 effects
 reveals–threea regime regimes.that isAtwell
 highreproduced
 (T > 100 K)bytemperature
 theEEcalculation from the fit ofduetheto INS data [29].
S=1/2 and orbital angular
 At momentum
 moderate (1 K < T L=3
 < 10 K) moment
 temperatures, ecalculated
 a
 (meV)
 (meV)
 plateau from of thedecreases
 QRR
 wavefunction
 E (meV)
 corresponds ofcrystal-field
 to the
 temperatures, the ground state doublet is thermally well isolated
 effects – a regime that is well reproduced by the calculation from excited crystal-field ~1.2 from X the fit of the INS datadipole
 [29].
mixed into a J=5/2 Figure 1of
 ground Inelastic
 multiplet neutron scattering (INS) data of Ce 2Sn 2O 7 probing the crystal-electric field
 levels, and therefore moment
 At moderate
 is sufficient as a minimalthe K < T analysis
 100
 effective
 (Figure K) is
 temperature
 magnetic
 3), thus further that moment
 hintingthe
 wavefunction
 decreases
 QRR ∝ T due of
 to crystal-field
 as a function of tem
 moment strengthen the expense of the dipole one, QRR as shown in Figure 2.
Transitions
 at cooperativewithin
 phenomenathisThe ground
 essential
 effects
 setting-in
 –multiplet
 1 Kelvin.are
 result
 a regime
 below reveals ofthat
 theisabove
 three wellthereproduced
 regimes. ground
 crystal-field
 At high state
 byanalysis
 (Tthe Kramers
 > 100 isK)that
 calculation doublet
 the
 from
 temperature wavefunctionisQRRof
 the fit ofdecreases
 the
 the ofINSthe ground
 datadue [29].
 to cr
easily seen in inelastic TheAt
 state
 moderateeffects
 essential
 Kramers result
 neutron
 (1 K of
 scattering
 doublet
 < –Tthe
 < 10
 isaofregime
 K) temperatures,
 above
 thegeneral crystal-field
 that
 general isform a plateau
 analysis
 well reproduced
 form |±⟩ = |±is3by
 of the
 that ~1.2
 ⁄2⟩ ± the X corresponds
 wavefunction
 calculation
 |∓ 3⁄2⟩,,from where
 where ofto
 the theground
 the
 fit
 andof
 dipole
 the
 areINS
 The magnetization curves are moment of the
 also instructive, asground
 At moderate state ‘dipole-octupole’
 (1 Kgeneral
 the powder-averaged < T < 10 doublet.
 K) temperatures,
 saturation at high field At low (T

that Ising moments of ~1.2 X on the pyrochlore lattice of Ce2Sn2O7 translate into classical
at
that
 Ising
 Ising
 moments
 moments of of
 ~1.2
 dipole–dipole X on
 ~1.2 X on
 couplings the
 the
 of pyrochlore
 pyrochlore
 about 0.025lattice
 K,lattice
 of of
 which Cesmall
 is Ce
 2Sn
 2Sn
 2O27Otranslate
 7 translate
 compared tointo
 into
 the classical
 classical
 scale of the dominant 11
pole–dipole
dipole–dipolecouplings
 couplingsof of
 interactions. about
 about
 This 0.025
 0.025
 simple K,comparison
 K,
 which
 whichis small
 is small
 compared
 compared
 indicates toto
 that thethe
 the scale
 scale
 of of
 thethe
 correlated dominant
 dominant
 state originates from
teractions.
interactions.quantum-mechanical
 This
 Thissimple
 simplecomparison
 comparison indicates
 indicates
 exchange that
 thatthethe
 interactions correlated
 [28]. correlated state
 stateoriginates
 originates
 All bulk measurements, asfrom
 from
 well as muon spin
quantum-mechanical
antum-mechanical exchange
 exchange
 spectroscopy [28],interactions
 interactions
 exclude the[28].
 [28].
 AllAll
 presencebulk
 bulk
 ofmeasurements,
 measurements,
 magnetic orderasin as
 well
 well
 Ce asas
 muon
 muon spin
 spin
 2Sn2O7 down to the lowest
spectroscopy
ectroscopy temperatures
 [28],
 [28],
 exclude
 exclude
 the
 the presence
 presenceof of
 magnetic
 magnetic order
 order
 in in
 CeCe Sn SnO O
 (0.02 K),aand instead suggest a highly frustrated magnet.
 2 2 2 2
 7 down
 7 down to to
 the
 the
 lowest
 lowest
mperatures
temperatures
 (0.02
 (0.02
 K),K),
 andand
 instead
 instead
 suggest
 suggest
 a highly
 a highly
 frustrated
 frustrated
 magnet.
 magnet. c
 a
 aa c
 cc

 b
 5T
 b
 5T
 bb
 5 T5 T
 0T
 0T
 0 T0 T

 Figure 3
 FigureEffective
 3 Effective magneticmoment
 magnetic moment QRR ∝ T (a) (a)and
 andheatheatcapacity
 capacity (b)(b)
 as as
 a function of of temperature in
 a function
igure
 Figure temperature
 3 Effective
 3 Effective magnetic
 magnetic inmoment
 the correlated
 moment QRR
 QRR∝regime
 ∝T 
 T (a)of(a)
 Ceand
 and2SnheatO7 capacity
 2heat below 1 Kelvin
 capacity
 (b)(b) [28-29].
 asas
 a functionTheofmagnetization
 a function of
 the correlated regime of Ce 2 Sn O
 2 7 below 1 Kelvin [28-29]. The magnetization curves as a func-
emperature
 temperature incurves
 in
 thetheas a function
 correlated
 correlated of field
 regime
 regime of ofCeare
 Ce
 2Sn2shown
 Sn
 2O27Obelow in panel
 7 below 1 Kelvin(c). [28-29].
 1 Kelvin All[28-29].
 experimental
 TheThe data are shown as open
 magnetization
 magnetization
urves
 curvesasasa function
 a functiontion
 or closeof of of
 field field
 circles
 field and
 areare are
 shown
 shownshown
 were used
 in in
 panel in
 paneltopanel
 (c).(c). (c).
 fitAllAll
 the All
 experimental experimental
 dipole–dipole
 experimental data
 data
 ee data are shown as open
 and
 areare
 shown octupole–octupole
 shown asas
 open
 open ggor close circles and

rorclose
 closecirclesexchange
 circlesand were
 andwereparameters
 wereused
 used
 used tousing
 totofitfitthe the
 fitthethe relevant
 dipole–dipole
 dipole–dipole
 d ipole–dipole Hamiltonian
 ee and
 and
 ee andforoctupole–octupole
 ‘dipole-octupole’ doublets
 octupole–octupole
 octupole–octupole gggg on the parameters using
 exchange
xchange
 exchange pyrochlore
 parameters
 parameters lattice.
 using
 usingthetherelevant
 relevant Hamiltonian
 Hamiltonian forfor
 ‘dipole-octupole’
 ‘dipole-octupole’ doublets
 doublets ononthethe
 the relevant Hamiltonian for ‘dipole-octupole’ doublets on the pyrochlore lattice.
yrochlore
 pyrochlore lattice.
 lattice.
 The set of bulk measurements presented in Figure 3 can be used to extract exchange constants
The
e setset
 of of
 bulk
 bulkmeasurements
 usingmeasurements
 the relevant presented
 presented
 Hamiltonian in in
 Figure
 Figure
 for 3rare-earth
 can
 3 canbebeusedused totoextract
 extract
 pyrochlores exchange
 exchange
 with constants
 constants doublets,
 ‘dipole–octupole’
 he further
using
ing decrease
 thetherelevant
 relevant of dipole
 Hamiltonian
 ∑Hamiltonian moment,
 l mm for m for observed
 m rare-earth
 rare-earth in eeQRRe when
 g gpyrochlores
 pyrochlores withwith‘dipole–octupole’
 ‘dipole–octupole’ doublets,
 doublets,
 ℋij ing = the soptoctupole o p moment
 + gg o pto+strengthen
 o pe + me at( the
 m e
 o p + crystal-field
 o p )r [30-31].levels,
 e m
 and therefore
 The doublet is is sufficient
 mmmK,mm ism due g gg g ee eee octupole–octupole
 ee e me me m me e e em m
ℋregime
ijij= ∑
 = ∑below
 sopt l mm1
 l 
 sopt expense 
 
 o op p 
 + + gggg
 
 of to
 
 o the dominant
 + +
 
 op p dipole 
 o po one.+
 p +
 In ( ( 
 oother + +
 
 op p owords, 
 )r )r
 mop pg e [30-31].
 [30-31].
 as The
 a Thedoublet
 doublet
 minimal islow-energy
 is m edescription of the
 modelled by pseudo-spin S = 1/2 operators ⃗o = ( o , o , o ), where the components o and o
 oment to
odelled
modelled bystrengthen
 by dominant
 pseudo-spin
 pseudo-spin
 transform
 at Sthe octupole–octupole
 =expense
 =S 1/2
 like magnetic
 1/2 operators of the ⃗dipole
 operators
 dipoles o ⃗= = , interactions
 mone.
 ( o( 
 owhile
 m g In
 , o o,eother
 o o, g
 g e
 where
 ),o ), where mix
 thethe degrees omof
 components
 components o freedom.
 m
 and oe oe
 and In the correlated regime,
 o behaves as an octupole moment. For the sake of
ole interactions
ansform
transform like
 like the
 mixotherwise
 magnetic
 magnetic thedipoles
 otherwise
 dipoleswhile degenerate
 degenerate
 while
 g g
 o obehaves
 behaves` abas=asan ±an 2c states
 ⁄octupole
 3octupole moment.
 moment.where
 ForFor the
 thethe
 sakemagnetic
 sakeof of susceptibility increases
 simplicity and in order to avoid over-parametrizing the fit, we consider mm = me = 0, which
ates – the
mplicity
simplicityand driving
 and to
 in in
 order form
 force
 order toto new
 being
 avoid
 avoid split eigenstates
 toover-parametrizing
 minimize the energy
 over-parametrizing thethe – fit,of the
 fit,wethe
 we driving
 consider
 consider mm slower
 mm me
 mean-field
 == than
 me
 = = expected
 which for
 which the bulk
 0,calculations,
 0, a simple paramagnet,
 still captures the essential physics of octupolar phases. Using
lldipole
still and the
 captures force
 octupole
 captures the being
 moment
 essential
 essential toofminimize
 sizes.
 physics
 physics the energy of the calculations,
 a hump is also observed in the heat capacity
 magnetic properties atof
 octupolar
 octupolar
 low phases.
 phases.
 temperature Using
 areUsingmean-field
 mean-field
 employed calculations,
 to extract valuesthethe
 for bulk
 bulk
 gg
 and ee (see
agnetic
magnetic system
 properties
 propertiesat at
 low due
 low to different
 temperature
 temperature are
 aremagnetic
 employed
 employedto dipole
 to
 extract
 extractand
 values
 values
 for(Figure
 for
 gggg 3),eethus
 and
 and
 (see
 ee further hinting at cooperative
 (see
 Figure 3) [29]. As already explained at a qualitative level, the drop of the effective magnetic
gure
Figure
 3) 3)
 [29].
 [29].
 AsAs
 octupole moment sizes.
 already
 already
 explained
 explained
 phenomena setting-in below 1 Kelvin.
 moment below 1 K can at
 beat
 aaccounted
 qualitative
 a qualitative
 level,
 for level,
 usingthe
 athe
 drop
 drop
 of of
 dominant the
 the
 effective
 effective
 magnetic
 magnetic
 octupole–octupole interaction gg .
 The magnetization curves are also instruc-
oment
moment below
 below1 K1 can
 K can
 bebeaccounted
 accounted forfor
 using
 using
 a dominant
 a dominant
 octupole–octupole
 octupole–octupole
 interaction
 interaction
 gg . .
 gg
 SIGNATURES OF CORRELATIONS IN tive, as the powder-averaged saturation at
 MACROSCOPIC MEASUREMENTS high field occurs at roughly half the value of
 the ground-doublet dipole moment, which is
 Our investigations of Ce2Sn2O7 started with expected for Ising moments on a pyrochlore
 measurements of the bulk magnetization and lattice due to the important noncollinear local
 heat capacity down to very low temperature anisotropy. The Ising anisotropy of the dipoles
ty calculated from the type of ground state wavefunction of
 [28]. As already exemplified with the plot of is also corroborated by calculations using the
he neutron data in Figure 1, i.e. |±⟩ = |± 3⁄2⟩ ± |∓ 3⁄2⟩.
 μefftheshown
ifferent values of and incoefficients.
 Figure 1b, signatures of a corre- wavefunction determined from the analysis
 lated state
N MACROSCOPIC MEASUREMENTS
 appear in this data below about 1 of the inelastic neutron scattering results. It
 Kelvin. At these temperatures, the ground state is interesting to note that Ising moments of
arted with measurements of the bulk magnetization and heat
 doublet is thermally well isolated from excited ~1.2 μB on the pyrochlore lattice of Ce2Sn2O7
rature [28]. As already exemplified with the plot of QRR shown
elated state appear in this data below about 1 Kelvin. At these
 doublet is thermally well isolated from excited crystal-field

12

 a b

 that Ising moments of ~1.2 X on the pyrochlore l
 that Ising moments of ~1.2 X on the pyrochlore dipole–dipole lattice of Ce2Sn 2O7 translate
 couplings into0.025
 of about classical
 K, which is sm
 that Ising moments of ~1.2 X on the pyrochlore lattice of Ce2Sn2O7 translate into classical
 dipole–dipole couplings of about 0.025 K, which is small comparedThis
 interactions. to the scale of
 simple the dominant
 comparison indicates th
moments
ments of of X on that Ising moments of ~1.2 0.025
 intoonclassical
 K,the pyrochlore lattice of Ce Sn2O 7 translate into classical
 ~1.2~1.2 X on
 thethe
 pyrochlore
 pyrochlore lattice
 lattice
 dipole–dipole of of
 Ce 2Ce
 Sn22Sn
 couplings O72Oof
 translate
 translate
 7 about X into classical
 which is small compared to2the scale of the dominant
 interactions. This simple comparison indicates that the correlated state
 quantum-mechanical originates
 exchange from [28]. All
 interactions
 couplings
 le couplingsof about
 of about 0.025
 0.025
 K, which
 K, which isdipole–dipole
 is small
 small This couplings
 compared
 interactions. compared to to
 thethe
 simple of about
 scale
 scale
 of the
 of the
 comparison 0.025
 dominant K, whichthat
 dominant
 indicates is small
 the compared
 correlatedtostate
 the scale of the dominant
 originates from
 quantum-mechanical exchange interactions [28]. Allspectroscopy bulk measurements, as wellthe
 [28], exclude as presence
 muon spinof magnet
This
 . Thissimple
 simplecomparison
 comparisonindicates quantum-mechanical exchange interactions [28]. All bulk measurements, as well as muon spinfrom
 indicates interactions.
 thatthatthethe This
 correlated
 correlated simple
 state
 statecomparison
 originates
 originates indicates
 fromfrom that the correlated state originates
 c spectroscopy [28], exclude the presence d of magnetic order in Ce(0.02
 temperatures 2Sn2O 7 down
 K), to thesuggest
 and instead lowest a highly fr
hanical
 echanicalexchange
 exchange interactions
 interactions[28]. Allquantum-mechanical
 [28]. All
 bulkbulk
 spectroscopy measurements,
 measurements,
 [28], exclude exchange
 asthe
 as
 well as interactions
 well as
 muon
 presence muon spin
 of spin[28]. Allorder
 magnetic bulk in
 measurements,
 Ce2Sn2O7 down as well as lowest
 to the muon spin
 temperatures (0.02 K), and instead suggest a highly frustrated magnet.
 y28],
 [28],
 exclude
 exclude thethepresence
 presenceof of
 magneticspectroscopy
 magnetic order
 temperatures orderin Ce[28],
 in
 (0.02 2Ce
 exclude
 Sn22Sn
 K), O
 and72Odown
 7 down
 the
 instead presence
 tosuggest
 to
 thethe
 lowest of magnetic
 lowest
 a highly frustratedorder in Ce2Sn2O7 down to the lowest
 magnet.
 a
 es
 (0.02
 (0.02
 K),K),
 and and
 instead
 insteadsuggest
 suggesta highly temperatures
 a highly
 frustrated
 frustrated (0.02
 magnet.
 magnet. K), and instead suggest a highly frustrated magnet.
 a c
 a c
 a a a c
 c c c

 b
 5T
 b
 5T
 b
 Figure 4 5T
 b Magnetic
 b b 0T
 dipoles respecting the ‘2-in-2-out’ ice rule on each
 5 T5 T 5 T tetrahedron (a) and octupoles
 0T
 obeying the ‘2-plus-2-minus’ rule (b), together with their respective neutron magnetic diffuse
 0T
 scattering patterns (c and d) calculated in the (HHL)
 0 T plane of reciprocal space using Monte Carlo
 0 T0 T
 simulations [29]. Note that the spin ice pattern (panel c) is displayed over a much larger area of
 Figure 3 Effective magnetic moment QRR ∝ T (
 reciprocal space than usual, but the typical features can be discerned in the central region.
 Figure 3 Effective magnetic moment QRR ∝ T (a) temperature
 and heat capacity in the correlated regime of
 (b) as a function of Ce2Sn2O7 b
 Figure 3 Effective magnetic moment curves
 (a) and asheata function
 capacityofThe field
 (b) as are shown in
 a function ofpanel (c).
 temperature in the correlated regime of QRR Ce2Sn ∝ 2T 
 O7 below 1 Kelvin [28-29]. magnetization
 temperature
 Figure in the
 3 Effective correlated
 magnetic regime of Ce Sn O or
 below close 1 circles
 Kelvin and
 [28-29]. were
 The used to
 magnetization fit the dipole–
 ctive
 fective magnetic
 magnetic moment
 moment QRR QRR ∝curves
 ∝ T T (a) as
 (a)
 and aandfunction
 heat heat of field
 capacity
 capacity (b)are (b) amoment
 asshown
 as function in panel
 a function of2 of(c).
 QRR 2∝ 7T 
 exchange
 (a) and heatdata
 All experimental capacity (b) as as
 are shown a function
 open of
 eeparameters using the relevant gg Hamiltoni
 re
 in the
 in thecorrelated translate
 correlated regime
 regime into
 of ofCe2Ce classical
 Sn22Sn orcurves
 close
 O72Obelow as
 temperature
 7 below
 dipole–dipole
 a
 circles function
 1 Kelvin
 1 Kelvinand in of
 were
 the
 [28-29]. field
 [28-29]. cou-
 used
 correlated
 TheTheare shown
 to fit
 regime
 magnetization in
 the The panel
 d
 of set
 Ce
 magnetization pyrochloreeelattice. 2 Snof
 (c).
 ipole–dipole 2 O bulk
 All7 below measurements
 experimental
 1 and
 Kelvin data arepresented
 octupole–octupole
 [28-29]. shown
 The as open
 magnetization
 
 or (c).
 exchangeclose
 curves circles
 parameters anddata
 as a function were
 using ofare used
 the
 field to shown
 relevant
 are fit as the indipole–dipole
 Hamiltonian panel for
 (c). and octupole–octupole
 All‘dipole-octupole’
 experimental doublets
 data are shown on the gg
 as open
 nction
 function of of field plings
 field
 areare
 shown of about
 shown in panel
 in panel 0.025
 (c). All K,
 All which
 experimental
 experimental is small data com- are
 shownshown inas Figure
 open open 3 can be used to extract exchange
 es
 rclesand and werewere usedused to to
 fit fit
 thethe exchange
 pyrochlore
 dipole–dipole or close
 dipole–dipole parameters
 lattice.
 circles
 ee
 andand
 ee and using
 octupole–octupolethe
 were used to 
 octupole–octupole relevant Hamiltonian
 fit the dipole–dipole
 gg gg for ‘dipole-octupole’
 ee
 measurements doublets
 and octupole–octupole on the gg
 pared to the scale ofexchange
 the dominant interac- The set of relevant
 bulk presentedfor in Figure 3
 parameters
 ameters using using
 thethe relevant
 relevant Hamiltonian pyrochlore
 Hamiltonian forfor lattice.
 parameters
 ‘dipole-octupole’
 ‘dipole-octupole’ using
 doubletstheconstants
 doublets relevant
 onon
 using the
 thetheHamiltonian for ‘dipole-octupole’ doublets on the
 Hamiltonian
 tions. This simple Thecomparison
 setpyrochlore
 of bulk measurements indicates that
 lattice. presented in Figure pyrochlores
 rare-earth 3 using
 can bethe used to extract
 relevant
 with exchange constants
 Hamiltonian
 ‘dipole–octupole’ for rare-earth pyro
 ttice.
 lattice.
 The set of bulk measurements presented in Figure 3 can be used to extract exchange constants
 the correlated stateThe using the originates
 set
 relevant
 of used
 bulk from quan- presented
 Hamiltonian for doublets,
 rare-earth pyrochlores
 ℋij3= ∑ with ‘dipole–octupole’
 l tomm m m
 o extract gg
 o doublets,
 p + exchange
 g g
 p + 
 ee e e
 o p + me
 ulk
 measurements
 measurements presented
 presented in Figure
 in Figure 3 can
 using 3the canberelevant
 be
 used to measurements
 to
 extract
 extract
 Hamiltonian exchange
 exchange
 for constants
 constants
 rare-earth in Figure
 pyrochlores can soptbewith used ‘dipole–octupole’ constants
 doublets,
 tum-mechanicalℋexchange ij = ∑ interactions
 l mm
 o p + [28].
 m m gg g g ee e
 o p + o p + modelled e me m e
 ( o p + by e m [30-31].
 )r [30-31].
 o pseudo-spin =The
 S The dou-
 1/2doublet
 operators is ⃗ = ( m ,
 elevant
 vant Hamiltonian
 Hamiltonian forfor rare-earth
 rare-earth usingsopt
 pyrochlores
 pyrochlores thewith relevant m Hamiltonian
 gg g g for rare-earth pyrochlores p with ‘dipole–octupole’ doublets,
 l with
 mm ‘dipole–octupole’
 ‘dipole–octupole’ doublets,
 doublets, o o
 All bulk measurements, ℋ ij = ∑as well as ommuon
 p + spin o p +blet ee oemodelled
 is pe + meg( omby pe + oe pm )r [30-31]. The doublet is
 e pseudo-spin S = 1/2 oper-
 sopt
 m m
 mm m m m gg gg
 mm g g g ee ee e eemodelled
 e me ℋ me m by = e
 m pseudo-spin
 ∑ e e l 
 m
 e mm
 m mS =
 m 1/2 operators
 + gg
 
 g
 
 g
 + 
 ⃗o ee= ( ,+
 e oetransform ), mwhere
 o , meo ( like
 e
 + the
 e mcomponents
 magnetic
 )r dipoles
 [30-31]. while
 The o and ooe behaves
 
 doublet
 g
 is
 l o po +
 p 
 + spectroscopy
 o po +
 p 
 + o [28],
 po + 
 +exclude
 ( ij( 
 o pbyo + [30-31].
 +o po presence
 the )r
 p )r o [30-31]. The
 of The o doublet
 p doublet
 ators ⃗o is=o is( 
 p m g e o p
 o , o , o ),, where where the components
 sopt
 pt pmodelled ppseudo-spin Sp = 1/2 operators g
 o p
 the components om and oe
 transform like magnetic dipoles while mo mbehaves as an
 simplicitym octupole
 g and e in moment.
 orderthe For the
 to components
 avoid sakemof
 over-parametrizing t
 mmodelled
 mg ge e by pseudo-spin S = 1/2 operators
 , 2 oSn , o2oO down tothe the low- while e ⃗eo = ( , , ), where o and oe
 yseudo-spin
 pseudo-spin S =Smagnetic
 1/2
 = 1/2operators order
 operators ⃗o = ⃗in =Ce
 ( o( o, 7o where
 ,like
 ), where
 ),magnetic the components
 components ooand
 o and and o o transform o like magnetic dipoles
 g
 otransform dipoles behaves as o an o octupole moment. For the sake of
 simplicity and in order to avoid over-parametrizing g the
 still fit,
 captures we consider
 the 
 essentialmm
 = me
 physics = 0,
 of which
 octupolar phas
 kemagnetic
 magnetic est temperatures
 dipoles
 dipoles whilewhile
 g g
 o behaves
 o behaves
 (0.02transform
 as as
 simplicity an K), an and
 octupole
 and
 likeinstead
 octupole
 in order
 magnetic
 moment.
 moment.
 to avoidsug-
 dipoles
 For For thewhile
 while
 thesake
 over-parametrizing sake o behaves
 of behaves
 of the fit, asasan
 weanoctupole
 octupole
 consider moment.
 mm = me =For
 moment. For the sake of
 0, which
 gest a highly still captures
 frustrated magnet. the essential physics of octupolar the sake phases.
 of magnetic Using properties
 simplicity mean-field
 and in calculations,
 at low mm
 order temperature
 to avoid the bulk are employ
 nd
 in order
 in order to to
 avoid
 avoidover-parametrizing
 over-parametrizing still simplicity
 the
 capturesthefit,fit, wethe and
 we in order
 consider
 consider
 essential mm
 to=avoid
 physicsmm
 me
 = of= meover-parametrizing
 0,
 = 0,
 octupolar which which phases. Using the fit,mean-field
 we consider = methe
 calculations, = 0, bulk which
 magnetic properties at low temperature are employed Figure to extract
 3) [29]. values As already for explained
 gg
 and eeat(see a qualitative
 he
 s theessential
 essential physics
 physics of of
 octupolar
 octupolar phases. still
 phases.
 magnetic captures
 Using Using the
 mean-field
 properties essential
 mean-field
 at low physics
 calculations,
 calculations,
 temperature of
 thethe octupolar
 are bulk employed to extract values for and (seebulk
 bulk phases. Using mean-field calculations,
 gg ee the
 Figure 3) [29]. As already explained at a qualitativemoment level, the below drop1of K canthe be effective
 accounted magnetic for using a dom
 erties
 operties at at lowlowtemperature
 temperature areare employed
 Figure magnetic
 employed 3)to[29]. to
 extractproperties
 extract values
 As already values at
 for low
 for temperature
 andand
 explained gg gg ee ee are employed to extract values for gg and ee (see
 (see
 at a qualitative (see level, the drop of the effective magnetic gg

iltonian
 onian forfor‘dipole-octupole’‘dipole-octupole’doublets doubletsononthe the
 e used to extract exchange constants
 ive magnetic moment QRR ∝ T (a) and heat capacity (b) as a function of 13
 nes thewith correlated‘dipole–octupole’regime of Cedoublets, 2Sn2O7 below 1 Kelvin [28-29]. The magnetization
 ure
 ee3 can 3 can bebe usedused toto extract
 thatextract
 Isingexchange
 exchange
 moments constants
 constants
 ction eof mfield are shown
 p + o p )r [30-31]. The doublet is
 in panel (c). Allofexperimental
 ~1.2 X on data the pyrochlore
 are shown as lattice
 openof Ce2Sn2O7 translate into classical
 pyrochlores
 spyrochlores
 and werewith with‘dipole–octupole’
 used ‘dipole–octupole’
 to fit
 dipole–dipolethe d doublets,
 ipole–dipole
 couplings doublets, ee
 and octupole–octupole gg
 of about 0.025 K, which is small compared to the scale of the dominant
 e m e
 ),mewhere
 ometers me m using me e e
 the the components
 em relevant
 m o and o
 Hamiltonian for ‘dipole-octupole’ doublets on the
 + ( ( o op p++ o op )r )r[30-31].
 [30-31].The
 pinteractions. The
 This doublet
 doublet
 simpleiscomparison
 is indicates that the correlated state originates from
 yice.
 (b) as a function
 octupole moment. ofFor the sake of the fit, we consider
 over-parametrizing at ILL, a weak diffuse signal appearing at high
 m mg ge e m m e e
=9].
 ( 
 o ,The
 o ,o ,o where
 ,o ),o ),
 magnetization where quantum-mechanical
 the thecomponents
 components o oand exchange
 and o o interactions [28]. All bulk measurements, as well as muon spin
 , we consider = = 0,, which
 measurements presented
 mm me in Figure which still captures the essen- scattering vectors was observed in high-sta-
 3 can be used to extract exchange constants
 ta are
 aves
 es asasan shown
 anoctupole as open
 octupole spectroscopy
 moment.
 moment.For [28],
 For the sakeofofthe presence of magnetic order in Ce2Sn2O7 down to the lowest
 theexclude
 sake
 ant mean-field
 Hamiltonian
 tupole–octupole
 sing tial physics
 for
 gg rare-earth
 calculations, ofthe octupolar
 pyrochlores
 bulk phases. Using mean- doublets,
 with ‘dipole–octupole’ tistics difference data between 5 K and several
 ng
 zingthe
 pole’ the fit,fit,wewe
 doublets consider
 on g thetemperatures
 consider mm mm
 == (0.02
 me me
 = =0,K),
 0, and
 which
 m which
 instead suggest a highly frustrated magnet.
 ggfield
 gcalculations, the bulk e magnetic properties lower temperatures ranging from 2 K to 0.05
 mm om pm +
 extract values for
 o p +
 gg ee e e
 and o p ee+ 
 (seeme
 ( 
 o p + e m
 
 o p )r [30-31]. The doublet is
 hases.
 phases.Using Usingmean-field
 mean-field
 at low calculations,
 calculations,the
 temperature the
 are bulkbulk
 g eemployed to extract K (Figure 5) [29]. The intensity distribution is
 , the dropSof
 eudo-spin the operators
 = 1/2 effective magnetic ⃗o = ( m
 , o , o ), where the components o and o m e
 gggga o
 act
mployed
 loyed exchange totoextract values
 constants
 extract values
 valuesfor
 for for and
 and and eeee
 (see (see
 (see Figure 3) [29]. zero at low scattering vectors q 0 coupling
 +0.03 ± 0.01 K), only
 corresponds not
 to be
 a studied
 gg in details due
 > 0 corresponds to a to powder aver-
 0T
 s forof octupoles
 ent
 gg
 frustrated =+0.03
 and eeand
 Although (see
 the isfitable
 arrangement ±to
 0.01
 is equally
 of K),
 goodonly
 for absence
 octupoles
 explain the
 gg
 and >0
 = corresponds
 is +0.48
 able to±explain
 of phase0.06 K or
 theaging.
 gg
 transition. In= First,
 absence of the
 −0.16 simple
 ± 0.02
 phase existence
 K (and In of this signal
 a small
 transition.
 the effective
 but to a frustrated
 magnetic
 finite dipole–dipole arrangement
 coupling of =
 ee octupoles ± and
 0.01 K),isonly
 likely to >
 gg be0an additionaltoand good reason
 ic chargethisdensity
 case, theof magnetic
 the octupolescharge
 ondensity +0.03
 of the octupoles
 a tetrahedron on a by
 is constrained tetrahedron
 a ‘2- is constrained by a ‘2- a
 corresponds
tupole interaction is able
 frustrated gg .to explainofthe absence ofisphase tran- to think thatofthe system is governed by quan-
e (Figure 4), leadingarrangement
 plus-2-minus’ icean
 to rule (Figureoctupoles
 extensively leadingand
 4),degenerate an able
 to manifold toof
 extensivelyexplain the
 degenerate
 octupole absence
 ice manifold phase transition.
 of octupole iceIn
 sition. In
 Figure this case,
 3 Effective the magnetic
 magnetic moment charge den- tum
 (a) exchange
 and heat capacity interactions. Second,
 (b) as abyfunction the mag-
 e present thiscontext,
 case, the
 configurations. magnetic
 In theand
 ‘plus’ charge
 present
 ‘minus’ density
 context,
 replace of‘plus’
 the QRR ∝ T 
 the octupoles
 and
 ‘in’ on astates
 an‘minus’
 ‘out’ tetrahedron
 replace the is constrained
 ‘in’ an ‘out’ a ‘2- of
 states
 sity of the octupoles on a tetrahedron is con- nitude of the signal, presented in absolute
 temperature in the correlated regime of Ce 2 Sn 2 O7 below 1 Kelvin [28-29]. The magnetization
 plus-2-minus’ curves ice rule (Figure 4), leading to an extensively degenerate manifold ofare
 octupoleandice
 characterizing
 Ising moments on the
 strained dipolebyasaIsing
 a function
 corner-sharing moments of field
 lattice
 ‘2-plus-2-minus’ are
 onofthe shown
 rule in
 tetrahedra
 ice inpanel
 corner-sharing
 (Figure (c).
 spin lattice
 ices, Alland
 units,experimental
 of tetrahedra
 iseein very data
 in spin
 good shown
 ices, as open
 agreement with that
 or
 configurations. closeIn circles
 the and
 present were used
 context, to fit
 ‘plus’ the
 and d ipole–dipole
 ‘minus’ replace and
 the octupole–octupole
 ‘in’ an ‘out’ states gg
e two possible 4),
 instead leading
 designate
 local the
 mean-values
 exchange toparameters
 an extensively
 two possible usingdegenerate
 local
 of the octupolar mean-values
 theoperator man-
 of
 relevantassociatedthe
 Hamiltonian expected
 octupolar
 with for a ground state
 operator associated
 for ‘dipole-octupole’ based
 with
 doublets on octupole
 on the
 o characterizing
 ifold dipole
 gg

14

 a d

 b

 c e

 Figure 5
 Diffuse octupolar scattering (blue points with error bars corresponding to ±1 standard error) ob-
 tained from the difference between neutron diffraction patterns measured at 5 K and at a lower
 temperature indicated on each panel. Measurements were performed on HRPT (λ = 1.15 Å, dark
 blue points on panels a-c) and D20 (λ = 1.37 Å, light blue points on panel c). Note the large scat-
 tering vectors required to observe scattering by magnetic octupoles. The increase of octupolar
 moment evaluated by the temperature dependence of the integrated diffuse scattering (d)
 matches with the drop of the dipole moment measured in bulk susceptibility (c.f. Figure 3a). The
 powder average of the diffuse scattering calculated for the octupole ice and spin ice configura-
 tions is shown respectively with red and green points in panel c, while the solid red line repre-
 sents the same calculation for the octupole ice but scaled (× 0.625) onto the experimental data.
 Note the different scales used to display the octupolar (left scale) and dipolar (right scale) scat-
 tering in Ce2Sn2O7.

in order to exclude spurious origins of the mation, high resolution in order to distinguish
hump observed at high q. The available neutron the signal from the shifts of Bragg peaks due
flux is actually not the most important instru- to thermal contraction in difference patterns,
mental characteristic to improve the statistics and optimal choice of wavelength in order to
here, but rather a combination of well under- favour a large detector angular range for the
stood instrument features, background, colli- region of interest in reciprocal space.

temperatures (0.02 K), and instead suggest a highly frustrated magnet.
ice of Ce2Sn2O7 translate into classical
re lattice oftoCe
 compared 2Sn
 the 2O7 translate
 scale into classical
 of the dominant 15
 a
 smallcorrelated
 the compared state
 to the originates
 scale of thefrom
 dominant c
lkthat the correlated
 measurements, state
 as well originates
 as muon spin from
All bulk
order in measurements,
 Ce2Sn2O7 downas
 towell
 the as muon spin
 lowest
 a b 0.4 0.4
netic order
trated in Ce2Sn2O7 down to the lowest
 magnet.
 y frustrated magnet. b 0.3 0.3
 5T

 Intensity (arb. u.)
 E (meV)
 0.2
 0.2
 c 0T

 0.1
 0.1

 on the pyrochlore
 thatlattice
 Ising of Ce2Sn2Oof
 moments 7 translate intothe
 ~1.2 X on classical 0 of Ce2Sn2O7 translate into classical
 pyrochlore lattice
 Figure 3 Effective magnetic moment (a) and heat capacity 0 of
 0.025 K, which isdipole–dipole
 small compared to the of
 couplings scale of the
 about dominant
 0.025 K, which is QRR
 small∝ compared
 T 0.2
 0 to the
 0.4 scale
 0.6 0.8 1(b)
 of the as
 1.2a function
 dominant 1.4
 temperature in the correlated regime of Ce2Sn2O7 below 1 KelvinQ[28-29]. ( ¹) The magnetization
arison indicatesinteractions.
 that the correlated
 This simplestate originates indicates
 comparison from that the correlated state originates from
 curves as a function of field are shown in panel (c). All experimental data are shown as open
nteractions [28].quantum-mechanical
 All bulk measurements,
 or close circles as and
 wellinteractions
 exchange as muon
 were used spin
 to fitAll
 [28]. thebulk
 dipole–dipole
 measurements, and
 ee as octupole–octupole
 well as muon spin gg
 exchange parameters using the relevant Hamiltonian for ‘dipole-octupole’ doublets on the
presence of magnetic order
 spectroscopy
 Figure in[28],
 Ce2Sn
 6 pyrochlore 2O7 down
 exclude the to the lowest
 presence of magnetic order in Ce2Sn2O7 down to the lowest
 lattice.
ad suggest a highly frustratedpart
 Imaginary
 temperatures magnet.
 ofK),
 (0.02 theand
 dynamic
 insteadspin susceptibility
 suggest (a, blue magnet.
 a highly frustrated points with error bars corresponding
 The set of bulk measurements presented in Figure 3 can be used to extract exchange constants
 to ±1 standard error). The difference map between low (correlated) and high (uncorrelated) tem-
 and heat capacity (b) as a usingfunction ofrelevant Hamiltonian for rare-earth pyrochlores with ‘dipole–octupole’ doublets,
 peratures (b),athe
 summarizes the wavevector dependence of the spin excitations as a function of
 w (a)
 1 Kelvin [28-29].
 and heat The magnetization
 capacity (b) as a function of g g
 experimental datac
 energy,
 are giving
 ℋ
 shown evidence
 ijas=open oma pmcontinuum
 ∑soptl mmfor c oe pe + mespin
 + gg o pof+fractionalized
 ee ( om excitations.
 e e m
 p + o p )r [30-31]. The doublet is
 The phenomenologi-
 7 below 1 Kelvin [28-29]. The magnetization
 ole ee
 and octupole–octupole
 cal form
 c). All experimental data are used to
 
 shown as
 gg
 fit the spectrum (red line) happens to captureg ethe features expected frommthe- e
 modelled by open m
 pseudo-spin S = 1/2 operators ⃗o = ( o , o , o ), where the components o and o
 for ‘dipole-octupole’
 le–dipole ee andory doublets on the gg
 octupole–octupole
 for spinon excitations in a quantum spin iceg (onset, peak and extent) [33,37-39].
 transform
 onian for ‘dipole-octupole’ doublets on the like magnetic dipoles while o behaves as an octupole moment. For the sake of

 be used to extract exchange simplicity and in order to avoid over-parametrizing the fit, we consider mm = me = 0, which
 constants
 b
elores
 3 canwith
 be
 5 Tused
 AN OCTUPOLAR
 to extract still doublets,
 captures
 exchange
 ‘dipole–octupole’ QUANTUM
 constants SPIN
 the essential physics ICE neutron-active
 5 T of octupolar phases. Using mean-field
 transitions calculations,
 between thethebulk
 two
 myrochlores
 e magnetic doublets,
 e m with ‘dipole–octupole’ states of the doublet split by
 properties at low temperature are employed to extract values for gg and ee (see [29]. Low-en-
o p + o p )r [30-31]. The doublet is
T me m e The exchange constants extracted from the ergy neutron spectroscopy data of Ce Sn O
 Figure
 oe pmcomponents
 )r [30-31]. m 3)doublet
 [29].
 e As isalready explained at a qualitative level, the drop of the effective magnetic
 The
 2 2 7
 , oe ),( where
 o p + the
 0T
 o and o
 g set of bulk measurements place Ce Sn O in indeed reveal the presence of low-energy
 moment below 1 Ke can be accounted for using a dominant octupole–octupole interaction .
 2 2 7 gg
 om , octupole
 an o , oe ), where
 moment. the components m
 For the sake oofand o
 the octupolar quantum spin ice regime [30-31] excitations that are dipolar in nature (Figure
esfit,as
 weanconsider
 octupole mmmoment.
 = me =For the sake of
 0, which
 – a quantum liquid built up from a manifold 6) [29], and their continuous character
ng
 mentthe fit,
 Using QRRwe ∝ consider
 mean-field (a) and
 Figure
 of octupole
 T 
 mm
 3 Effective
 calculations, =ice the
 heat me
 0, which
 =bulk
 capacity
 magnetic
 configurations (b) as aset
 momentfunction QRRof∝
 by the dom- T (a)matches
 and heat capacity (b) as afor
 expectations function of
 the fractionalized
egime Using
hases. of Ce2Sn 2O7temperature
 below
 mean-field 1 Kelvin
 calculations, inee [28-29].
 the correlated
 the The magnetization
 bulk regime of Ce2Sn2O7 below 1 Kelvin [28-29]. The magnetization
 to extract values inantfor andgg
 exchange (seeand allowed to quantum spinon excitations of a quantum spin liquid.
 shown in panel (c). All experimental
 curves as a functiondata of fieldare are
 shownshown as open
 in panel (c). All experimental data are shown as open
 loyed
vel,
d fittodrop
 tothe the fluctuate
 extract
 dof values
 the for
 effective
 ipole–dipole
 or close thanks gg
 circles andto
 and
 eemagnetic
 and ee
 acting
 (see usedas
 octupole–octupole
 were toafit
 transverse
 the
 gg Gapless
 dipole–dipole ee andexcitations of a quantum
 octupole–octupole gg spin ice
 relevant
 ive level, Hamiltonian
 the dropexchange
 perturbation.
 of for
 the ‘dipole-octupole’
 parameters
 effective magnetic doublets on
 gg using the relevant Hamiltonian the for ‘dipole-octupole’
 known as ‘photon’ doublets on the
 excitations, however, are
ant octupole–octupole interaction .
 pyrochlore lattice.
 In a (dipolar)
dominant octupole–octupole interaction . quantum ggspin ice, excitations expected to follow an octupolar form factor in
 esented in Figure are 3expected
 The can
 set be used
 of bulk tomeasurements
 beextract
 to of twoexchangetypes [33]. Gapped
 constants
 presented in Figure 3 can thebepresent case. exchange
 used to extract And, given the bandwidth
 constants
 for rare-earthexcitations,
 pyrochlores
 using the relevant akin ‘dipole–octupole’
 with toHamiltonian
 spinons, for correspond
 rare-earth to
 doublets, expected
 pyrochlores with for such excitations,
 ‘dipole–octupole’ doublets, in the μeV
 ee e e defects me mcreated by osingle spin-flips + in ao ‘2-in- range, resolving their energy spectrum using
 g e e mm
 m g g
 p + o p +ℋ ij( =o ∑ psopt
 + l o p )r
 m
 [30-31].
 m
 p + 
 gg The
 o p doublet
 ee e
 isp + ( 
 e me m e e m
 o p + o p )r [30-31]. The doublet is
 2-out’m gmanifold e – a quantum version m ofe the m neutron
 g e scattering appears impossible in view
 operators ⃗o = ( modelled ), where
 o , o , o by the components
 pseudo-spin o and ⃗o = ( o , o , o ), where the components om and oe
 S = 1/2 operators
 g
 magnetic monopoles of classical spin
 g
 ice. of the incident energies of the order of 50 meV
 while o behaves as an octupole
 transform like magnetic moment. dipolesFor while
 the sake of
 o behaves as an octupole moment. For the sake of
 Equivalently, in an octupolar quantum spin required to reach far enough reciprocal space.
over-parametrizing ice the fit, and
 simplicity
 [30-31], weweconsider
 in order
 still expect mmavoid
 to = me
 such = 0, which to the fit,The
 over-parametrizing
 excitations wecontinuum
 consider mmis=peaked
 me = 0, which 0.05 meV,
 around
cs of octupolararise phases. Using
 still captures
 in the formmean-field
 the of essential
 dipolar calculations,
 physics
 low-energy the bulk
 of octupolar
 inelas- phases. Using mean-field
 which calculations,
 is approximately thedominant
 the bulk ex-
perature are employedticmagnetic
 neutron scattering,
 to properties
 extract values
 at low because
 for gg and ee are
 temperature allows
 (see employed change
 to extract interaction
 values for gg ,and
 and extends
 ee
 (see up to at
ned at a qualitative level,
 Figure the drop
 3) [29]. of the explained
 As already effective magnetic
 at a qualitative level, the drop of the effective magnetic
nted for using a moment
 dominantbelow
 octupole–octupole interaction
 1 K can be accounted for gg
 using . a dominant octupole–octupole interaction gg .