Geometrical frustration and quantum origin of spin dynamics

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dynamics

Alexandre Bertin

To cite this version:

Alexandre Bertin. Geometrical frustration and quantum origin of spin dynamics. Other

[cond-mat.other]. Université Grenoble Alpes, 2015. English. �NNT : 2015GREAY014�. �tel-01172689�

Pour obtenir le grade de

DOCTEUR DE L’UNIVERSIT ´E DE GRENOBLE

Sp ´ecialit ´e :

Physique de la mati `ere condens ´ee et du rayonnement

Arr ˆet ´e minist ´eriel : 7 ao ˆut 2006

Pr ´esent ´ee par

Alexandre BERTIN

Th `ese dirig ´ee par

Pierre Dalmas de R ´eotier

et codirig ´ee par

Bj ¨orn F ˚ak

pr ´epar ´ee au sein du

Service de Physique Statistique, Magn ´etisme et

Supraconductivit ´e (CEA-Grenoble/INAC/SPSMS)

et de l’

Ecole Doctorale de Physique, Grenoble

Geometrical frustration and

quantum origin of spin dynamics

Th `ese soutenue publiquement le

Jeudi 21 Mai 2015,

devant le jury compos ´e de :

M. Olivier ISNARD

Professeur `a l’Universit ´e Joseph Fourier, Pr ´esident

M. Hans-Henning KLAUSS

Professeur `a l’Universit ´e technique de Dresde, Rapporteur

M. Andrew WILDES

Chercheur `a l’Institut Laue-Langevin, Rapporteur

M. Fabrice BERT

Professeur `a l’Universit ´e Paris XI, Examinateur

M. Pierre DALMAS DE REOTIER

Chercheur au CEA-Grenoble, Directeur de th `ese

M. Bj ¨orn F ˚

AK

Enn la page de remer iements, la page la plus simple à é rire (et la plus lue par ertains d'entre vous, lefrançais s'arrête àla n de ette page!!) marquantla n de la réda tion!!

Entoutpremierlieu,jeremer ietrès haleureusementmondire teurdethèsePierre Dalmas de Réotier pour toute ton aide si pré ieuse tant dans le domaine de la physiquequedel'informatique,toute equetuaspum'apprendreau oursde esquatre années, mais surtout pour ta patien e, ta gentillesse et ton impli ation quotidienne. Bien évidemment, je remer ie également très sin èrement mon en adrant Björn Fåk pour tadisponibilité,tonomnis ien e du neutron ettasympathie(etpourles bureaux à l'ILL!!). Un immensemer i àvous deux pour m'avoirpermis d'en arriverjusque là!! Je remer ie tout parti ulèrement les membres du jury qui m'ont fait l'honneur d'évaluer ma thèse. Je ommen erai bien sûr par Olivier Isnard qui a a epté de présider le jury, mais surtout mon premier maitre de stage à l'Institut Néel au ours duquelj'aipudé ouvrirlemondedelare her he, etquim'adonnélegoûtdepoursuivre dans ette voie. Un grand mer i à mes deux rapporteurs, Hans-Henning Klauss et Andrew Wildes, qui ont eu la tâ he d'évaluer le manus rit. Enn, je remer ie vive-mentFabri e Bert pour son oeil ritique en tant qu'examinateur.

Jeremer ie Jean-Pas al Brisondem'avoira epté auseindu servi edephysique statistique, magnétisme,et supra ondu tivité (SPSMS),ainsi quetoutes les personnes que j'ai pu y toyer, en ommen ant bien évidemment par Alain Yaouan . Mer i pour ton impli ation, pour tout e que tu as pu m'apprendre sur le muon et le mag-nétisme, mais surtout pour m'avoir poussé durant toutes es années! Un immense mer iégalementàChristophe Marinpoursagentillesse,pour lesé hantillons (mer i également à Anne Forget du CEA-Sa lay), mais également pour m'avoir a ompagné régulièrement durant les manips d'aimantation et de DRX. Un très haleureux mer i à Marielle Perrier, pour entre autre s'être o uper de toute sorte de problèmes ad-ministratifs, mais surtout pour les moments très sympathiques au C5. Je remer ie

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les mesures d'aimantation,sans oublier Frédéri Bourdarot pour ses nombreux on-seils, s ientiques ounon. Ennjeremer ie les"jeunes",postdo southésards,que j'ai pu ren ontrer auSPSMS, en parti ulier lenéo-papa Ahmad SULTAN (Féli itations!!!)  Mer i pour ton soutien et tes nombreux onseils!  mais aussi Driss, Caro, Ni o, Justin, Alex, Benoit, Vladimir, Mounir...

Un ertainnombre d'expérien es ontété réaliséesauxseinsdes grandsinstruments. Je remer ie don tout les onta ts lo aux qui m'ont a ueilli et a ompagné, tout d'abord àl'Institut Laue-Langevin,PeterFouquet (IN11), Bernhard Fri k(IN16), Clemens Ritter (D1B); àl'Institut PaulS herrer, Chris Baines (LTF),Alex

Am-ato(GPS), etDenis S heptyakov (HRPT); et enn à ISIS, Peter Baker (MuSR),

Jon Taylor et Ross Stewart(MARI).

Un lin d'oeil bien évidemment aux ollègues que j'ai ren ontré à AITAP et qui ont rendu ette aventure très agréable. Je pense tout parti ulièrement, (et dans le désordre), à Heimanu,Raph, Seb B etSeb G, Larissa, Emilie, Kevin, Ja ques, Pierre, Matteo,Guillaume,Emanuella,Lu ia,Thomas, Clément,Benoit, Marion...ettous eux quej'oublie (désolé).

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Ennjetermineraipar eux quimesontleplus her,quim'onttoujourssoutenu etsupporté! , et sur qui j'ai toujours trouvé soutien et ré onfort: mes parents et la petitefratrie(pasordre d'âgedé roissant, pasde jaloux!!): Fred,Charles, Camilleetle petit Raphoupour qui ilfaudra attendre quelques annéesavantde feuilleter es pages. Mer i millefois!! Unepenséeparti ulière pour toi papaqui t'es toujours a harné àme remettre sur pied quand le globulese faisaitla malle.

Glossary 6

1 Introdu tion 15

1.1 Geometri alfrustration . . . 15

1.2 The pyro hlore ompounds . . . 17

1.3 The lassi alspin-i e . . . 20

1.3.1 The water i e model . . . 20

1.3.2 The dipolarspin-i e model(DSM). . . 21

1.3.3 Magneti monopoles . . . 24

1.3.4 Experimentaleviden e for magneti monopoles . . . 26

1.4 The quantum spin-i e. . . 27

1.4.1 Beyond the lassi al spin i e . . . 27

1.4.2 The ex hange Hamiltonian . . . 29

1.5 A large variety of magneti groundstate . . . 31

1.5.1 Tb

2

Ti

2

O

7

vs Tb

2

Sn

2

O

7

. . . 31

1.5.2 Yb

2

Ti

2

O

7

vs Yb

2

Sn

2

O

7

. . . 32

1.5.3 Er

2

Ti

2

O

7

vs Er

2

Sn

2

O

7

. . . 35

1.5.4 Gd

2

Ti

2

O

7

vs Gd

2

Sn

2

O

7

. . . 36

1.6 Content of the manus ript . . . 38

2 Experimental te hniques 40 2.1 Bulk measurements . . . 41

2.1.1 Spe i heat . . . 41

2.1.2 Magnetometry. . . 43

2.2 Fa ilitiesfor mi ros opi probemeasurements . . . 45

2.2.1 Institut Laue Langevin (ILL), a ontinuous neutron sour e . . . 45

2.2.2 ISIS, a muon and neutron pulsed sour e . . . 46

2.2.3 A neutron and muon pseudo- ontinuous sour e at PSI. . . 46

2.2.4 A third generation syn hrotronat PSI . . . 46

2.3 Dira tion experiments . . . 47

2.3.1 Introdu tiontodira tion . . . 47

2.3.2 Nu lear or harges attering . . . 47

2.3.3 Magneti s attering . . . 49

2.3.4 Powder dira tometers . . . 50

2.3.5 X-ray experiments . . . 50

2.3.6 Neutron experiments . . . 51

2.4 Neutron time-of-ightspe tros opy . . . 54

2.4.1 The MARI spe trometer . . . 54

2.4.2 Energy resolution . . . 55

2.5 Neutron ba ks attering spe tros opy . . . 56

2.5.1 The IN16 spe trometer . . . 56

2.5.2 The ba ks attering pro ess . . . 57

2.5.3 Spe tros opy . . . 59

2.6 Muon spe tros opy . . . 59

2.6.1 Introdu tion . . . 60

2.6.2 Experimentaldetails . . . 60

2.6.3 Pseudo- ontinuous versus pulsed sour e . . . 61

2.6.4 Muon spe trometers . . . 62

2.6.5 Polarisationfun tions. . . 63

2.6.6 Muon Knight shiftmeasurements . . . 65

3 CEF study of the pyro hlore series

R

2

M

2

O

7

68 3.1 Introdu tion . . . 69

3.1.1 Rare earthproperties . . . 69

3.1.2 The Stevens Hamiltonian. . . 71

3.1.3 Neutron ross se tion . . . 74

3.2 CEF of the titanate series

R

2

Ti

2

O

7

. . . 75

3.2.1 Published CEF parameters. . . 75

3.2.2 Proposalof a single CEF solution . . . 75

3.2.3 Analysis of Tb

2

Ti

2

O

7

. . . 79

3.2.4 Analysis of Er

2

Ti

2

O

7

. . . 81

3.2.5 Analysis of Ho

2

Ti

2

O

7

. . . 83

3.2.6 Con lusions . . . 86

3.3 CEF of the stannate series

R

2

Sn

2

O

7

. . . 88

3.3.1 Published CEF parameters. . . 88

3.3.2 Analysis of Ho

2

Sn

2

O

7

. . . 90 3.3.3 Analysis of Tb

2

Sn

2

O

7

. . . 94 3.3.4 Analysis of Er

2

Sn

2

O

7

. . . 95 3.3.5 Con lusions . . . 98 4 Experimental study of Nd

2

Sn

2

O

7

100 4.1 Introdu tion . . . 101 4.2 Powder synthesis . . . 101

4.3 Crystal stru ture analysis . . . 102

4.4 Neutron time-of-ightspe tros opy . . . 103

4.5 Bulk measurements . . . 104

4.5.1 Spe i heat . . . 104

4.5.2 Magnetisation . . . 107

4.6 Determination of the magneti stru ture . . . 110

4.7 Neutron ba ks attering measurements. . . 114

4.7.1 Spin Hamiltonianfor

143

Nd . . . 114

4.7.2 In oherent s attering ross-se tion . . . 115

4.7.4 Data analysis . . . 117

4.8

µ

SR spe tros opy . . . 120

4.8.1 Eviden e of long-range order . . . 121

4.8.2 Persisten e of spin dynami s . . . 125

4.8.3

λ

Z

behaviour inthe paramagneti phase . . . 127

4.8.4 Anomalouslyslowparamagneti u tuations . . . 130

4.9 Con lusions . . . 132

5 Insights into Tb

2

Ti

2

O

7

133 5.1 Introdu tion . . . 133

5.2 Tb

2

Ti

2

O

7

: a Jahn-Teller transition? . . . 136

5.2.1 Context . . . 136

5.2.2 X-ray syn hrotron radiationmeasurements . . . 139

5.3 Tb

2

Ti

2

O

7

: a quantum spin-i e realisation? . . . 140

5.3.1 The ex hange Hamiltonian . . . 140

5.3.2 Predi tion of a magnetisationplateau . . . 145

5.3.3

µ

SRfrequen y shiftmeasurements . . . 148

5.4 Con lusions . . . 152

6 General on lusions 154 6.1 Beyond the Stevens Hamiltonian . . . 154

6.2 Observation of spontaneous os illations . . . 156

6.3 Originof spin dynami s . . . 157

6.4 A magneto-elasti mode: solving the Tb

2

Ti

2

O

7

ase . . . 159

6.5 New perspe tives: the spinel ompounds . . . 159

A Crystallography of the pyro hlore ompounds 161 B The point harge model 164 C Neutron absorption orre tion 168 C.1 Re tangular geometry . . . 168

C.2 Annular geometry . . . 168

D Complements to magneti dira tion 171 D.1 Elements of group theory . . . 171

D.2 BasIREPS vs SARAh. . . 173

D.3 Analyti al eviden efor IR

Γ

3

sele tionin Nd

2

Sn

2

O

7

. . . 173

E Complements to

µ

SR 176 E.1 Derivation of the spin latti e relaxationrate . . . 176

E.2 Relaxationby ex itations. . . 178

E.2.1 Ferromagneti magnons . . . 178

E.2.2 Antiferromagneti magnons . . . 182

List of publi ations 185

Symbol Denition

a

†

_i

, a

i

Boson reationand annihilation operatorsat rare earth site

i

a

as

(ε)

Asymmetry parameter varying with the kineti energy

ε

a

bg

Time-independentba kground term

a

d

Distan e between the enter of two neighbouring tetrahedra

a

lat

Latti e parameter

a

mag

(q)

Amplitudeof the magneti intera tion

a

0

Initialmuon asymmetryor Bohrradius,

depending onthe ontext

A

h

(

≡ A)

Absorptionfa tor

A

inc

Weighingfa tor for in oherent nu lear intensity

A

mag

Weighingfa tor quasielasti magneti intensity

A

m

n

(

≡ A

m

n

(R))

CEF parameters of rare earth

R

A

143

hyp

Hyperne onstantof isotope

143

Nd

b

g,i

Ba kgroundintensity at the experimental point

i

b

j

Fermi length of atom

j

B

dem

Demagnetising eld

B

_dip

Dipolarmagneti eld

B

′

_dip

Dipolarmagneti eld arising frommagneti moments

inside the Lorentzsphere

B

hyp

Hyperne magneti eld

B

int

Internaleld

B

j

Parameter des ribing the amplitude of the isotropi t

displa emen aroundthe atomi mean position, and involved inthe Debye-Waller fa tor

B

J

(x)

Brillouinfun tion

B

loc

Lo al magneti eld

B

Lor

Lorentzmagneti eld

B

max

Maximum amplitude of the lo aleld

B

loc

B

m

n

CEF parameters:

B

m

n

= A

m

n

hr

n

iΘ

n

c

a

, c

x

Heat apa ity of the platform and of the sample, respe tively

c

p

Heat apa ity at onstant pressure

C

Constant

C

el

Ele troni spe i heat

C

ex

Spe i heat of magnon-like ex itations

Symbol Denition

C

p

Spe i heat at onstant pressure

C

ph

Latti e ontributionto the spe i heat

C

sh

Constant

C

v

Spe i heat at onstant volume

C

α,β

_(q)

Analyti al fun tionof

q

d

Dimensionof a system/matrix/representation

d

hkl

(

≡ d)

Interplanar spa ing

d

pair

Ve tor joininga magneti ionto one of itsnearest neighbours

d

1

, d

2

Interla ed sublatti es des ribing aHeisenberg

ollinearantiferromagnet

d

(µ)

ν

(g

i

)

Matrixrepresentation of the symmetry element

g

i

inthe representation

Γ

(µ)

ν

ˆ

d

(µ)

ν

(g

i

)

Matrixrepresentation of the symmetry element

g

i

inthe representation

Γ

ˆ

(µ)

ν

D

Dipolarenergy s ale

D

c

(B

loc

)

Fielddistribution

D

diff

Diusion oe ient

D

_DM

Dzyaloshinskii-Moriyave tor

D

nn

Dipolarenergy s ale between twonearest neighbours

D

αβ

r

i

Componentsof the eld dipole tensorasso iated with site

r

i

D

Constant

D

t

S ale of the distortion

e

+

Positron

E

ex

Ex itationenergy

E

f

Neutronnal energy

E

i

Neutronin identenergy or CEF energy levels,

depending onthe ontext

E

m

Nu lear energy levels

E

max

Maximalenergy of a magnonex itation

f

Frustration index orlling fa tor, depending onthe ontext

f

j

(q)

Atomi form fa tor (Fouriertransform of the ele troni density)

f

mag

(q)

Magneti form fa tor

F (x)

Fun tiondes ribing a CEF transitionand taken asthe onvolution of aGaussian and a Lorentzian fun tion

F

mag

(q)

Magneti stru ture fa tor

F

n

(q)

Neutronstru ture fa tor

F

′

n

(q)

Unit- ellstru ture fa tor

F

p

(q)

X-ray stru ture fa tor

g

Spe tros opi splittingfa tor or orderof

G

k

,

depending onthe ontext

g(ω)

Density of states

g

eff

Ee tive spe tros opi fa tor

g

i

Symmetry operation

Symbol Denition

g

m

(E)

Magneti density of states

g

_k

Longitudinal spe tros opi fa tor

g

_⊥

Transverse spe tros opi fa tor

G(x)

Gaussian fun tion

G

k

Little group: subgroup of the spa e group leaving

the magneti propagation waveve tor invariant

G

αβ

r

i

Components of the tensor

G

representing the oupling

between the muon spin and the spins of the system

h

Label of the Braggpeakspositions atthe angle

θ

h

,

or translationalpart of a symmetry operator, depending onthe ontext

~

Redu ed Plan k onstant (or Dira onstant)

H

applied

Realapplied magneti eld atthe sample

H

c

Criti almagneti eld indu ingaphase transition

H

_ext

External magneti eld

H

G

Fullwidth at halfmaximum of the Gaussian fun tion

H

L

Fullwidth at halfmaximum of the Lorentzian fun tion

H

AF

Heisenberg ollinear antiferromagneti Hamiltonian

H

CEF

CEF Hamiltonian

H

(J)

CEF

CEF Hamiltoniana ting onthe multipletdened by a

total angularmomentum

J

H

(J),mix

CEF

CEF Hamiltoniana ting onthe multipletdened by a total angularmomentum

J

taking into a ount the

J

-mixing ee t arising fromthe ouplingwith other multiplets

H

CSI

Classi alspin-i e Hamiltonian

(longitudinal ex hange Hamiltonian)

H

DB

Dipolarspin-i eHamiltonianin terms of

the dumbell modelnotation

H

DSM

Dipolarspin-i eHamiltonian

H

ex

Anisotropi ex hange Hamiltonian

H

FM

Hamiltonianfor a ferromagneti system

H

per

PerturbativeHamiltonian

H

Q

QuadrupolarHamiltonian

H

QSI

Quantum spin-i eHamiltonian(XXZ model)

H

XYZ

Anisotropi ex hange Hamiltonianof the XYZ model

H

Z

ZeemanHamiltonian

H

⊥

Transverse ex hange Hamiltonian(XXZmodel)

I

Nu lear spin ve tor operator

I

bg

Ba kground ontribution

I

c

Criti al urrent ina Josephson jun tion

I

h

Intensity at the Bragg position

h

I

Isotropi ex hange oupling onstant

I

eff

Ee tive nearest-neighbour isotropi ex hange oupling onstant

Symbol Denition

I

0

S alingfa tor

{I

1

, ...,

I

4

}

Anisotropi ex hange onstants involved in

H

ex

Notations

{I

zz

,

I

±

,

I

±±

,

I

z±

}

are alsoused

I

⊥

Transverse ex hange oupling onstant

J

i

Totalangular momentum ve tor operator of rare earth atsite

i

J

1

(x)

Besselfun tion of the rst kind

J

_±

Raising and lowering spin operators

{ ˜

J

x

, ˜

J

y

, ˜

J

z

}

Ex hange onstants involved in

H

XYZ

{J

1

, ...,

J

4

}

Anisotropi ex hange onstants involved

inthe ee tivespin-1/2 ex hange Hamiltonian

k

Ve tor in the re ipro al spa e

k

B

Boltzmann onstant

k

_i

, k

f

In ident and nal waveve tors, respe tively

k

mag

Magneti propagation waveve tor

K

Disso iation onstantfor the nu leation of magneti monopoles

K

exp

Normalisedmuon frequen y shift

K

′

dip

Muon Knight shift that arises onlyfrom the dipolareld

reated by the magneti moments insidethe Lorentz sphere

K

0

Complex onjugation operator

K

1

, K

2

Thermal ondu tan e between the ryostat and the platform,

and between the platform and the sample,respe tively

K

µ

Muon Knight shift

L Neutronight path

L(x)

Lorentzian fun tion

L

i

Totalorbital momentum ve tor operator of rare earth atsite

i

L

p,h

Lorentz fa tor

m

e

Ele tronmass

m

n

Neutronmass

m

pm

Paramagneti moment

m

sat

Saturationvalue of the magneti moment

m

sp

Spontaneous magneti moment

m

111

Proje tion of the spontaneousmagneti moment

overthe [111℄ axis

m

µ

Muon mass

M

Bulk magnetisation

M

_d

Divergen e-free part of the Helmholtzde omposition

M

h

Multipli ity of the ree tion

h

M

Lor

Magnetisationinside the Lorentz sphere

M

_m

Curl-freepart of the Helmholtzde omposition

M

_⊥

(q)

Proje tion of the Fouriertransform of the

total magnetisationdensity ona plane perpendi ular to

q

n

Orderof the operatorsor number of freeparameters,

depending onthe ontext

Symbol Denition

n(4f )

Number of

4f

ele trons

n

b

Number of bound magneti monopoles

n

BE

(x)

Bose-Einsteindistribution fun tion

n

FD

(x)

Fermi-Dira distributionfun tion

n

P

(x)

Plan k distribution

n

u

Number of disso iatedmagneti monopoles

n

0

n

0

= n

b

+ n

u

N

Number of magneti ions in the system

N(t)

Positron ountsin a dete tor

N

Demagnetising eld tensor

N

c

Number of unit ells in the system

N

Cu

Number of Cu nu lei in the sample holder

N

f

Number of formulaunit in the unit ell

N

L

Number of magneti moments inside the Lorentz sphere

N

mag

Number of magneti ells

N

Nd

Totalnumberof

143

Nd nu lei in the sample

N

p

Number of experimental points

N

0

S ale of the positron ount

N

_±

Positron ountsin the forward (+)/ba kward (-)dete tors

N

ZZ

Longitudinal omponentof the diagonaltensor

N

N

A

Avogadro number

O

m

n

Stevens operators

p

Magneti s attering lengthfor a magneti moment of

1 µ

B

at

q = 0

p

Pressure orproton, depending onthe ontext

p

i

Relative abundan eof isotope

i

p

m

n

Prefa tor

P

Thermalpower

P (θ)

Polarisation fa tor

P

n

(x)

Legendre polynomials

P

m

n

(x)

Asso iated Legendre polynomials

P

X

(t)

,

P

Y

(t)

Transverse muon polarisationfun tions

P

_X

exp

(t)

Experimentallymeasured transverse muon

polarisationfun tion

P

Z

(t)

Longitudinal muon polarisationfun tion

P

_Z

exp

(t)

Experimentallymeasured longitudinal muon

polarisationfun tion

P

stat

Z

(t)

Stati longitudinalmuon polarisationfun tion

q

S attering ve tor

q

BZ

Radius of the rst Brillouinzone onsidered as asphere

q

i

Ele tri harge

q

m

Magneti harge arisingfrom the fragmentationof

the magneti moment

Q

Quadrupolarmoment

Symbol Denition

Q

gs

Quadrupole moment of the Mössbauer ground state

Q

h

Heatinput brought tothe sample

Q

α

(

≡ Q)

Totalmagneti monopole harge ina tetrahedron

α

˜

Q

eff

Ee tive magneti harge arriedby amagneti monopole

r

Spin anisotropy ratio:

r = g

⊥

/g

k

r

i

Ve tor linkingthe muon tothe rare earth site

i

r

_ij

Ve tor linkingrare earth sites

i

and

j

r

nn

distan e between nearestneighbours

hr

n

_i

Expe tation values of the

n

th powerdistan e between the nu leus of the magneti ion and the

4f

ele troni shell

R

Idealgas onstant or rare earth ion, dependingon the ontext

R(x)

Instrumentalresolution fun tion

R

exp

,

R

p

,

R

wp

Prole,weight prole, and expe ted weightprole fa tors, respe tively

R

i

Distan e between anele tri hargeand the rare earth

S(q, ω)

S attering fun tion

S

′

(

_{≡ S}

µ

₎

Ee tive spin-1/2

S

el

Ele troni entropy

S

i

Totalspin ve tor operator of rare earthat site

i

S

iso

(q, ~ω)

Isotope-in oherent s attering fun tion

S

mag

(q, ~ω)

Magneti s attering fun tion

S

spin

(q, ~ω)

Spin-in oherent s attering fun tion

S

µ

Muon spin

T

Temperature

T

C

Curietemperature

T

c

Transitiontemperature

T

0

, T

a

, T

x

Temperatures of the ryostat, the platform,and the sample, respe tively

{U, V, W }

Half-width freeparameters des ribing the resolution fun tion

U

αβ

Anisotropi displa ement parameters involved in

the Debye-Waller fa tor

v

c

(

≡ v

0

) Volume of the unit ell

v

⋆

c

Volume of the rst Brillouinzone

v

D

Doppler velo ity

v

ex

Ex itationvelo ity

v

i

, v

f

Neutronin identand nal velo ity, respe tively

v

mag

Volume of the magneti ell

v

Tb

Volume per terbiumion

V (r

αβ

)

Magneti Coulomb intera tion between

two magneti monopoles separated by a distan e

r

αβ

V

CEF

CEF potential

V

F

(r)

Fermi pseudo-potentialat the

r

real spa e position

V

mag

Potentialof magneti intera tion

Symbol Denition

V

zz

Prin ipal omponent of the ele tri -eld gradient tensor

W (θ)

Probability of the positronto beemitted ina dire tion

θ

x

Position of oxygen atom O1

X

Isotropi strain parameter

y

c,i

Cal ulated intensity atthe experimentalpoint

i

y

c,0

S aling fa tor

y

o,i

Observed intensity atthe experimental point

i

Y

Isotropi size parameter

Y

m

n

(x)

Spheri al harmoni s

z

Quantisationaxis [111℄

z

nn

Number of nearestneighbours

Z

i

(

≡ Z)

Partition fun tion of isotope

i

Z(θ)

Peak prole fun tion

Z

m

n

(x)

Tesseral harmoni s

α

Parameter set involvingthe

n

freeparameters:

α = (α

1

, ..., α

n

)

α

c

Criti alexponent involved inthe riti albehaviourof

C

el

α

d

Instrumental balan eparameter

α

m

α

m

= n

u

/n

0

β

c

Criti alexponent involved inthe riti albehaviourof

m

sp

α

D

Constant

β

se

Exponent of the stret hed exponential fun tion

δ

i

Unit ve tor belonging toa <111>axis atrare earthsite

i

χ

Bulk magneti sus eptibility

χ(q, ~ω)

Dynami al sus eptibility

χ

′

ac

Realpart of the a. . magneti sus eptibility

χ

′′

_{(q, ~ω)}

Imaginary part of the dynami al sus eptibility

≡ Im{χ

αβ

_{(q, ω)}

_}

χ

′

_(q)

_q

-dependent stati sus eptibility

δ(x)

Dira fun tion

δ

CEF

Energy splitting between the low-lying CEF energy levels

δ

i,j

Krone kersymbol

∆

Anisotropi energy gap

∆

a

Strength of the spin anisotropy

∆

G

Standard deviation of aGaussian eld distribution

∆

N,i

Energy splitting between nu learlevelsof isotope

i

∆

Q

Nu lear quadrupole splitting

∆

so

Energy splitting between the CEF groundstate

and the rst CEF ex ited energy level

∆S

elec

Ele troni entropy variation

∆t

Time s ale

∆

X

Standard deviation of the eld distribution

η

Mixing parameterinvolved in the pseudo-Voigtfun tion

ϕ

Phase shift

Symbol Denition

Φ

αβ

_(t)

Symmetrised orrelation fun tionof the u tuatingpart of the lo almagneti eld atthe muon site

Φ

±

0

Groundstate wavefun tions

γ

i

Gyromagneti ratio of isotope

i

γ

µ

Muon gyromagneti ratio

γ

_∞

Sternheimer oe ient

Γ(x)

Gammafun tion

Γ

i,i

′

Linewidths of the Lorentzian fun tiona ounting forthe

lifetimeof the

i

′

CEF energy level duringthe transition

i

→ i

′

Γ

q

Quasielasti Lorentzian linewidth

Γ

Z

Inverse lifetimeof the nu learlevel

Γ

(µ)

ν

(

≡ Γ

ν

)

Irredu ible representation of order

µ

and labelled by the index

ν

ˆ

Γ

(µ)

ν

Loadedirredu ible representation

κ

α

m

Magneti ondu tivity illustrating

the motionof the magneti monopoles

λ

so

Spin-orbit oupling onstant

λ

X

Transverse (orspin-spin) relaxationrate

λ

Z

Spin-latti erelaxation rate

λ

exp

_Z

Expe ted spin-latti e relaxationrate

λ

Z,0

Constant

Λ

αβ

_{(q, ω)}

Symmetrised spin orrelation fun tion

µ

Magneti moment or hemi alpotential,

depending onthe ontext

µ

0

Permeability of free spa e

µ

B

Ele troni Bohrmagneton

µ

CF

CEF magneti moment

µ

CF

k

CEF magneti moment alongthe

z

axis

µ

CF

⊥

CEF magneti moment perpendi ular tothe

z

axis

µ

n

Magneti moment of the neutron

µ

N

Nu lear Bohrmagneton

µ

+

Muon with positive ele tri harge

ν

e

Neutrinoasso iated tothe positron

ν

ext

Muon pre ession frequen y around the external

magneti eld

B

ext

ν

FC

Fermi hopper frequen y

ν

M

Relaxationrate of the magnetisation

ν

0

Selfenergy a ounting for the dipolarand ex hange energy

between nearestneighbours

ν

α

m

Relaxationrate for re ombination of the

nu leated magneti monopoles

ν

µ

Muon neutrino or muon pre ession frequen y

aroundthe lo almagneti eld

B

loc

, depending onthe ontext

¯

Symbol Denition

ω

µ

Muon pre ession angularfrequen y

Ω

Solid angle

Ω

m

Number of mi rostates

Ψ

j

_i

(

≡ Ψ

i

)

Basis ve torsof the irredu ible representations taken atatom

j

(the index

i

labels the dierent basis ve tors)

Ψ

±

CEF wavefun tions of agiven doublet state

π

+

Positivepion

σ

Neutron spin

σ

a,i

Neutron absorption ross se tion of atom

i

σ

i

Standard deviation of

y

i

σ

i

spin

, σ

iso

i

Spin-in oherentand isotope-in oherent ross se tions of atom

i

σ

2

S reening oe ient

Σ, Σ

′

In identand naltotal absorption ross se tions, respe tively

τ

Redu ed temperature:

τ =

T −T

c

T

c

τ

c

Magneti orrelationtime:

τ

c

= 1/ν

c

τ

0

Spin u tuation time:

τ

0

= 1/ν

0

τ

1

Relaxation time of the sampletemperature

τ

µ

Muon lifetime

ˆ

θ

Odd time-reversal symmetry operator

θ

CW

Curie-Weiss temperature

θ

h

Bragg peak angle

Θ

D

Debye temperature

Θ

n

Stevens multipli ativefa tors

ξ(x)

Riemannfun tion

|ii

Eigenve tors of

H

CEF

|mi

Zeemanstates (

−I ≤ m ≤ I

,

I

nu lear spin)

|m

J

i

Zeemanstates (

−J ≤ m

J

≤ J

,

J

total angularmomentum)

dσ

dΩ

Dierential neutron ross se tion

dσ

coh

(q)

dΩ

Dierential oherent neutron ross se tion

dσ

inc

(q)

dΩ

Dierential in oherent neutron ross se tion

dσ

mag

(q)

dΩ

Dierential magneti neutron ross se tion

d

2

σ

dΩdE

′

Double dierentialneutron ross se tion



d

2

_σ

dΩdE



inc

Double dierentialin oherentneutron ross se tion



d

2

_σ

dΩdE



mag

Double dierentialmagneti neutron ross se tion



d

2

_σ

dΩdE



se

Double dierentialneutron ross se tion

from the sampleenvironment

[A, B]

Commutatorof operators

A

and

B

:

[A, B] = AB

− BA

{A, B}

Symmetrised orrelation fun tion of operators

A

and

B

:

2

_{{A, B} = AB + BA}

Introdu tion

Contents

1.1 Geometri al frustration . . . 15

1.2 The pyro hlore ompounds. . . 17

1.3 The lassi al spin-i e . . . 20

1.3.1 Thewateri emodel . . . 20

1.3.2 Thedipolar spin-i e model(DSM) . . . 21

1.3.3 Magneti monopoles . . . 24

1.3.4 Experimental eviden efor magneti monopoles . . . 26

1.4 The quantum spin-i e . . . 27

1.4.1 Beyondthe lassi al spin i e. . . 27

1.4.2 Theex hange Hamiltonian . . . 29

1.5 A large variety of magneti ground state . . . 31

1.5.1 Tb

2

Ti

2

O

7

vs Tb

2

Sn

2

O

7

. . . 31

1.5.2 Yb

2

Ti

2

O

7

vsYb

2

Sn

2

O

7

. . . 32

1.5.3 Er

2

Ti

2

O

7

vsEr

2

Sn

2

O

7

. . . 35

1.5.4 Gd

2

Ti

2

O

7

vs Gd

2

Sn

2

O

7

. . . 36

1.6 Content of the manus ript . . . 38

A general introdu tion on magneti geometri al frustration and a non exhaustive review of the dierent exoti magneti states en ountered in the two pyro hlore series

R

2

M

2

O

7

(

M

=Ti,Sn) of interestin this workare provided inthe following. Moreover, a briefdes ription of the ontent of the manus riptis given atthe end of this hapter.

1.1 Geometri al frustration

Magneti ompounds usually undergo a transition to establish at low temperatures a long-range magneti order and stabilise in a well-known magneti state su h as ferro-magneti order whereallthe spinsare parallel,antiferromagneti order wherespinsare antiparallel or ferrimagnetism order where magneti moments of dierent magnitudes

are antiparallel. For instan e, the ferromagneti order should appear below the Curie temperature

T

C

≈ θ

CW

, where

θ

CW

is the Curie-Weiss temperature hara terising the natureand strength of the magneti intera tions.

The notion of frustration in magnetism refers to the inability to simultaneously satisfy all the magneti intera tions. This originates from the ompetition of several ex hange pathsbetween twomagneti ions, i.e.frustrationof intera tions, orfromthe topology of the latti ewhere the spatial arrangement of the magneti atoms pre ludes thesatisfa tionofthemagneti intera tionssimultaneously. Thelatter ase,of interest here, is alled geometri al frustration. An example is given in Fig. 1.1 where Ising spins,i.e.spins allowed topointup ordown, with nearest-neighbour antiferromagneti intera tionsare lo atedatthe ornerof asquareand a triangle. In the former ase,all theantiferromagneti intera tionsaresatisedwhereasinthetriangular ase,ifone an-tiferromagneti intera tionissatisedwithtwospinsantiparallel,theorientationofthe thirdspin isun ertain sin e it annotsatisfy simultaneouslythe twoantiferromagneti bonds with itstwoneighbours.

AF

AF

AF

AF

AF

AF

AF

?

Figure1.1: Isingspins are lo ated at the orner of a square latti e (left) where allthe antiferromagneti intera tions between the rst neighbours an be satised and on a triangle(right)whereone of the AFbonds displayed by the blue bond isnot satised.

Geometri al frustration has fo used a lot of attention from an experimental and theoreti al point of view in the past de ades in front of the ri hness of the magneti groundstates. This on ept leads to un onventional magneti states, su h as omplex magneti stru turesorpreventionofthelong-rangemagneti order. Frustrationusually forbids the establishment of a single state, and the lowest energy spin onguration is realisedbyminimisingtheintera tionenergiesinseveralmanners,i.e.thegroundstates of frustrated ompounds are usually highly degenerated. The degree of frustration an be evaluated through the ratio

f =

|θ

CW

|/T

c

, where

T

c

denotes the temperature of the transition, if any, to a magneti order or a glassy state. Among the various latti es leading to frustration, the most popular two-dimensional stru tures are the triangularand the Kagome latti e, illustrated in the left and right panels of Fig. 1.2, respe tively. Wannier[1℄rstlyintrodu edthis on eptnoti ingthatferromagneti and antiferromagneti intera tions between Ising spins have very dierent properties on a triangularlatti e: in the latter ase, no magneti transition is predi ted down to the lowest temperatures. Three-dimensional geometri ally frustrated latti e are displayed

Figure1.2: Twodimensionalgeometri allyfrustratedsystems: the triangular(left)and Kagome(right) latti e.

Figure 1.3: Examples of three-dimensional geometri ally frustrated systems: the py-ro hlore latti e omposed of orner-sharing tetrahedra. Magneti ions are drawn by bla k spheres lo ated at the orners of tetrahedra. Reprinted gure with permission from Ref. [2℄. Copyright 2015 by the Ameri an Physi al So iety. Right: hyperk-agomé latti e ( orner-sharing triangles) as found in the gadoliniumgarnet ompound Gd

3

Ga

5

O

12

[3℄.

inFig. 1.3inthe ase of a orner-sharing tetrahedra (left)ortriangles(right) network.

1.2 The pyro hlore ompounds

A realisationof a three-dimensional frustrated network is the pyro hlore latti e, illus-trated in the left panel of Fig. 1.3 wheremagneti ions are lo ated inthe verti es of a orner-sharing tetrahedra network. We will fo us on insulator ompounds of hemi al formula

R

2

M

2

O

7

where

R

isarare earthmagneti ion,and

M =

TiorSn inthiswork. They rystallise in the fa e entred ubi latti e of spa e group

F d¯3m

. More details of the unit ell rystallography are provided in App. A. However, we need to noti e that the [111℄ dire tion is a lo al trigonal symmetry axis whi h will be taken as the quantisation axis

z

in the following. Some rare earth properties will be given in the introdu tion of Chapter 3.

nearest-neigh-Figure 1.4: The three spin ongurations of the irredu ible representation

Γ

7

dened by the basis ve tors

Ψ

4

(left),

Ψ

5

(middle), and

Ψ

6

(right), see Tab. D.1. Pi ture reprodu ed fromRef. [9℄with kind permissionof IOP Publishing.

bour ex hange Hamiltonian:

H = −I

X

i,j

S

i

· S

j

,

(1.1)

where

I

isthe nearest-neighbour ex hange oupling, i.e.

I > 0

in the aseof ferromag-neti intera tions and

I < 0

for antiferromagneti ones, and

S

i

is a Heisenberg spin lo atedatsite

i

. Inthe antiferromagneti ase,the authorsofRefs.[46℄ showthrough MonteCarlosimulationsthat thesystem remainsdisorderedatany nitetemperature, i.e.a lassi alspinliquid. Notethattheferromagneti ase doesnotlead tofrustration sin e the minimalenergy ongurationis a hieved when allthe spins are parallel.

Nevertheless, still onsidering lassi alHeisenbergspinsintera tingthrough nearest-neighbour antiferromagneti intera tions,and taking intoa ount dipolarintera tions, Palmer and Chalker [7℄ show that the degenera y asso iated to the innite number of spin ongurations, previously predi ted in Ref. [8℄, is lifted. For a spe i range of the ratio of the dipolar energy s ale over the ex hange energy, the system enters a four-sublatti e long-range magneti order with a magneti propagation waveve tor

k

_mag

= (0, 0, 0)

and a oplanar spin onguration illustrated in Fig. 1.4 by the three basis ve tors of the

Γ

7

irredu ible representation (see Tab. D.1).

Howeverother aspe ts need tobe onsidered. One importantfeature of the investi-gated pyro hlore ompounds is the strong spin-orbit oupling, larger than the rystal-ele tri -elda tingattherareearthsiteand reatedbythesurroundingele tri harges. Aswewill see inChapter 3,the rystal eld perturbationsplits the groundstate mul-tiplet,leadinginmost ases toagroundstatemagneti doublet. Thisenfor esastrong anisotropy of the spin. Withregard to the lo al axis [111℄ atthe rare earth site, spins ouldliealongorperpendi ulartothisaxis,i.e.theIsingorXYanisotropy,respe tively. ConsideringIsing lassi alspins, the Hamiltonianis writtenas:

H

ex

=

−I

X

i,j

S

i

· S

j

− ∆

a

X

i

(δ

i

· S

i

)

2

,

(1.2)

where

∆

a

> 0

s alesthe strength of the anisotropy and

δ

i

isa unit ve tor belongingto a<111>axis. Monte-Carlo al ulationspredi t[10,11℄,withintheapproximationthat a strong anisotropy enfor es spins to lie along the <111> axis (

|I| ≪ ∆

a

), that with nearest-neighbourantiferromagneti intera tionsalong-rangemagneti ordero ursat

Figure1.5: Spins ongurationforaplanaranisotropyinasingletetrahedron: the non oplanar

Ψ

2

state (left) and the oplanar

Ψ

3

state (right). Blue spheres indi ate rare earth ions sitting on the orner of a tetrahedron and red arrows show the orientation of the spins. Reprinted gure with permission from Ref. [14℄. Copyright 2015 by the Ameri an Physi alSo iety.

T

c

≈ |I|

with a magneti propagation waveve tor

k

mag

= (0, 0, 0)

and a onguration where all the spins are pointing into or out the enter of the tetrahedra; the rst experimental realisation of this magneti order has been found in the orner-sharing tetrahedra ompound FeF

3

[12℄. On the ontrary, in the ase of nearest-neighbour ferromagneti intera tions,the systemdoesnot displayany long-rangemagneti order: two spins are pointing intoand twospins are pointing out the enter of a tetrahedron, i.e. the lassi al spin-i e ase (see below) [13℄. This absen e of order results from the high degenera y ofthe ground statesin e several energy equivalentspin ongurations full the "two-in/two-out" onstraint, see Se . 1.3.

Inthe aseofanXYspinanisotropywithnearest-neighbourantiferromagneti inter-a tions,twomagneti stru tures anbea hievedwherespinslieinaplaneperpendi ular to the lo alaxis [111℄, as shown in Fig. 1.5: a non oplanar spin onguration dened as the

Ψ

2

state (left panel) and a oplanar spin arrangement hara terised by the

Ψ

3

state (right panel). Note that these two states are the basis ve tor of the irredu ible representation

Γ

5

allowed by the spa e group

F d¯3m

, see Tab. D.1. These states are energyequivalentleadingtothedegenera yofthegroundstate. However, inaso- alled orderbydisorderme hanism[15℄,thermalu tuationssele tthe

Ψ

2

states,i.e.whereas the internal energy of the twostates are equal, minimisingthe freeenergy whi htakes into a ount thermal u tuations will sele t the aforementioned state [16℄. Therefore a rst-order magneti transition is predi ted to o ur with a magneti propagation waveve tor

k

mag

= (0, 0, 0)

. Whenquantumu tuationsare onsidered,ase ond-order magneti transitionis predi ted [14, 17℄.

In summary, the magneti ground state of the pyro hlore is ruled by numerous physi al aspe ts: the nature of the nearest-neighbour ex hange intera tion and the hara ter of the spin anisotropy need to be onsidered, but also dipolar and further neighbour intera tions, anisotropi ex hange intera tions, and whether the spins are

Figure 1.6: Illustration of the analogy between the spin-i e and the water i e model. Left: Water i e stru ture where the oxygen ions

(O2−)

are displayed by the empty spheres andthe protons(H

+

)by the bla k ones. Arrows showthe protondispla ement fromthemiddleoftwooxygenatomswheretwoarenearthe entraloxygenionwhereas theothertwoarefarfromit. ReprintedgurewithpermissionfromRef.[24℄. Copyright 2015by theAmeri anPhysi alSo iety. Right: Single tetrahedronobeyingthe i erule: twoIsingspinsarepointingintothe enterofthetetrahedronandtwospinsarepointing out. Reprinted gurewith permissionfrom Ref.[18℄. Copyright2015 by the Ameri an Physi alSo iety.

lassi alorquantum. Thesubtlebalan ebetweenthese onsiderationsisattheoriginof thevariousexoti magneti statesen ounteredinthepyro hloreseries. Inthefollowing, we endeavour ourselves to summarise briey dierent magneti ground states at play inthe

R

2

M

2

O

7

familieswhere

M =

Ti orSn.

1.3 The lassi al spin-i e

Theterminologyof spin-i e wasrst introdu edbyHarriset al.[18℄ for the pyro hlore ompoundHo

2

Ti

2

O

7

wherenolong-range orderwaseviden ed down to50mK by

µ

SR spe tros opy [19℄. Otherpyro hlore ompounds, namelyDy

2

Ti

2

O

7

[20℄, Ho

2

Sn

2

O

7

[21℄ andDy

2

Sn

2

O

7

[22℄ have alsobeen unambiguously lassiedas lassi alspin-i e. Inthe following, wewillpresent some pe uliarproperties of these ompounds.

1.3.1 The water i e model

The rystal-ele tri -eld a ting on the rare earth site onstrains the spins to lie along the lo al [111℄ dire tion, i.e. dening the Ising model. The onguration on a single tetrahedronistwospinspointingintothe enterofthetetrahedraandtwospinspointing out,deningtheso- alledi erule. Thisdenominationoriginatesfromtheanalogymade with the model of the water i e

I

h

originally proposed by Bernal and Fowler [23℄, as illustrated in Fig. 1.6, where two protons are lose to the entraloxygen position and two farfrom it.

The degenera y of the ground state of frustrated materials is a onsequen e of the pe uliar latti e topology. For a given tetrahedron obeying the i e rule, only six ongurationsareavailableasillustratedinFig.1.7. The orrespondingentropy anbe

=

=

=

=

=

Figure 1.7: The six possible spin ongurations obeying the i e rule illustrate the degenera y of the groundstate ina spin-i e ompound.

al ulated [25℄: a system of

N

spins orresponds to

N

2

tetrahedra sin e a spin belongs to two tetrahedra. As Ising spins are onsidered, i.e. up or down,

2

4

ongurations should be onsidered fora single tetrahedronbut only

6

of them are available inorder to satisfy the i e rule. Thus the number of mi rostates a essible to the spin-i e is al ulated as

Ω

m

= 2

N

₍

6

16

)

N

2

and the entropy per spin is

S

el

/N = k

B

ln Ω

m

=

k

B

2

ln

3

2

, orresponding to Pauling's result for water i e [26℄. The magneti entropy is dedu ed from spe i heat measurements down to 0.2 K on Dy

2

Ti

2

O

7

[20℄, illustrated in the left panel of Fig. 1.8, and down to 0.34 K on Ho

2

Ti

2

O

7

[27℄, after subtra tion of the nu lear ontribution arising from strong hyperne intera tions a ting on the nu leus, and is in agreementwith this predi tion. The sibling stannate ompounds present the same residual magneti entropy inHo

2

Sn

2

O

7

[28℄and Dy

2

Sn

2

O

7

[29℄.

Thespin-i e ompoundsdonotexhibitanymagneti long-rangeorderasforinstan e in Ho

2

Ti

2

O

7

where no spontaneous os illations and no drop in the initial asymme-try of the muon polarisation fun tion are resolved by zero-eld

µ

SR experiments [19℄. The ele troni spe i heat exhibits a broad hump roughly around

T = 1

K below whi h it drops to almost zero, indi ative of a spin freezing in Ho

2

Ti

2

O

7

[27℄ and Dy

2

Ti

2

O

7

[20℄. This property was onrmed by magnetisationmeasurements with the presen eof anhysteresisee t betweenzero-eld andeld oolingpro eduresat0.65K for Dy

2

Ti

2

O

7

[30℄, and 0.75 K for Ho

2

Sn

2

O

7

[31℄, the latter ase being illustrated in the rightpanelof Fig.1.8. An additionalproof ofthis spin freezingliesinthe presen e of apeak inthe real part ofthe a. . sus eptibility inDy

2

Ti

2

O

7

[30℄and Dy

2

Sn

2

O

7

[22℄ indi ativeof the development of spin orrelations.

1.3.2 The dipolar spin-i e model (DSM)

As dis ussed above, the ase of lassi al spins with a strong Ising anisotropy, see the HamiltonianinEq.1.2, leads tothe spin-i e ongurationif ferromagneti intera tions are atplay, whi h is in agreement with the positive Curie-Weiss temperature dedu ed from sus eptibility measurements:

θ

CW

≈ 1.9

, 0.5, 1.8, and 1.7 K for Ho

2

Ti

2

O

7

[18℄, Dy

2

Ti

2

O

7

[20℄, Ho

2

Sn

2

O

7

[31℄, and Dy

2

Sn

2

O

7

[33℄, respe tively.

There-Figure 1.8: Left: Temperature dependen e of the magneti entropy of Dy

2

Ti

2

O

7

re-vealing the same residual entropy as explained by Pauling in water i e [26℄. A t to the data is a hieved using the dipolar spin-i e model, see Eq. 1.3. Experimental data are fromRef.[20℄. Reprinted gurewith permissionfrom Ref.[32℄. Copyright2015 by the Ameri an Physi al So iety. Right: Temperature dependen e of the magnetisation of Ho

2

Sn

2

O

7

re orded in ZFC-FC and showing a spin freezing behaviour. Copyright IOP Publishing. Pi ture reprodu ed from Ref. [31℄ by permission of IOP Publishing. Allrights reserved.

fore, dipolar intera tions are not negligible ompared to the weak ex hange intera -tion inferred from the Curie-Weiss temperature. An estimation of the dipolar energy s ale between two nearest neighbours is given by

D

nn

=

5

3

µ

0

4π

µ

2

r

3

nn

≈ 2.4

K [34℄, where

r

nn

= a

lat

√

2/4

isthe nearest-neighbourdistan e and

µ = 10 µ

B

. Therefore, anee tive nearest-neighbourenergys aleisputforwardtotakeintoa ountboththeee tofthe ex hangeanddipolarintera tions:

I

eff

≡ I

nn

+ D

nn

,where

I

nn

isthenearest-neighbour ex hange onstant. Analysing spe i heat data, a negative value of the ex hange onstantisinferred indi ativeofnearest-neighbour antiferromagneti ex hange intera -tions, i.e.

I

nn

=

−0.52

and -1.24 K for Ho

2

Ti

2

O

7

[27℄ and Dy

2

Ti

2

O

7

[32℄, respe tively. Therefore, dipolar intera tions are of prime importan e sin e they restore the ferro-magneti nature of the net nearest-neighbour intera tions, a mandatory ondition to re overthe spin-i e ase.

The dipolarspin-i e Hamiltonianwas introdu ed in order todes ribe the low tem-peratureproperties of the lassi alspin-i e ompounds[32℄:

H

DSM

=

−I

X

<i,j>

S

i

S

j

z

i

· z

j

+ Dr

nn

3

X

j>i

S

i

S

j

 z

i

· z

j

|r

3

ij

|

−

3(z

i

.r

ij

)(z

j

.r

ij

)

|r

5

ij

|



,

(1.3)

wheretherstterma ountsforthenearest-neighbourex hangeintera tion(

I = 3I

nn

) 1

and the ve tor

z

i

refers to lo al

h111i

dire tion of spin

S

i

lo ated at the rare earth site

i

. The se ond term arises from the dipolar intera tion (

D = 3D

nn

/5

).

2 The

1

The fa tor

3

omes from the s alar produ t between the lo al

h111i

dire tions of two nearest neighbourIsingspinslo atedatsites

i

and

j

.

2 The

3

5

fa tor omesfromthes alarprodu tbetweenthe

h111i

dire tionsandtheve tordire tion onne tingtwonearestneighbours.

Figure 1.9: Zero-eld phase diagram of the dipolarspin-i e modelpredi ted by Melko et al. [38℄ with Monte Carlo simulations.

J

nn

and

D

nn

have been dened in the main textand refertothe nearest-neighborex hange and dipolarenergy s ales,respe tively. Here,

J

nn

≡ I

nn

. Copyright IOP Publishing. Pi ture reprodu ed from Ref. [38℄ by permission of IOPPublishing. All rightsreserved.

role of the long-range dipolarintera tions was at stake for these frustratedsystems to understand why they do not lift the degenera y to establish a long-range ordering. If therstMonteCarlosimulationsfailtodes ribethe spe i heatandmagneti entropy results[35,36℄,duetoatrun atedsumoverthedipolarterm[37℄,bulkpropertiesofthe spin-i e ompound were nally onsistent with simulations using the dipolar spin-i e Hamiltonian for Ho

2

Ti

2

O

7

[37℄ and Dy

2

Ti

2

O

7

[32℄, the latter ase being illustrated in the left panel of Fig. 1.8.

The orresponding phase diagram of the Hamiltonian written in Eq. 1.3 has been omputed in Refs. [32, 38℄, see Fig. 1.9. When the nearest neighbour ex hange energy be omes su iently large ompared to the dipolar one, we re over the all-in-all-out antiferromagneti statewithamagneti propagationwaveve tor

k

mag

= (0, 0, 0)

. Above thisvalue,theferromagneti spin-i e aseiseviden edwheretheupperdottedlinerefers to the broad peak in spe i heat measurements orresponding to a slowing down of the spin u tuations. De reasing the temperature, the spin-i e ompoundis predi ted to undergo a rst order transition at

T /D

nn

≤ 0.08

with

k

mag

= (0, 0, 1)

, whi h has never been eviden ed experimentally.

Theexperimentaleviden eofasignatureoftheexisten eofdipolarspin orrelations was a hallenge over the past few years. Dipolar orrelations in the real spa e are hara terised by a

1/r

3

de ay, whi h orresponds in the re ipro al spa e by Fourier transformation to[39℄:

hS

i

(

−k)S

j

(k)

i ∝



δ

ij

−

k

i

k

j

k

2



,

(1.4)

where

k

is a ve tor of the re ipro alspa e. This leads to singularities at the Brillouin zone entres, the so- alled pin h points in neutron s attering measurements. Whereas these pin h pointswere hardly seenwith unpolarisedneutron experiments onthe spin-i e ompounds Ho

2

Ti

2

O

7

[34, 40℄ and Dy

2

Ti

2

O

7

[41℄, Fennell et al. [42℄ su eeded to

Figure 1.10: Diuse magneti s attering map re orded on the spin-i e ompound Ho

2

Ti

2

O

7

at1.7Kinthe (hhl)planeinordertoeviden e pin hpoints. FromRef. [42℄. Reprinted with permission fromAAAS.

eviden ethesepe uliarpin hpointsonHo

2

Ti

2

O

7

usingpolarisedneutrons,seeFig.1.10, revealing the dipolarnature ofthe spin orrelations. The omparisonof data re orded in the spin ip and non spin-ip hannels explains why previous measurements ould not resolvethese pin h pointswith unpolarised neutrons.

1.3.3 Magneti monopoles

The notion of magneti monopoles was rstly introdu ed by Ryzhkin [43℄ in order to des ribeex itationsinspin-i e. Then,thedumbbellmodel,seeforinstan eRef.[44℄,has beendeveloppedinordertoillustratetheDSMHamiltonianandtodes ribethethermal u tuations breaking of the i e rule with emergent quasiparti les, i.e. the magneti monopoles [43℄. The prin iple lies on the fragmentation of the magneti dipole into twomagneti monopolesof opposite harges

±q

m

(dumbbell)asillustratedinFig.1.11, and separated by a length

a

d

=

√

3a

lat

/2

whi h is the distan e separating the enter of two neighbouring tetrahedra. Thus, the magneti moment arried by the dipole

µ = q

m

a

d

is re overed. Therefore, the total magneti harge in a tetrahedron

α

is

Q

α

=

P

i

q

m,i

,wherethesumrunsoverthefourmagneti hargesinsidethetetrahedra. This resultingtotal magneti harge is the so- alled magneti monopole. Note that in the i e rule ground state

Q

α

= 0

and if a spin is ipped

Q

α

=

±2q

m

. A ording to Refs.[44, 45℄, the magneti Coulombintera tion between twomonopolesis writtenas:

V (r

αβ

) =

(

_µ

0

4π

Q

α

Q

β

r

αβ

if α

6= β

ν

0

Q

2

α

2

if α = β,

(1.5)

Figure1.11: (a)Twoneighbouringtetrahedraobeyingthei erule. (b) Thespin shared by the two tetrahedra is thermally ipped to reate a pair of magneti monopoles of opposite harge. Panels( ) and (d) are the illustrationof panel(a)and (b) intermsof the dumbbellmodel: amagneti momentisrepla ed by twoopposite magneti harges

±q

m

. (e) Propagation of two magneti monopoles along a Dira string. Reprinted by permission fromMa millanPublishers Ltd: Nature [44℄, opyright2015.

where

r

αβ

denotes the distan e between two monopoles. The rst lineof Eq. 1.5refers to the dipolarintera tion of the DSM and the introdu tionof the self energy

ν

0

inthe se ond line a ounts for the dipolarand ex hange energy between nearest neighbours. The DSM Hamiltonian an berewritten in terms of the dumbbellnotationsu h as:

H

DB

=

µ

0

4π

X

α6=β

Q

α

Q

β

r

αβ

+

ν

0

2

X

α

Q

2

_α

(1.6)

When the i e rule is satised, the spin-i e state is dened as a Coulomb phase sin e the three riteria stated by Henley [46℄ are fullled: (i) the system is highly disorderedsin enolong-rangeorderisestablished,(ii)ea hdumbbellisasso iatedtoa magneti ux, and(iii)themagneti uxatthe entreofthetetrahedronvanishes. The last ondition an be rewritten as a divergen e free oarse-grained eld, i.e.

∇

· B =

µ

0

P

_α

Q

α

= 0

in the spin-i e ground state. We should noti e that in a more usual ooperativeparamagnet,thesystementersinaphasewithoutlong-rangemagneti order withspin orrelationsde reasingexponentially,whereasintheso- alledCoulombphase spin orrelations are algebrai .

Therefore, this modelallows to des ribe spin dynami s in su h a system: toa spin thermally ipped orresponds the nu leation of two magneti monopoles of opposite

harge lo ated in two orner-sharing tetrahedra. These monopoles intera t through a magneti Coulomb potential. The divergen e-free ondition is broken, i.e. the i erule isnot fullled anymore. Thus, on e magneti monopoles are nu leated, their diusion alonga pathof reversed spins, i.e.the so- alledDira string,see panel (e) of Fig. 1.11, orrespondstothepropagationofazeroenergy ostspinreversalalongthestring,sin e ea htetrahedrontendstore overthegroundstatedenedbythei erule onguration.

1.3.4 Experimental eviden e for magneti monopoles

Bramwelletal.[47℄have re entlyproposedbymuon spe tros opy the presen eof mag-neti monopoles intera ting through a magneti potential in the spin-i e pyro hlore ompound Dy

2

Ti

2

O

7

. The prin iple lies on the in rease of the magneti monopoles densitywhenapplyingamagneti eld,inspiredfromOnsager'swork[48℄onthese ond Wienee t whi h predi ts the in rease of the disso iation onstant of water mole ule intoH

3

O

+

and OH

−

ions underan appliedele tri eld whi h over omes the Coulomb energy barrier. Pursuing this analogy, the disso iation onstant

K

for the nu leation of magneti monopoles was assumed to take a similar form as in Onsager's theory for weakmagneti eld

B

[47℄:

K(B) = K(0)



1 + b +

b

2

3

...



,

(1.7) where

b =

µ

0

Q

3

B

8πk

2

B

T

2

. 3

Atthe equilibrium,i.e.without appliedmagneti eld, the number

of bound magneti monopoles

n

b

is predominant ompared tothe disso iated ones

n

u

. A ording to Ref. [47℄, the disso iation onstant is writtenas:

K(0) = n

0

α

2

m

1

− α

m

,

(1.8)

where

n

0

= n

b

+ n

u

and

α

m

= n

u

/n

0

. The re ombination of nu leated magneti monopoles follows an exponential de ay with a relaxation time

1/ν

α

m

. Sin e

ν

α

m

∝

κ

α

m

, where

κ

α

m

is the magneti ondu tivity (illustrating the motion of the magneti monopoles)proportionaltothedensityofmagneti monopoles,andre allingthat

α

m

≪

1

, it follows [47℄:

ν

α

m

(B)

ν

α

m

(0)

=

κ

α

m

(B)

κ

α

m

(0)

=

α

m

(B)

α

m

(0)

=

s

K(B)

K(0)

≈ 1 +

b

2

.

(1.9)

Furthermore, Bramwell et al. [47℄ put forward that the u tuations of the magneti monopole density produ es u tuations of the lo al eld. Therefore after a magneti eld perturbation, the relaxation rate of the magnetisation

ν

M

is proportional to the relaxationrate of the magneti monopoledensity

ν

α

m

.

ν

α

m

(B)

ν

α

m

(0)

=

ν

M

(B)

ν

M

(0)

(1.10) 3

Note that the index

α

labelling a tetrahedronhas been dropped now,

Q

refers to the magneti hargeofanee tivemonopole.

Figure 1.12: Temperature dependen e of the al ulated value of the ee tive mag-neti harge

Q

˜

eff

inferred from the eld dependen e of the muon spin relaxation rate in the ase of Dy

2

Ti

2

O

7

. Reprinted by permission from Ma millan Publishers Ltd: Nature [47℄, opyright 2015.

In the transverse eld muon spin relaxation te hnique, see Se . 2.6, the muon polari-sation fun tion is hara terised by os illations illustratingthe pre ession of the muon spinaroundthelo aleld,andanenvelopegivinginformationondynami softhe lo al eld at the muon site: in the ase of slowu tuations of the lo aleld, the relaxation rate

λ

, hara teristi of the exponential de ay of the envelope, is proportional to

ν

M

. Therefore, Bramwell et al. nd an ingenious way to measure the magneti harge ar-ried by the magneti monopoles. Hen e, measuring the eld dependen e of

λ

allows to extra tthe ee tivemagneti harge arriedbythe monopoles,see Fig.1.12. A typi al value of

Q

˜

eff

= 5 µ

B

Å

−1

has been inferred in good agreement with Ref. [44℄ within the temperature range

T

lower

≤ T ≤ T

upper

where Onsager's theory remains valid. The authors of Ref. [49℄ draw the same on lusions with

µ

SR experiments on the spin-i e ompoundHo

2

Ti

2

O

7

.

Whereas these results were strongly debated [50, 51℄ in a rst instan e, additional experimental proofs eviden ed a signature of magneti monopoles in spin-i e as for instan e the observation of Dira strings in Dy

2

Ti

2

O

7

with neutron s attering experi-ments under a magneti eld applied along [100℄ [52℄. Existen e of su h strings were previously suggested in Ref. [42℄ from the broadening of pin h points. Furthermore, the temperature dependen e of the relaxation time inferred from a. . sus eptibility on Dy

2

Ti

2

O

7

[30℄,previously misunderstood,has been des ribed intermsof themotionof magneti monopoles [53℄.

1.4 The quantum spin-i e

1.4.1 Beyond the lassi al spin i e

The quantum spin-i e is dened by the same properties as its lassi al ounterpart: Ising spins along the trigonal axis [111℄ full the i erule onstraint,dening the same