Neutron scattering investigation of the spin correlations in frustrated spinels

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Thesis

Reference

Neutron scattering investigation of the spin correlations in frustrated spinels

GAO, Shang

Abstract

Spinels of the chemical formula AB2X4 are important systems in the study of frustrated magnetism. The A and B sites of spinels constitute diamond and pyrochlore lattices, respectively, whose special geometries can give rise to many exotic phenomena. In this thesis, we investigate the spin correlations in three types of frustrated spinels: the spin ice candidates CdEr2X4 (X = Se, S), the spiral spin-liquid candidate MnSc2S4, and the spin-Peierls system MgCr2O4.

GAO, Shang. Neutron scattering investigation of the spin correlations in frustrated spinels . Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5095

DOI : 10.13097/archive-ouverte/unige:95576 URN : urn:nbn:ch:unige-955764

Available at:

http://archive-ouverte.unige.ch/unige:95576

Disclaimer: layout of this document may differ from the published version.

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UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES Section de Physique

Department of Quantum Matter Physics Professeur Ch. Rüegg

PAUL SCHERRER INSTITUT

Laboratory for Neutron Scattering and Imaging Dr. O. Zaharko

_______________________________________________________________

Neutron scattering investigation of the spin correlations in frustrated spinels

THÈSE

présenté à la Faculté des Sciences de l'Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique

par

Shang Gao

de Suqian (China)

Thèse Nº 5095

GENÈVE

Atelier de reproduction de la Section de Physique 2017

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To do anything substantial it will take three years.

— P. W. Anderson

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Abstract

Spinels of the chemical formulaAB2X4are important systems in the study of frustrated magnetism. TheAandBsites of spinels constitute diamond and pyrochlore lattices, respectively, whose special geometries can give rise to many exotic phenomena. In this thesis, we investigate the spin correlations in three types of frustrated spinels: the spin ice candidates CdEr2X4

(X=Se, S), the spiral spin-liquid candidate MnSc₂S₄, and the spin-Peierls system MgCr₂O₄. For CdEr₂X₄(X =Se, S), we provide microscopic evidence for their spin ice states. Using inelastic neutron scattering to study the crystal electric field transitions and neutron diffuse scattering to study the short-range spin correlations, we reveal the Ising character of the Er³⁺ spins and their ferromagnetic couplings, which are the two pre-requisites to realize the spin ice state. With our magnetic susceptibility and neutron spin echo experiments, we explore the monopole dynamics in CdEr₂X₄(X=Se, S) and explain their fast dynamics with an increase of the monopole hopping rates.

In theA-site spinel MnSc₂S₄, we directly observe the spiral correlations using neutron diffuse scattering and thus experimentally confirm the predicted spiral spin-liquid state. With neutron diffraction, we clarify the multi-step ordering process of MnSc2S4 and discover a vortex- like triple-q phase under magnetic field. Spin excitations in MnSc₂S₄ are studied using inelastic neutron scattering, and the obtained spin wave dispersions confirm the importance of perturbations from third-neighbour couplings.

MgCr₂O₄is interesting because of its strong spin-lattice coupling that relieves the ground state degeneracy and leads to an ordered state with a distorted lattice. Using synchrotron X-ray and neutron diffraction, we clarify the lattice and magnetic structures of MgCr₂O₄in the distorted phase. With inelastic neutron scattering, the low-energy spin wave excitations are also studied.

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Zusammenfassung

Spinell Materialien mit der chemischen FormelAB2X4sind wichtige Systeme für das Studium von frustriertem Magnetismus. DieAundBStellen der Spinell Struktur bilden ein Diamant- und Pyrochlore-Gitter, deren besondere Geometrien zu vielen exotischen Zuständen führen können. In dieser Arbeit untersuchen wir die Spin-Korrelationen in drei Arten von frustrierten Spinell Materialien: Die Spin-Eis Kandidaten CdEr₂X₄(X =Se, S), den Kandidat für eine Spirale Spin-Flüssigkeit MnSc₂S₄, und das Spin-Peierls-System MgCr₂O₄.

Im Falle von CdEr₂X₄(X=Se, S) präsentieren wir den mikroskopischen Nachweis für ihre Spin- Eis-Zustände. Unter Verwendung von inelastischer Neutronenstreuung werden einerseits die Kristallfeld Übergänge studiert, während andererseits durch diffuse Neutronenstreuung die kurzreichweitigen Spin-Korrelationen untersucht werden, mit dem Ziel den Ising-Charakter der Er³⁺Spins und ihre ferromagnetischen Kopplungen nachzuweisen, welche beides wichtige Voraussetzungen für die Realisierung des Spin-Eis-Zustandes sind. Mit den magnetischen Suszeptibilitäts- und Neutronenspin-Echo-Messungen erforschen wir die Monopol-Dynamik in CdEr₂X₄(X=Se, S) und erklären ihre schnelle Dynamik mit der Erhöhung der Monopol- Hopping-Raten.

In demA-Stellen Spinell MnSc₂S₄, untersuchen wir direkt die Spiral-Korrelationen mit dif- fuser Neutronenstreuung und bestätigen experimentell den vorhergesagten Spirale-Spin- Flüssigkeits Zustand. Mit den Daten der Neutrondiffraktion klären wir den Prozess der mehr- stufigen Ordnung in MnSc₂S₄und entdecken die Entstehung einer vortexartigen Triple-q Phase im externen Magnetfeld. Spin-Anregungen in MnSc₂S₄werden mit inelastischer Neu- tronenstreuungen untersucht, wobei die resultierende Spinwellen-Dispersion die Bedeutung von Störungen der drittnächsten Nachbar Kopplungen nachweist.

MgCr2O4ist wegen seiner starken Spin-Gitter-Kopplung interessant, welche die Grundzu- stands Entartung aufhebt und zu einem geordneten Zustand mit einem verzerrten Gitter führt.

Unter Verwendung von Synchrotron-Röntgen- und Neutrondiffraktion zeigen wir die magnetischen sowie die Gitterstrukturen in MgCr2O4in der verzerrten Phase. Zusätzlich werden mit inelastischer Neutronenstreuung auch die niederenergetischen Spinwellen Anregungen untersucht.

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Résumé

Les spinelles de formule chimique AB2X4 sont des systèmes importants pour l’étude du magnétisme frustré. Les sites A et B des spinelles forment respectivement un réseau diamant et un réseau pyrochlore, dont les géométries spéciales peuvent donner lieu à de nombreux phénomènes exotiques. Dans cette thèse, nous investiguons les corrélations de spin dans trois types de spinelles frustrés : les candidats de glace de spin CdEr₂X₄(X= Se, S), le candidat de liquide de spin chiral MnSc₂S₄, et le système spin-Peierls MgCr₂O₄.

Pour CdEr₂X₄(X = Se, S), nous présentons des preuves microscopiques de leurs états de glace de spin. En utilisant la diffusion inélastique des neutrons pour étudier les transitions de champs électriques cristallins et la diffusion neutronique diffuse pour étudier les corrélations de spin à courte portée, nous révélons le caractère Ising des spins de Er³⁺anisi que leurs couplages ferromagnétiques, qui sont deux prérequis pour réaliser l’état de glace de spin. Avec nos expériences de susceptibilité magnétique et de spectroscopie neutronique à écho de spin, nous explorons la dynamique des monopoles dans CdEr₂X₄(X= Se, S) et nous expliquons leur dynamique rapide avec une augmentation du taux de déplacement des monopoles.

Dans le spinelle de site A MnSc2S4, nous observons directement les corrélations spirales en utilisant la diffusion neutronique diffuse et nous confirmons expérimentalement l’état de liquide de spin spiral prédit par la théorie. Avec la diffraction des neutrons, nous clarifions le processus d’ordre par étape de MnSc2S4et découvrons une phase de type vortex triple-qsous l’application d’un champ magnétique. Les excitations de spin dans MnSc₂S₄sont étudiées avec diffusion inélastique des neutrons, et les ondes de spin obtenues confirment l’importance des perturbations dues au couplage au troisième voisin.

MgCr₂O₄est intéressant à cause de son fort couplage entre le réseau et les spins qui réduit la dégénérescence de l’état fondamental et qui mène à un état ordonné avec un réseau déformé.

En utilisant la diffraction de rayons-X synchrotron et la diffraction de neutrons, nous clarifions les structure cristalline et magnétique de MgCr₂O₄dans la phase avec le réseau déformé. Avec la diffusion inélastique des neutrons, les ondes de spin de basse énergie sont aussi étudiées.

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List of Figures

1.1 One-dimensional quantum Ising model . . . 2

1.2 Unfrustrated square plaquette and frustrated triangular plaquette . . . 3

1.3 kagomé lattice . . . 3

1.4 Exchange of identical particles . . . 5

1.5 Spinon excitations in 1D AFS=1/2 chain . . . 6

1.6 Affleck-Kennedy-Lieb-Tasaki (AKLT) state . . . 7

1.7 One representative Valence-Bond-Solid (VBS) state in 2D. . . 8

1.8 The Kitaev honeycomb model . . . 10

1.9 The pyrochlore lattice . . . 11

1.10 The hyperhoneycomb lattice . . . 12

1.11 The second- and third-order perturbation terms in quantum spin ice . . . 13

1.12 Excitations of the pyrochlore U(1) quantum spin-liquid state . . . 14

2.1 Energy-momentum relations of different probes . . . 17

2.2 Triple-axis spectrometer . . . 20

2.3 Time-of-flight spectrometer . . . 21

2.4 Spin-echo spectrometer . . . 21

3.1 Refinement for the synchrotron diffraction data of CdEr2Se4 . . . 31

3.2 Q-E map for the crystal field transitions . . . 32

3.3 Fits of the crystal field transitions in CdEr₂Se₄ . . . 33

3.4 Fits of the crystal field transitions in CdEr₂S₄ . . . 35

3.5 Fitted CEF levels in CdEr₂Se₄and CdEr₂S₄ . . . 36

3.6 Magnetization measurements for CdEr₂Se₄and CdEr₂S₄ . . . 37

3.7 Non-polarized neutron diffuse scattering results for CdEr2Se4. . . 38

3.8 x yzpolarization analysis for the diffuse scattering of CdEr2Se4. . . 39

3.9 Mean-field calculation results for the magnetic scattering data. . . 42

3.10 Monte Carlo simulation results for the magnetic scattering data. . . 43

3.11 Reverse Monte Carlo simulation results for the magnetic scattering data. . . 44

3.12 Refinement of the field-induced long-range magnetic order in CdEr₂Se₄. . . 45

3.13 Non-polarized neutron diffuse scattering results for CdEr2S4. . . 46

3.14 Mean-field calculations for the magnetic scattering in CdEr2S4. . . 47

3.15 AC susceptibility of CdEr₂S₄ . . . 49

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3.16 Monopole dynamics in the low temperature regime. . . 50

3.17 Angular dependence of the perturbation effect. . . 51

3.18 Neutron spin echo results for the high temperature dynamics in CdEr₂Se₄and CdEr₂S₄. . . 52

3.19 Frequency-reversed temperature relation of the spin dynamics in CdEr2Se4and CdEr₂S₄. . . 53

4.1 Diamond lattice and the spiral surface. . . 56

4.2 Diffuse neutron scattering map of CoAl2O4. . . 58

4.3 Refinement of the MnSc₂S₄synchrotron X-ray diffraction data. . . 59

4.4 Temperature dependence of the magnetic diffuse scattering in MnSc₂S₄. . . 60

4.5 Monte Carlo simulation results for the spin correlations in MnSc₂S₄. . . 61

4.6 Magnetic susceptibility of MnSc₂S₄measured with field along the [111] direction. 62 4.7 Magnetic susceptibility of MnSc2S4measured with field along the [001] direction. 63 4.8 Temperature dependence of the (0.75 0.75 0) Bragg peak. . . 64

4.9 2D map around the magnetic Bragg peak. . . 65

4.10 Refinements of the MnSc2S4neutron diffraction datasets. . . 66

4.11 Temperature dependence of the SNPP_{y y}element. . . 70

4.12 Relieving the ground state degeneracy by the third-neighbour couplingJ₃. . . . 71

4.13 Easy-axis anisotropy caused by the dipolar interactions. . . 71

4.14 Domain re-distribution under a [001] magnetic field . . . 72

4.15 Phase diagram of MnSc₂S₄under a magnetic field along the [001] direction. . . 73

4.16 Refinement of the triple-qstructure. . . 75

4.17 Domain re-distribution under a [111] magnetic field. . . 75

4.18 Vortex structures of the triple-qphase . . . 76

4.19 Comparison between vortex and skyrmion lattices for q = (0.25 0.25 0). . . 77

4.20 Inelastic neutron scattering results of MnSc₂S₄powder sample. . . 79

4.21 Spin wave excitations in the helical phase of MnSc2S4. . . 80

4.22 Comparison of the excitations in the helical, collinear, and triple-qphases. . . . 81

4.23 Typical scans measured in the helical, collinear, and triple-qphases. . . 82

5.1 Excitations for the spin-lattice-coupled pyrochlore lattice. . . 87

5.2 Resonance-like molecular excitations in MgCr₂O₄ . . . 88

5.3 Refinement of the 20 K synchrotron X-ray diffraction data of MgCr₂O₄. . . 89

5.4 Refinement for the 5.5 K synchrotron diffraction data of MgCr2O4. . . 90

5.5 Refinement for the 1.6 K neutron diffraction data of MgCr₂O₄. . . 92

5.6 Magnetic strucutres for theq₁=(0.5 0.5 0) component . . . 92

5.7 Magnetic strucutres for theq2=(1 0 0.5) component . . . 94

5.8 Spin dynamics in MgCr₂O₄measured on the triple-axis spectrometer EIGER. . 98

5.9 Spin wave excitations of theq₁component. . . 99

5.10 Spin wave excitations of theq2component. . . 101

6.1 The crystal electric field levels for the CdRE₂Se₄compounds . . . 105

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List of Tables

3.1 CdEr2Se4crystal electric field parameters . . . 34

3.2 CdEr₂S₄crystal electric field parameters . . . 36

4.1 Intensities measured by spherical neutron polarimetry . . . 67

4.2 Measured spherical neutron polarimetry matrices for MnSc2S4. . . 68

4.3 Calculated spherical neutron polarimetry matrices for MnSc₂S₄ . . . 69

5.1 Refinement results of the MgCr₂O₄structure . . . 89

5.2 MgCr2O4SNP matrices of theq1components . . . 96

5.3 MgCr₂O₄SNP matrices of theq₂components . . . 97

6.1 CdRE₂Se₄cystal electric field parameters . . . 105

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1 Introduction to frustrated magnetism

The study of frustrated magnetism started to flourish in the 1970s when Fazekas and Anderson proposed the resonating-valence-bond (RVB) model on the triangular lattice [1], although many of its basic ideas, including short-range correlations, spontaneous symmetry breaking, and the concept of quasiparticles, shall date back to the 1950s when the foundation of solid state physics was being laid.

Through its four decades of development, two questions have been constantly raised for different frustrated models: What is the ground state, ordered or disordered? What are the excitations, confined or fractionalized? If we remember that even the very existence of the staggered spin alignment in antiferromagnets was uncertain for some time due to quantum fluctuations [2], we will understand why these two questions continue to be puzzling even today.

Before continuing the discussion of frustrated magnetism, it is worthwhile to first clarify the concept of quantum fluctuations, which is the origin of many exotic phenomena. One of the textbook examples of quantum fluctuations is the quantum harmonic oscillator [3]. There, the noncommutativity of the momentum and position operators [q,p]=i is transferred to the commutation rule of the creation and annihilation operators [a,a^†]=1, leading to zero-point fluctuations with energyE=1/2ħω, whereωis the characteristic oscillation frequency. Similar principle holds for spin systems, but the noncommutation relation comes from the spin operators: [S^x,S^y]=iS^z. This commutation rule, after the Holestein-Primakoff transformation, leads to the canonical commutation relations of the boson creation and annihilation operators [a_i,a^†_j]=δi j, where the indicesi,j indicate the spin sites. Similar to the situation for the harmonic oscillator, this noncommutation relation leads to zero-point quantum fluctuations that will renormalize the ground state energy or even destroy the long-range ordered state [4].

One elegant example to illustrate the effect of quantum fluctuations in spin systems is the one-dimensional quantum Ising model withS=1/2 [5] that is illustrated in Fig. 1.1. Its

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g T

g_c

Ordered state Quantum disordered state Quantum Critical

Ordered ferromagnetic state

Quantum disordered state

H_x

a b

Figure 1.1 –(a)Ground states of one-dimensional quantum Ising model under zero field (top) and a dominating transverse field (bottom).(b)Quantum fluctuations in quantum Ising model are controlled by the applied field, driving the system through a quantum phase transition.

Hamiltonian can be written as:

H = −J gX

i

S^x_i −J X

〈i j〉

S_i^zS^z_j, (1.1)

whereJ >0 is the ferromagnetic exchange constant,g>0 is a dimensionless variable that can be tuned with an applied magnetic field, and spins are pointing either up or down due to their Ising character. Under zero field, the state| ↑〉 =Q

i| ↑〉ior| ↓〉 =Q

i| ↓〉iis the eigenstate of the Hamiltonian (1.1) and is obviously the ground state. However, under a strong enough magnetic field, the−J gP

iS^x_i term will dominate and, due to the noncommutation relation betweenS^xandS^y, the ground state should become an overlap between the two states:

| →〉i=Y

i

(| ↑〉i+ | ↓〉i)/p 2,

| ←〉i=Y

i

(| ↑〉i− | ↓〉i)/p

2. (1.2)

Thus the noncommutation relation ofS^xandS^y enables the manipulation of the strength of quantum fluctuations by a magnetic field, which drives the system through a quantum phase transition.

Back to the frustrated magnetism, it is immediately clear why smaller spin size is favored in the search of novel spin correlations out of the classical paradigm. Basically, the effect of quantum fluctuation scales with 1/S. Let us take an antiferromagnetically coupled spin pair as an example. The Hamiltonian can be written asH =JS1S2. In the simplest case, we have

|S₁| = |S₂|and its quantum ground state energy can be easily solved to beE_q= −JS²(1+1/S) through the following transformation:

H =1/2J[(S₁+S₂)²−(S₁)²−(S₂)²]; (1.3)

while the classical solution gives a ground state energy ofE_c= −JS². Thus the system gains an

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energy∝1/Sby allowing quantum fluctuations [6].

Another route of increasing the effect of quantum fluctuations is by geometric frustration. The logic behind this route is a little bit different from that of small spins. With small spins, we directly increase the quantum fluctuations; while with geometric frustration, we increase the energy of the classical Néel state and thus make the quantum state more favorable. Fig. 1.2 compares the classical ground state spin configurations for the nearest-neighbour antiferromagnetically coupled square and triangular plaquettes. For the un-frustrated square plaquette, all the four nearest-neighbour pairs are anti-parallel and the energy per spin is−|J|S². For the frustrated triangular plaquette, neighbouring spins define a 120^◦angle, and the energy per spin becomes−1/2|J|S², which is higher than that of the square plaquette. Therefore, classical configurations for frustrated lattices constituting of triangles will be energetically less favorable, leaving more ‘room’ for quantum fluctuations to play a role.

J J

a b

Figure 1.2 – Classical ground state configurations for nearest-neighbour antiferromagnetically coupled square(a)and triangular(b)plaquette.

For most geometrically frustrated lattices, even if quantum frustration is not strong enough to remove the long-range order, an enormous degeneracy can still emerge for the ground state. We may take the kagomé lattice as an example, which is constituted of corner-sharing triangles as is shown in Fig. 1.3. Assuming an equal antiferromagnetic coupling strength for all the nearest-neighbour bonds, the spin HamiltonianH =JP

〈i j〉S_iS_j, wherei,j denote different spin sites, can be transformed toH =1/2JP

∆(S_∆1+S_∆2+S_∆3)²+const, where∆ indexes different triangles andS_∆iwithi=1, 2, 3 denote the three spins of the triangle∆. As

Figure 1.3 – kagomé lattice constituted of corner-sharing triangles. Three sublattices are indicated with different grayscale.

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long as the state has a total triangular moment of zero, it will be the ground state. Such a degeneracy will be highly susceptible to thermal and quantum fluctuations and might give rise to order-by-disorder transitions [7, 8].

Besides being of fundamental interest, frustrated systems are also promising for potential applications. One field that is under vigorous investigations currently is the multiferroics [9, 10]. Its overlap with frustrated magnetism comes from the discovery that long-range ordered cycloidal spin structures, which are common for frustrated systems, might host a ferroelectric polarizationP∝q×(a₁×a₂), whereqis the propagation vector,a₁anda₂are two basis vectors lying in the cycloidal plane [11, 12]. This mechanism strongly couples the ferroelectric polarization with the magnetization and lends an efficient way to control the dielectric properties with a magnetic field [13] and vice versa [14].

A more tantalizing application of frustrated systems comes from the quantum spin-liquid state.

Similar to the fractional quantum Hall state [15–17], a quantum spin-liquid might possess fractional excitations that behave as non-abelian anyons [18–20]. Information can be encoded with the braiding of anyon worldlines if we can stabilize and manipulate the excitations by perturbations like impurities. More information on this rapidly developing field can be found in ref [21].

In the following, we shall briefly go through archetypical frustrated systems of different spatial dimensions. Spatial dimension plays an important role in frustrated magnetism. One well- known effect is from the Mermin-Wagner theorem [22], which claims that due to thermal fluctuations, systems with dimensionD≤2 cannot spontaneously break their continuous symmetry to enter the long-range ordered state at finite temperature ofT >0.

Another crucial effect of spatial dimension that is often neglected is its constraint on the quantum statistics. For one-dimensional systems, it is not possible to exchange two particles without crossing each other, so the statistics (whether the particle is a boson or fermion) is not well defined [23]. For two-dimensional systems, the statistics can be quite rich. Let us consider the case of two free hard core identical particles [24]. Their configuration space can be expressed as direct multiplication of two subspaces ofV₊andV₋:

configuration space={(r₁,r₂)|r16=r2,(r1,r2)∼(r2,r1)}

={(r₊,r₋)|r₋6=0,(r+,r₋)∼(r+,−r−)}=V₊⊗V₋, (1.4) wherer₊=1/2(r₁+r₂),r₋=1/2(r₁−r₂),V₊={r₊}, andV₋={r₋|r₋6=0,r−∼−r−}. The removal of ther₋6=0 point makes theV₋space topologically non-trivial: exchanging two identical particles forms a non-contractible loop aroundr₋=0. Similar to the Aharonov-Bohm effect for charged particles, a global phase factor could be added to the wavefunction after the exchange process: e^iθn^w, whereθcontrols the quantum statistics andnwis the winding number of the loop. It is immediately clear thatθ=0(π) leads to a phase factor of 1 (−1) and represents boson (fermion) statistics; while other values ofθwill lead to anyons with fractional statistics.

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1.1. One-dimensional systems

a

b

c

Figure 1.4 –(a,b)Exchange of two identical hardcore particles in two dimensions with winding numbernw=1 (a) and−1 (b).(c)Loops with winding numbernw=1 and −1 are equivalent in three dimensions through a rotation operation along the dashed line.

The situation is very different in three dimensions. Fig. 1.4a and b show two loops with the winding numbern_w=1 and−1 in two dimensions. These two loops become equivalent in three dimensions by a rotation operation shown in Fig. 1.4c. Thus we have the constraint that eⁱ^θ=e⁻ⁱ^θ, meaning thatθcan only have two possible values: 0 for the bosons andπfor the fermions. Therefore, particle-like anyons cannot exist in three dimensions and there are only bosons and fermions in our three-dimensional real world.

1.1 One-dimensional systems

Although frustration is not the main driving force behind the many exotic phenomena in one-dimensional systems [23], many concepts,e. g.fractional spinon excitations, are essential for frustrated systems of higher dimensions. Here we will start with Haldane’s conjecture for spin chains and then utilize theS=1/2 andS=1 chains to illustrate the concepts of fractional spinon excitations and valence-bond solids.

The Haldane conjecture states that properties of integer and half-odd-integer spin chains are very different [25]. For integer spin chains, excitations are gapped and the spin correlations are exponential; while for half-odd-integer spin chains, excitations are gapless and the spin correlations are algebraic. This argument has a topological origin — the Berry phase. Using the non-linear-σ-model [26], it is found that the Berry phase contribution to the actionSTcan be expressed asS_T=2ħπSQ_pand the Pontryagin indexQ_pis always an integer. This causes an additional component for the propagatore^ħⁱ^S^T=(−1)^2SQ. For half-odd-integer spin chains, e^ħⁱ^S^T= ±1; while for integer spin chains,e^ħⁱ^S^T ≡1, meaning that the Berry phase has no effect at all!

As an example for the half-odd-integer case, we take a look at the antiferromagnetic Heisenberg spin chain withS=1/2. An analytical solution for this model exists, which is known as the Bethe-ansatz [27, 28]. Noticing that the total spin along thezdirectionP

iS_i^zcommutes with the Hamiltonian, Bethe uses the fully polarized ferromagnetic state as the reference, and constructs all the eigenstates by flipping a finite number of spins. Thus the ground state has

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E ( π|J |/2 )

a b

1 2

π q

−π 0

spinon continuum

Figure 1.5 – Classical picture for the spinon excitations in a 1DS=1/2 chain.(a)The upper part shows the∆S=1 excitation by flipping one spin.(b)Applying the hopping term ofS⁻_iS⁺_i₊₁ moves the spinon by two lattice sites and separates the spinon pairs.

PiS^z_i =0 and can be constructed through a combination of all the states with half of the spins flipped. Analysis reveals that such a state exhibits critical correlations and has a total energy of

−0.44N|J|, whereN is the total number of spins [23]. This ground state energy is much lower than that of the classical Néel state of−1/4N|J|, from which we see how quantum fluctuations prevent the long-range order.

More exciting phenomena distinguish theS=1/2 chain when considering its low-energy excitations. Through a dimensional analysis [29], it is found that the solution of the∆S=1 excited state possesses two independent momentum parameters, in contrast with the single parameter expected for spin waves. This means that the magnons in theS=1/2 chain should be split into two fractional excitations known as spinons! This fractionalization process is illustrated in Fig. 1.5a. By flipping one spin, we create a∆S=1 excitation. However, operators likeS⁻_iS⁺_i+1can separate this excitation into two parts, where each part carries a∆S=1/2. The dispersion relation of the spinon is calculated to be gapless [23]:

E=π|J|

2 cos(q), q∈(−π/2,π/2] (1.5)

where the reduced Brillouin zone of (−π/2,π/2] comes from the fact that the spinons always move by two lattice sites. An immediate consequence of the fractionalization is the continuum

∆S=1 spectra shown in Fig. 1.5b, which obeys:

q=q₁+q₂, E(q)=π|J|

2 [cos(q₁)+cos(q₂)] . (1.6)

Such a continuum has been observed experimentally by inelastic neutron scattering [30, 31]

and has become the most convincing signature for fractional excitations.

Properties of the integer spin chains are very different from those of the half-odd-integer

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1.2. Two-dimensional systems

singlet projector i

_L

i

_R

i+1

_L

i+1

_R

Figure 1.6 – For the AKLT state, spins are decomposed into twoS=1/2 sites, which then form valence-bonds that connect to neighbouring sites.

ones. Renormalization group analysis of the pure non-linear-σ-model shows that for integer spin chains, excitations are always gapped and the spin correlations of the ground state are exponential [32]. An enlightening example is theS=1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state [33], which is the ground state of the Hamiltonian:

HAK LT=X

i

P_i,i+1

=X

i

1 3+1

2Si·Si+1+1

6(Si·Si+1)², (1.7)

where theP_i,i+1 operator projects theS=1 pairs into theS=2 subspace and selects the S_tot=0,1 as the ground state. Fig. 1.6 presents a schematic for the AKLT state. In such a state, eachSi=1 is decomposed into twoS=1/2 sites ofiL andiR, and the neighbouring spins are connected through a valence-bond singlet. Thus the AKLT state can be viewed as a one-dimensional valence-bond solid and its singlet-triplet excitation gap has been confirmed experimentally [34, 35].

1.2 Two-dimensional systems

With a simple extension of the phenomena discussed for the one-dimensional systems, many exotic properties of two-dimensional frustrated magnets can be readily understood. One example is the valence-bond solid state, which can be viewed as a two-dimensional analog of the AKLT state. Fig. 1.7 presents a representative valence-bond solid state realized on theS=1/2 kagomé lattice of Rb₂Cu₃SnF₁₂[36]. Here, the neighbouring spins form ordered singlets, and the low-energy dynamics originates from the gapped singlet-triplet excitations.

Different alignments of the valence-bonds give rise to different valence-bond solid states and it is sometimes a challenge to find the real ground state [37].

Another possibility also exists: instead of a unique singlet configuration, the ground state might be a superposition of different valence-bond solid states. That means, the singlets are no longer fixed but constantly fluctuating, giving rise to a resonating valence-bond state [1,38].

Theoretically, such a state can be constructed from the Bardeen–Cooper–Schrieffer (BCS)

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Figure 1.7 – One possible periodic structure realized by the singlets on the kagomé lattice.

wavefunction using the Gutzwiller projectionP_G [38]:

|ΦRV B>=P_GY

k

(u_k+v_kc_k↑^† c^†_−k↓), (1.8)

whereu_kandv_kare coefficients determined by the Hamiltonian,c_k↑^† andc^†_−k↓are electron creation operators andc_k^†_↑c₋^†_k_↓creates a spin-singlet cooper pair. The role of the Gutzwiller projectionP_Gis to remove the components with doubly occupied sites and reduce the enlarged Hilbert space to the physical region [4].

With the ground state clear, we can move on to the excitations. One possible excitation are again the spinons: these are unpairedS=1/2 spins similar to those of the half-odd-integer spin chains. Since the spinons are fractionalized, theoretical descriptions of their dynamics in the resonating valence-bond state usually employ the parton technique, with the expectations that the actual low-energy dynamics correspond directly to the dynamics of the partons [39].

Similar to the slave-boson technique [40], here one spin operatorSis decomposed into two fermion operatorsf:

~S=1 2

X

αβ

f_α^†~σαβf_β, (1.9)

where~σis the vector of Pauli matrices and the physical constraint ofP

αf_α^†f_α=1 should be imposed. Bosonic representation for the partons, i.e. the Schwinger boson technique [41], follows similar derivations. Then the exchange term can be expressed as:

~S_i·~S_j = 1

4f_i^†_α~σ_αβf_iβ·f_j^†_γ~σ_γδf_jδ (1.10)

= 1

2f_iα^† f_jβ^† f_jαf_iβ−1 4,

where the relation~σ_αβ·~σ_α⁰_β⁰=2δ_αβ⁰δ_α⁰_β−δ_αβδ_α⁰_β⁰has been used. To tackle the four-fermion

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1.2. Two-dimensional systems

interaction, we can employ the mean-field approximation by introducing [26]:

χi j = X

α

Df_iα^† f_jαE ,

∆i j = X

αβ²_αβ f_iαf_j_β®

, (1.11)

where²αβis the Levi-Civita tensor. Thus the Hamiltonian is transformed into a quadratic form with the main components:

Hmean=X

i j−3 8J_{i j}

"

χi jX

α f_iα^† f_jα+∆i jX

αβ

f_iα^† f_jβ^† +h.c.

#

. (1.12)

We see thatχi jand∆i jare just the effective coefficients for the kinetic and potential energies.

Specifically, when∆i j=0 andχi j=χ¯i jis free of fluctuations, this Hamiltonian just describes the dynamics of free spinons.

What happens if there are fluctuations? Staying with the case of∆i j=0, since the magnitude fluctuation of ¯χi jis negligible under the saddle-point approximation, the only fluctuation we need to consider is the phase fluctuation of ¯χi je^ia^{i j} [24]. The Hamiltonian 1.12 becomes:

Hmean=X

i j−3 8J_{i j}

·

χ¯i je⁻^ia^{i j}X

α f_iα^† f_jα+h.c.

¸

(1.13)

Such a Hamiltonian is invariant under the U(1) gauge transformation:

a_{i j} → a_{i j}+θi−θj, (1.14)

f_i → f_ie^iθⁱ, (1.15)

which might lead to a U(1) quantum spin-liquid state. On the other hand, if∆i jhas a non- zero value, the Hamiltonian (1.12) will instead possess a Z₂ symmetry under f_iα→ −f_iα, f_iα^† → −f_iα^†, which gives rise toZ₂gauge fluctuations and a possibleZ₂quantum spin-liquid state [26, 39].

Now we can see why the excitation gap is so important in the study of quantum spin-liquids.

If there is a finite gap for the parton excitation, then the matter field can be integrated out and the low-energy dynamics becomes a pure gauge theory, being either U(1) orZ2depending on the specific mean-field ansatz [26]. However, when partons have gapless excitations, the low-energy dynamics should involve both the matter and gauge fields. Examples for the latter case includes the recently identified U(1) quantum spin-liquid phase in YbMgGaO4[42, 43], where the observed spinon surface is a direct confirmation of the Fermi liquid behavior of the gapless spinon dynamics. Further classification of quantum spin-liquids is possible with the projective symmetry group method and can be found in Ref. [44].

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y x z

gapless gapped Jz = 1, Jx = Jy = 0

J_x = 1,

J_y = J_z = 0 J_z = 1,

J_x = J_y = 0

a b

Figure 1.8 –(a)The Kitaev honeycomb model with bonds divided into three typesx, y, z depending on their directions. The shaded triangular block illustrates the decomposition of the spin operator into four Majorana fermions.(b)The phase diagram of the Kitaev honeycomb model. Excitations in the shaded central phase are gapless while those in the three phases boarding the corner have a finite gap [48].

The existence of an excitation gap is also important for the definition of the topological order [45], which arises from the long-range entanglement of the ground state and determines the fractional statistics of the excitations [46, 47]. One way to define the topological order is to use the topological degeneracy of the ground state. For example, on a compact Riemann surface with genusg, the ground state degeneracy of the chiral spin liquid is 2k^g, wherek is just the topological invariant that characterizes and classifies different chiral spin liquid states [45]. Considering that the existence of a finite gap is the prerequisite for the definition of ground state degeneracy, topological order is thus expected only for quantum spin-liquids with a finite gap [47].

One example that illustrates many of the exotic properties of two-dimensional quantum spin-liquids is the Kitaev honeycomb model [48]. It is constructed with anisotropic exchange interactions:

H =Jx X

〈i j〉∈xσ^x_iσ^x_j+Jy X

〈i j〉∈yσ_i^yσ^y_j+Jz X

〈i j〉∈zσ^z_iσ^z_j. (1.16)

Here, the honeycomb bonds are divided into three types ofµ=x,y,z depending on their directions, which is illustrated in Fig. 1.8a. Kitaev found that a decomposition of the complex fermion parton into Majorana fermionsf =c⁰±ic" makes the Hamiltonian (1.16) quadratic:

H = i 4

X

〈i j〉

A_{i j}c_ic_j, (1.17)

whereA_{i j} is a coefficient that depends on the exchange couplings. Diagonalization of this Hamiltonian leads to aZ2quantum spin-liquid as the ground state, and its fermionic excitations²can be either gapped or gapless depending on the exchange parameters, which is summarized in Fig. 1.8b. Of special interest is the gapless phase, which becomes gapped

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1.3. Three-dimensional systems

a b c

Figure 1.9 – The pyrochlore lattice consists of corner-sharing tetrahedra. When viewed along the [111] direction, it can be seen as a stacking of the 2D kagomé and triangle lattices. The dashed lines emphasize the kagomé lattice in the [1¯11] direction.

under a magnetic field and gives rise to non-abelian anyonsσthat obey the fusion rule of σ×σ=1+², where 1 denotes the vacuum state [48]. This means that when twoσanyons are brought together, they will either annihilate each other or survive as an²fermion, with a possibility depending on the braiding of anyon worldlines. Thus the Kitaev honeycomb model provides a candidate implementation for topological qubits.

1.3 Three-dimensional systems

There is no reason that the effect of frustration discussed above should be excluded from three-dimensional systems. One ‘trivial’ situation is that a three-dimensional system can be somehow decomposed into two-dimensional layers with perturbational couplings in-between.

The pyrochlore lattice is just such a candidate. As is shown in Fig. 1.9, this lattice of corner- sharing tetrahedra can be viewed as the stacking of kagomé and triangle layers along the [111]

direction. And the valence-bond solids of the kagomé layers, if exist, shall naturally give rise to gapped excitations [49]. In some special cases, even a dimensional reduction to 1D chains might be possible. For example, in the spinel CuIr2S4, Ir ions have an average valence of +3.5 and constitute a pyrochlore lattice [50]. However, due to the orbital degree of freedom of Ir⁴⁺, charge-ordered chains of Ir⁴⁺-Ir⁴⁺-Ir³⁺-Ir³⁺will arise in thex yplane, with spins of the Ir⁴⁺-Ir⁴⁺pairs forming a valence-bond singlet [51].

Another route to realize a three-dimensional frustrated system is through the ‘distortion’ of a two-dimensional lattice [52, 53]. One typical example is the hyper-honeycomb lattice [54, 55], where the local coordination for each site, despite its 3D geometry, remains the same as the original 2D honeycomb lattices. Thus a three-dimensional analog of the Kitaev model can be introduced for the hyper-honeycomb lattice and the possibility of a similar quantum spin-liquid ground-state has been predicted [56]. However, the statistics of the excitations

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c b a

Figure 1.10 – The hyperhoneycomb lattice realized inβ-Na2IrO3[54]. The local coordination remains the same as for the two-dimensional honeycomb lattice.

in the honeycomb and hyper-honeycomb lattice cannot be the same. As is discussed in the beginning of this chapter, quasiparticles in three dimensions should be either boson or fermion. Therefore, non-trivial statistics in the hyper-honeycomb lattice should rely for example on loop excitations [57–59].

As an example for the frustration effect in three-dimensions, we take a look at the U(1) quantum spin-liquid state realized on the pyrochlore lattice with nearest neighbour interactions [60].

The main components of the Hamiltonian are the dominant easy-axis exchange interaction Jz>0 and the transversal perturbationJ_⊥>0:

H =HI+H⁰, HI= J_z

2 X

t (S_t^z)², H⁰=J_⊥

2 X

〈i j〉

(S⁺_iS⁻_j +h.c.), (1.18)

whereS^z_t is the summation of easy-axis spin componentS^z over the tetrahedrat and〈i j〉 denotes the nearest-neighbouring links of the pyrochlore lattice. We note that the easy-axis directionzmight be defined locally and varies at different sites, which is the case for spin ice [61] (see Chapter 3). The ground states ofHIare constrained only locally byS^z_t=0 and thus possess O(N_t) degeneracy, whereN_tis the total number of tetrahedra. Obviously,H⁰is not commutative withHI and is the orgin of quantum fluctuations.

Now we can use the textbook degenerate-state perturbation theory to treat theH⁰term [3].

The first-order perturbation term inevitably creates a pair of tetrahedra withS^z_t= ±1 and shall not contribute to any virtual excitation process. Thus the effective perturbational Hamiltonian projected to theHIground-state space should start from the second-order:

H_eff=Ph

−H⁰1−P

HI H⁰+H⁰1−P

HI H⁰1−P

HI H⁰+ ···i

P, (1.19)

whereP projects onto theHI ground state manifold. When treatingS^z= ±1/2 as the pres- ence/absence of a boson, the effect ofH⁰can be viewed as the hopping of bosons. As is shown

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1.3. Three-dimensional systems

a b

c

Figure 1.11 –(a)The second order perturbation term involves virtual hoping forwards and backwards of bosons. (b,c)Two possibilities of the three-order perturbation: three-step hopping of one boson along the triangle (b) and one-step hopping of three bosons along a hexagon (c).

in Fig. 1.11a, the second-order perturbation involves a continuous forwards/backwards hopping along one nearest-neighbour bond. A similar process also holds for the third-order perturbations, where the boson hops along the bonds of a triangle. Of special interest is the third-order perturbation process shown in Fig. 1.11c, where the three bosons on a hexagon collaboratively hop by one step. Existence of such a collaborative hopping relies on the fact that the six bonds of the hexagon are exactly shared by six corner-sharing tetrahedra, thus theS^z_t =0 constraint is maintained under such a process. Now the effective Hamiltonian becomes:

Heff=(J²_⊥/Jz)(J_⊥/Jz−1)Nt

+J_ringX 7

(S⁺₁S⁻₂S⁺₃S⁻₄S₅⁺S⁻₆+h.c.), (1.20) whereJ_ring=3J_⊥³/2J²_z and the summation is over all of the hexagons.

Next, we transform the spin model (1.20) to a field theory using the quantum rotor model [5].

Since the field variables are defined on links instead of sites, we need to introduce the dual diamond lattice that connects the center of the pyrochlore tetrahedra. The site variable S_i on the pyrochlore lattice now becomes a link variableS_{r r}⁰ on the dual diamond lattice, wherer r⁰denotes the link that cross the sitei. With the correspondence ofS^z=n−1/2 and S^±=exp(±iφ), the ring exchange Hamiltonian can be expressed by the rotor variablesn∈Z andφ∈(−π,π]:

Hp=U 2

X

〈r r⁰〉

(n_{r r}⁰−1/2)²−KX

7^cos(^φ¹⁻^φ²⁺^φ³⁻^φ⁴⁺^φ⁵⁻^φ⁶^), ^(1.21) where the first term is a softened version of the hard-core boson condition withn=0,1.

With the bipartite character of the diamond lattice, we can assign a direction to each link as pointing from one sublattice to the other and define the field parameters:

E_{r r}0= ±(n_{r r}0−1/2), A_{r r}0= ±φr r⁰. (1.22)

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J_zz electric charge Q_e

J_ring magnetic charge Q_m photon Energy

Figure 1.12 – Excitations of the pyrochlore U(1) quantum spin-liquid consist of a massless photon mode of the gauge field, gapped magnetic charge excitations at an energy scale of Jring=3J³_⊥/2J_z², and gapped electric charge excitations at an energy scale ofJzz.

Then the Hamiltonian can be expressed as:

Hp=U 2

X

〈r r⁰〉

E_{r r}² 0−KX 7^cos

³ X

r r"⁰∈7^A^{r r}⁰^´

=U 2

X

〈r r⁰〉

E_{r r}² 0−KX

7^cos(^{∇ ×}^A^). ^(1.23)

Here,∇ ×A is the lattice version of curl and the fieldA is defined on the links of a dual plaquette lattice that connects the hexagon center of the original dual diamond lattice. The Hamiltonian (1.23) just describes a compact U(1) lattice gauge field theory, whereE_{r r}⁰ and B_p .

= ∇ ×A correspond to the electric and magnetic field, respectively. Taylor expansion of the cosine term up to the second order leads toH ≈U/2P

〈ab〉E_ab² +K/2P

pB_p², which naturally reminds us of the total energy of continuum electromagnetism in vacuum: H ≈ Rd³x£

²/2|~E|²+1/2µ|B~|²¤

. Therefore, the low-energy excitation of the model (1.23) should be an analog of the photon excitation in the standard U(1) quantum electrodynamics [60].

Besides the photon excitation of the gauge fields, there are also matter field excitations of electric and magnetic charges that can be defined as the divergence of the fields:

Q_e(r)=(∇ ·E)_r=X

r⁰

E_{r r}0, Q_m(r^∗)=(∇ ·B)_r∗=X

r^∗0

B_r∗r^∗0, (1.24)

whereQ_e andQ_m live on the sites of the diamond latticer and dual plaquette latticer^∗, respectively. From the spin model, we know that flipping one 1/2 spin creates a∆S=1 excitation that is shared by a pair of positive and negative electric charges that act as the source and sink of the electric field. Thus each electric chargeQ_ecarries a fractional spin 1/2 excitation and corresponds to a deconfined spinon in the language of the spin liquid.

Neutron scattering investigation of the spin correlations in frustrated spinels

Thesis

Reference

Neutron scattering investigation of the spin correlations in frustrated spinels

GAO, Shang

GAO, Shang. Neutron scattering investigation of the spin correlations in frustrated spinels . Thèse de doctorat : Univ. Genève, 2017, no. Sc. 5095

DOI : 10.13097/archive-ouverte/unige:95576 URN : urn:nbn:ch:unige-955764

_______________________________________________________________

Neutron scattering investigation of the spin correlations in frustrated spinels

THÈSE

Shang Gao

Abstract

Zusammenfassung

Résumé

Contents

List of Figures

List of Tables

1 Introduction to frustrated magnetism

1.1 One-dimensional systems

a b

singlet projector i

i

i+1

i+1

1.2 Two-dimensional systems

1.3 Three-dimensional systems

c b a