Thermal fluctuations in low-dimensional quantum antiferromagnets

Texte intégral

(1)Thesis. Thermal fluctuations in low-dimensional quantum antiferromagnets. KESTIN, Noam. Abstract Dans cette thèse, nous nous intéressons aux effets thermiques dans des systèmes quantiques de spin fortement corrélés en basse dimension. Ces systèmes peuvent être étudiés à l'aide de matériaux dans différentes phases via l'application de forts champs magnétiques externes. La façon dont le matériau se comporte dans chaque phase peut-être décrit à l'aide de fonction de corrélation sur des modèles effectifs quantiques. La technique de prédilection utilisée dans cette thèse est la méthode numérique de renormalisation de la matrice densité (DMRG) qui peut être directement comparée à l'étude expérimentale des propriétés magnétiques via diffraction inélastique de neutron (INS) parachevée par deux différents groupes de recherche. Ces méthodes indépendantes (théorie et expérience) corroborent et rendent la physique particulièrement attrayante. Nous étudierons à l'aide de ces techniques les déviations en température de théories analytiques établies à basse dimension sous le nom de liquide de Luttinger (TLL).. Reference KESTIN, Noam. Thermal fluctuations in low-dimensional quantum antiferromagnets. Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5414. DOI : 10.13097/archive-ouverte/unige:127613 URN : urn:nbn:ch:unige-1276138. Available at: http://archive-ouverte.unige.ch/unige:127613 Disclaimer: layout of this document may differ from the published version..

(2) UNIVERSITÉ DE GENÈVE Section de Physique Département de physique de la matière quantique. FACULTÉ DES SCIENCES Professeur T. Giamarchi. Thermal Fluctuations in Low-dimensional Quantum Antiferromagnets. THÈSE présentée à la Faculté des Sciences de l’Université de Genève pour obtenir le grade de docteur ès Sciences, mention Physique par. Noam Kestin du Grand-Saconnex (GE). Thèse n◦ 5414. GENÈVE Atelier d’impression ReproMail 2019.

(3) UNIVERS|TÉ DE GENÈVE FACUTTÉ DES SCIENCES. DOCTORAT ES SCIENCES, MENTION PHYSIQUE. Thèse de Monsieur Noam KESTIN intitulée. <<Thermal. Fluctuations in Low-dimensional Quantum Antiferromagnets>. La Faculté des sciences, sur le préavis de Monsieur T. GIAMARCHI, professeur ordinaire. et directeur de thèse (Département de physique de la matière quantique), Monsieur C. RUEGG, professeur titulaire (Section de physique, MaNEP (Materials with Novel Electronic Properties). -. Research with Neutrons and Muons, Paul Scherrer lnstitute,. Villigen, Switzerland), Monsieur F. MILA, professeur (lnstitut de physique, EPFL. Polytechnique Fédérale. de. - Ecole. Lausanne, Suisse), Monsieur J.-S. CAUX, professeur. (lnstitute of Theoretical Physics, lnstitute. of Physics, University of Amsterdam,. The. Netherlands), autorise I'impression de la présente thèse, sans exprimer d'opinion sur les propositions qui y sont énoncées. Genève, le 20 novembre 2019. Thèse -5414Le Doyen. N.B.. -. La thèse doit porter la déclaration précédente et remplir les conditions énumérées dans les "lnformations relatives aux thèses de doctorat à I'Université de Genève".. ii.

(4) ‫לכל המשפחה‬ Je vous aime, sogar abwesend.. ‫شكرا لك دهلية‬.

(5) iv.

(6) Acknowledgements This PhD end has been an invaluable experience for me. I am grateful to all the people who made it possible. I would like to thank in particular :. ♣. T. Giamarchi for giving me the opportunity to work by his side. I am thankful for his expert scientific advice, constant kindness and humility.. ♦. S.E. Tapias-Arze for the physics passion we shared during our studies together; E. Coira and L. Foini for the great collegial atmosphere they created during my time here. I miss you every day.. ♠. D. Pantelic, F. Hartmeier, G. Augé-Freytag and the permanent of the university for the great personal support; N. Bachar and I. Kapon for their guidance.. ♥. B. Normand, C. Rüegg, D. Blosser and A. Zheludev for their collaboration on experimental systems; S. Ward and B. Wehinger for their explanation, friendly collaboration and arranging lab visits.. ♣. C. Berthod and P. Bouillot for giving me access to cluster responsibilities and for constantly solving my data issues.. ♦. A. Kantian, N.A. Kamar, S. Uchino, S. Furuya, S. Takayoshi, A. Borin, A. Goremykina, I. Protopopov and D.A. Abanin for the shared discussions they facilitated.. ♠. The Swiss National Foundation (FNS) under Division II and the Department of Quantum Matter Physics (DQMP) of University of Geneva (UNIGE) that provided financial and academical support; I am happy to remain altruist and in good health after this experience.. ♥. Finally, I want to thank my family, friends, especially Dehlia, for their fundamental support. This thesis would not have been possible without them! And I wish the next generation great success growing our knowledge further. ♥♥♥♥♥♥. v.

(7) vi.

(8) Abstract - Résumé. Dans cette thèse, nous nous intéressons aux effets thermiques dans des systèmes quantiques de spin fortement corrélés en basse dimension. Ces systèmes peuvent être étudiés à l’aide de matériaux dans différentes phases via l’application de forts champs magnétiques externes. La façon dont le matériau se comporte dans chaque phase peut-être décrit à l’aide de fonction de corrélation sur des modèles effectifs quantiques. Avec les meilleurs outils de calcul connus à ce jour pour l’étude des corrélations dynamiques à température finie en basse dimension, nous comprenons l’enjeu qu’il y a dans les propriétés magnétiques. La technique de prédilection utilisée dans cette thèse est la méthode numérique de renormalisation de la matrice densité (DMRG) qui peut être directement comparée à l’étude expérimentale des propriétés magnétiques via diffraction inélastique de neutron (INS) parachevée par deux différents groupes de recherche. Ces méthodes indépendantes (théorie et expérience) corroborent et rendent la physique particulièrement attrayante. Nous étudierons à l’aide de ces techniques les déviations en température de théories analytiques établies à basse dimension sous le nom de liquide de Luttinger (TLL). ———————————. vii.

(9) ——————————— In this thesis, we investigate thermal effects of strongly correlated lowdimensional spin systems. Their behavior could be studied in different phases by the use of large external magnetic fields on real compounds. The description of the materials requires focus on the correlation functions of the effective quantum models. We used the best numerical tools to date to study dynamical correlations in low-dimensional systems at finite temperature and end with a greater insight into magnetic materials. We used the Density Matrix Renormalization Group (DMRG) to study the different microscopic models and compared with Inelastic Neutron Scattering (INS) experiments on various compounds performed by different groups. Those two different independent methods (experience and theory) make the physics so exciting. In this work, we study both numerical and analytical quasi-one-dimensional spin systems which offer rich physics and show deviations to expected analytical behavior such as the low temperature bosonization prediction or Tomonaga-Luttinger Liquid theory (TLL). ———————————. viii.

(10) Contents. 1 Introduction - thesis plan 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Plan of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1 3. 2 Brief introduction to quantum magnetism 2.1 Sorry, what is a spin ? . . . . . . . . . . . . 2.1.1 Spin-1/2 . . . . . . . . . . . . . . . . 2.1.2 Effective magnetic moment . . . . . 2.2 Magnetism in condensed matter . . . . . . . 2.2.1 Electrons in a solid . . . . . . . . . . 2.2.2 Many-body models . . . . . . . . . . 2.2.3 Magnetic exchanges . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 5 5 6 7 8 9 11 12. 3 Low-dimensional antiferromagnetic 3.1 Spin-1/2 models . . . . . . . . . . 3.1.1 The XXZ spin chain . . . . 3.1.2 The dimerized model(s) . . 3.2 Synthetized compounds . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 15 15 15 17 21. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 25 25 27 28 29 29. models . . . . . . . . . . . . . . . . . . . .. 4 Methods 4.1 Tomonaga-Luttinger Liquid . . . . . . . . 4.1.1 Bosonization of the spin-1/2 chain 4.1.2 Correlations in TLL theories . . . 4.2 Density Matrix Renormalization Group . 4.2.1 Numerics . . . . . . . . . . . . . .. ix. . . . . ..

(11) CONTENTS. 4.3. 4.2.2 Algorithms . . . . . . . . . . . . 4.2.3 Outputs . . . . . . . . . . . . . . Inelastic Neutron Scattering . . . . . . . 4.3.1 Scattering surface and dynamical 4.3.2 Analysis of the spectra . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . structure factor . . . . . . . . .. 5 Finite-temperature correlations in a quantum spin uration 5.1 Experimental setup . . . . . . . . . . . . . . . . . . . 5.1.1 Sample K2 CuSO4 Cl2 and experimental setup 5.1.2 Measured quantities . . . . . . . . . . . . . . 5.2 Data analysis and numerical calculations . . . . . . . 5.2.1 Spin Hamiltonian . . . . . . . . . . . . . . . . 5.2.2 Effective 1D spin chain . . . . . . . . . . . . 5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Components of the excitation spectrum . . . 5.3.2 Scaling of critical fluctuations . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 32 36 37 37 39. chain near sat. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 41 42 42 42 43 43 44 46 49 50 54. 6 Low-dimensional correlations under thermal fluctuations 6.1 Models under consideration . . . . . . . . . . . . . . . . . . 6.2 Correlations for ladders, dimers and chains . . . . . . . . . 6.2.1 Chains C . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Ladders L . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Dimers D . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Results for the correlation functions . . . . . . . . . . . . . 6.3.1 Discussion of the T-DMRG results . . . . . . . . . . 6.4 Comparison with field theory . . . . . . . . . . . . . . . . . 6.4.1 Extraction of TLL parameters at T = 0 . . . . . . . 6.4.2 Bosonization and T-DMRG comparison . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 55 55 56 56 57 57 62 63 65 67 69 72. 7 Thermal fluctuations in the ladder compound 7.1 Experimental setup . . . . . . . . . . . . . . . . . 7.1.1 (C5 H12 N)2 CuCl4 compound . . . . . . . . 7.1.2 Effective model . . . . . . . . . . . . . . . 7.2 Experimental measurement . . . . . . . . . . . . 7.3 Results and discussions . . . . . . . . . . . . . . 7.3.1 Magnetic deviation and additional spectra 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 73 74 74 74 75 78 81 82. 8 Conclusion and perspective. . . . . . . .. . . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . . .. 85. x.

(12) CONTENTS. Appendix A Mapping to fermions and the XY spin-1/2 model A.1 Jordan-Wigner transformation . . . . . . . . . . . . . . . . . . A.2 The XY spin-1/2 model . . . . . . . . . . . . . . . . . . . . . . A.3 Correlations and magnetization in the XY model . . . . . . . . A.3.1 Transverse correlation Six Sjx . . . . . . . . . . . . . . A.3.2 Longitudinal correlation Siz Sjz . . . . . . . . . . . . . A.4 Animation details . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. 89 89 90 90 91 92 92. Appendix B Mapping to bosons and spin-wave theory B.1 Hard-core bosons . . . . . . . . . . . . . . . . . . . . . B.2 Linear spin-wave theory . . . . . . . . . . . . . . . . . B.2.1 Spin-waves on polarized configurations . . . . . B.2.2 Spin-waves on unpolarized configurations . . . B.3 Semi-classical mapping . . . . . . . . . . . . . . . . . . B.3.1 Spin rotation symmetry . . . . . . . . . . . . . B.3.2 Spin coherent mapping . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 93 93 94 95 96 97 98 98. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. Appendix C Global conservations and details of numerical algorithms101 C.1 Conservation of quantum number . . . . . . . . . . . . . . . . . . . 101 C.1.1 Implementation details . . . . . . . . . . . . . . . . . . . . . . 102 C.2 Publication details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 C.2.1 Previously used code . . . . . . . . . . . . . . . . . . . . . . . 103 C.2.2 Code based on our implementation . . . . . . . . . . . . . . . 103 C.3 Convergence and precision . . . . . . . . . . . . . . . . . . . . . . . . 105. xi.

(13) CONTENTS. xii.

(14) CHAPTER. 1. Introduction - thesis plan. 1.1. Introduction. Nowadays, experiments are able to realize well-controlled microscopic Hamiltonians 1 using optical lattices 2,3 and quantum magnets. 4 In particular, low-dimensional materials exhibit such strong and complex quantum fluctuations that we cannot be immune to a new quantum application at any time. Indeed, the rich number of phases that materials provide (supraconductors, metals, semiconductors, band, Mott and Anderson insulator) arising from different dimensionality, connectivity, interaction, orbitals, effective masses and spin sizes, is broadening the picture. For this reason, one needs to start with simplified models. One class of systems which presents a very rich set of phases, depending on the precise microscopic interactions, is the one of quantum one-dimensional or quasione-dimensional magnets. 5,6 Indeed, such systems possess ground states ranging from quasi-long-range magnetic order to spin liquids. 7 A famous illustration is the Haldane “conjecture” 8 that predicts different ground states between integer and half-integer spin chains. Furthermore, the coupling of several one-dimensional chains such as ladders lead to a very rich phase diagram as a function of the number of legs. 9 The correlations in these systems can be probed by, e.g., Inelastic Neutron Scattering (INS) 10 or Nuclear Magnetic Resonance (NMR) 11,12 experiments, giving a complete access to the spatial or time dependence of the spin-spin correlations. Theoretically, the computation of correlations in a low-dimensional system is a challenge by itself. At the numerical level, the Density Matrix Renormalization. 1. Correlation hS x (q, ω)S x i (see Eq. (A.4) with Jxy = 1 and various hz ). Enjoy the thesis!.

(15) 1. INTRODUCTION - THESIS PLAN. Group (DMRG) 13–19 algorithms showed auspicious outcome since its start. In particular, two-leg ladders have extensively been studied at zero temperature. 20,21 Quite recently, finite temperature correlations 22–24 have attracted more attention using similar theoretical techniques 25–28 which open a direct comparison opportunity with real compounds in the strongly correlated regime. In addition to the numerics, the correlation functions at T = 0 could be computed analytically using the Betheansatz formalism. 29,30 In this thesis, we used numerical and analytical tools in order to extract dynamical correlation functions in quantum low-dimensional spin-1/2 systems at finite temperature. The quantities under consideration can be confronted with various experimental measurements on different compounds. The overall thesis is based on mainly three different projects and a more detailed outline of each chapter can be found in the Plan of the thesis section below. In the first part, 31 we collaborated with A. Zheludev’s group (see Chap. 5) and studied the correlations 32–35 within the Quantum Critical Point (QCP) 36,37 region of weakly coupled spin-1/2 chains. Using INS experiment on the compound K2 CuSO4 Cl2 , 38,39 we compared our low-dimensional results with the real compounds. A finer study with the DMRG tools shows that the temperature dependence of weakly coupled chain in that phase would need more than one model description to describe fully the measurement. This work opens the road to a finer study of dimensionality and temperature dependence in low-dimensional quantum spin-1/2 systems. The second part 40 was then largely motivated by dimerized spin-1/2 compounds such as the one of Cu(NO3 )2 · 2.5D2 O, 41 (C5 H12 N)2 CuCl4 (BPCC) 42 and the similar bromium substitution (C5 H12 N)2 CuBr4 (BPCB). 42–45 In those systems, the different energy scales provide an appropriate test of the range of validity of the low-dimensional theory expectation, namely the bosonization prediction or Tomonaga-Luttinger Liquid theory (TLL). 5 In fact, those theories describe perfectly spin chains at low temperature and frequency. In this thesis, we study various low-dimensional models by increasing the dimensionality and the temperature and check the similarities or divergences to the well established TLL predictions. At intermediate frequencies (beyond those predictions), we additionally studied the behavior of the modes once other states than the ground states starts to be thermally populated. 46,47 As a result, one sees some structures appearing on the top of washedout temperature effects. In the third part, 48 we compared our theoretical expectation with experiments in a collaboration with C. Rüegg’s group (see Chap. 7) on the BPCC compound. We extracted from noisy INS excitation spectrum – of only one order of magnitude stronger than the background noise – data which validates our previous theoretical findings. Our work gives access to a finer study of the higher energy modes, where the thermal effects are now breaking down the standard T = 0 picture. This finding opens questions in related models such as the t − J model describing the T = 0. 2.

(16) 1.2 Plan of the thesis. dispersion of the triplet zero state of a dimer. 21 To summarize, we used DMRG based algorithms to study dynamical correlations in various low dimensional models at finite temperature. The models under consideration can be related to each other in some regime of the phase diagram and can be directly compared with real experimental probes!. 1.2. Plan of the thesis. You are now in the chapter 1 (Introduction - thesis plan) of the thesis. We start and end this thesis with additional material on quantum magnetism (see Chap. 2 and App. B) that is nearly related to the main work of this thesis. Then, the first chapters (see Chap. 3 and 4) introduce the models and methods under consideration whereas the following chapters discuss the main results (see Chap. 5 to 7). We end with a brief perspective (see Chap 8) and supplementary material (see App. A and C). Here is a short overview of the organization of this thesis. • in the chapter 2 (Brief introduction to quantum magnetism), we introduce textbook concepts that we learned and used during the thesis time. This chapter represents our personal digest of quantum magnetism. • in the chapter 3 (Low-dimensional antiferromagnetic models), we introduce the main models and respective phase diagram encountered in this work. We then briefly illustrate some of them in real synthesized compounds. • in the chapter 4 (Methods), we will present the analytical, numerical and experimental techniques that we used to extract correlations in the models of the previous chapter. • in the chapter 5 (Finite-temperature correlations in a quantum spin chain near saturation), we study weakly coupled spin chains at various temperatures and confront the DMRG techniques to INS experiment on the compound K2 CuSO4 Cl2 done by A. Zheludev’s group. Due to the relevance of the interchain coupling, we end with a good description of the 1D framework by introducing two low-dimensional models. Additionally, we checked the validity of the exponent in the QCP region. • in the chapter 6 (Low-dimensional correlations under thermal fluctuations), we study at finite temperature different low-dimensional models, from the XXZ model to dimerized systems. The low-frequency modes show a TLL description from which one deviates once the temperature gets relevant. Intermediate frequency modes show formation of incoherent band at finite temperature due to the decoherence of the weakest coupling. The thermal correlations become overall very weak in intensity as one increases the temperature.. 3. Correlation hS x (q, ω)S x i (see Eq. (A.4) with Jxy = 1 and various hz ). Enjoy the thesis!.

(17) 1. INTRODUCTION - THESIS PLAN. • in the chapter 7 (Thermal fluctuations in the ladder compound), we use the previous methods and apply them to the study of the temperature effects in the BPCC compound made of weakly coupled strong-rung two-leg ladders. The comparisons between the theoretical results and the experimental ones show an excellent agreement. One however ends with signal of the order of the background noise but it follows the theory in an impressive agreement. As a result, the experiment validates our previous analysis and opens the door of a finer study of the physics of the propagation of higher energy modes. • in the chapter 8 (Conclusion and perspective), we end with a retrospective of the last three chapters and briefly propose perspectives within the acquired techniques. • in the appendix A (Mapping to fermions and the XY spin-1/2 model), we introduce the mapping from spin-1/2 to spinless fermions. One can compute correlations in the XY spin model using Pfaffian operators. • in the appendix B (Mapping to bosons and spin-wave theory), we follow personal notes on linear spin-wave theory which opens predictions to other models restricted to simpler configuration of the phase diagram. • in the appendix C (Global conservations and details of numerical algorithms), we give some details of the implementation of the DMRG algorithms. The main result in this thesis is based on our own implemented code. Algorithm with global quantum numbers at finite temperature are less common than the initial T = 0 DMRG formulation by S. White. 13,14. 4.

(18) CHAPTER. 2. Brief introduction to quantum magnetism. This first chapter does not aim to a general introduction to solids and magnetism. It is a personal part crossing different advanced student references. 1,4,5 Those notes explain briefly the main mechanism of quantum magnetism and strongly correlated system which are always present throughout the thesis. The main introduction to the published works can be found in Chap. 3 where we introduce the low-dimensional models under consideration of this thesis.. 2.1. Sorry, what is a spin ?. Even though it relies on abstract mathematical structure, a spin is easily understood as a quantized angular momentum. 1 It took time for physicists to understand the underlying quantum mechanic mechanism as well as the interplay role of symmetry and preserved quantities of the fundamental particles. 1 Modern theories rely on Lie group theory and their algebra representation. Spins and angular momentum possess different group structure even though they share the same algebra (see the commutation relation Eq. (2.1)). 1 One can see it by the generator rotating by 2π 2π z around the ẑ or quantification axis; the ei ~ Ŝ |↑i picks up a sign ei2πsz for half spins (sz ∈ 21 Z) while rotation remains untouched ei2πN Ω̂z (counting the number of rotation N ∈ Z around Ω̂z ). As a result, half spin operators and rotations differ in their global group structure – half spins are sometimes said to double cover the. 5.

(19) 2. BRIEF INTRODUCTION TO QUANTUM MAGNETISM. sphere since there is a surjective homomorphism from SU (2) to SO(3).1 Unlike the rotations, no spherical harmonics representation exists for generic spin-s and one needs to introduce a spinorial representation of discrete dimension 2s + 1 on which the spin acts. The Lie algebra of a spin-s is an angular momentum algebra su(2) ' so(3) as described above. There are 3 generators in this algebra whose dimension depends on the spinorial s representation (dimension of the matrix basis will be 2s + 1). As a rule of thumb, the generic spin-s can be found in any textbook 1 as • an operator Ŝ = (Ŝ x , Ŝ y , Ŝ z ) following an angular momentum commutation relation h i δ Ŝ α , Ŝ γ = i~εαγ (2.1) ;δ Ŝ where α, γ, δ ∈ {x, y, z} and εαγ ;δ denotes the Levi-Civita symbol • decomposed in a spinorial representation |s, sz i quantized by the quantum number discretization along the longitudinal axis sz ∈ {−s, −s + 1, . . . , s − 1, s} • used as a basis to build the following operators and observables Ŝ 2 |s, sz i = ~2 s(s + 1) |s, sz i Ŝ z |s, sz i = ~sz |s, sz i p Ŝ ± |s, sz i = ~ s(s + 1) − sz (sz ± 1) |s, sz ± 1i where we introduced the lowering and raising spin operator Ŝ ± = Ŝ x ± iŜ y along the z-axis. This algebra have the particularity to be invariant under a two sign transformation  x     −Ŝ x  −Ŝ x  Ŝ  Ŝ x y y y → → → (2.2) −Ŝ Ŝ −Ŝ y Ŝ   z   z z Ŝ Ŝ −Ŝ −Ŝ z which is heavily used in quantum magnetism. In the following we omit the hat operator.. 2.1.1. Spin-1/2. ~ α α The most famous illustration is the spin-1/2 operators S = 2 σ acting on the 1 0 spinorial s = 1/2 representation |↑i = and |↓i = . The three σ α generators 0 1 1 Mathematician. introduced the group Spin(n) who corresponds to the double covering of the SO(n) group as a generalization (in particular Spin(3)Z2 ∼ = SU (2)Z2 ∼ = SO(3)).. 6.

(20) 2.1 Sorry, what is a spin ?. above are the Pauli matrices σ = (σ x , σ y , σ z ) 0 1 0 −i σx = σy = 1 0 +i 0. σz =. +1 0. 0 −1. δ who obeys the previous commutation relation σ α · σ γ = δ αγ + iεαγ ;δ σ . Spin-1/2 is the smallest basis for spins and thus the Pauli matrices are the irreducible representation of rotations! Nowadays, it is known that many fundamental particles such as the electrons, the neutrons, the protons as well as all leptons, quarks and baryons possess a spin-1/2 !. 2.1.2. Effective magnetic moment. The fundamental particle properties listed above were actually discovered phenomenologically. They share the same spin-1/2 property but does not carry the same magnetic moment. This picture is sometimes related to the classical kinetic ~ = ~r × p~ picture – with a bigger mass, it becomes harder to spin. We momentum L ~ (where µi = e~ is an atomic magnetic list the effective moment m ~ i = −gi µi S 2mi moment scale) of few particles encountered in the thesis. particle electron neutron proton. gi -factor2 ge ≈ 2.002 gn ≈ −3.8261 gp ≈ 5.5856. µi µB µN µN. mass me mn mp. m ~i m ~e m ~n m ~p. Table 2.1: Relevant parameters of local magnetic moment of isolated electron, neutron and proton. All carry a spin-1/2. The µN is the Nuclear magneton, when the µB is commonly referred to the Bohr magneton of the electron. The Nuclear moment µN is order of magnitude smaller than the Bohr magneton µB . Thus, as it is written in any book of quantum mechanics, 1 the total magnetic ~ + S, ~ moment of an atom m ~ J is given by the total electron orbitals and spins J~ = L where the orbit of the electrons follows some radial and spherical harmonics (m ~L= ~ Each state of the clouds of electrons is described by quantum numbers 1 −ge µB L). |n, `, m, sz i with n, ` and m ∈ {−`, −` + 1, ..., `} labeling the energy and orbitals quantification. The electron carries a spin |↑i = |sz = +1/2i or |↓i = |sz = −1/2i. Let’s remind shortly the common Hund’s rule for isolated atoms 1 that are mostly valid (not always!). I maximize the total spin angular momentum s of all electrons. This means that electrons are symmetric in the spin sector and thus antisymmetric in the 2 https://en.wikipedia.org/wiki/G-factor_(physics). 7.

(21) 2. BRIEF INTRODUCTION TO QUANTUM MAGNETISM. orbital sectors. The electrons are in average further between themselves and screen the ion potential. II maximize the total angular momentum `. This means that they are orbiting in the same direction and encounter themselves less. This reduces the electron– electron Coulomb interaction. III minimize (respectively maximize) the total angular momentum J = L + S if the last layer is less (respectively more) than half filled. One reduces the energy due to spin orbit coupling HSOC = ξ L · S when the orbital and spin moments are opposite. Once the orbitals are filled, we remain with a local effective moment mJ for the relevant energy scale (in general, the outermost occupied shell) mJ = −gJ µB J where we have introduced the Landé factor gJ . In compounds, the effective gyromagnetic factor can vary due to more complex orbital structure or environment (see g = 2.26 in Sec. 5.2.1 and Sec. 7.1.2). For instance, compounds based on the Copper atom Cu has an outermost filling 4s1 3d10 which leads to an effective localized spin-1/2. The Nickel Ni is a prime candidate for effective spin-1 made of two spin-1/2 with his 4s1 3d9 outermost shell (See Table. 3.1). Other atoms such as Iron Fe and Cobalt Co have on site effective spin-3 and spin-2 and thus are good ferromagnets. In opposition with the last two itinerant metallic magnets, this thesis is about strongly correlated low-dimensional models in localized magnets where the relevant electrons are confined to their orbitals.. 2.2. Magnetism in condensed matter. In solids, atoms are not isolated and orbitals can for instance hybridize. Many adaptations have to be taken into account mainly due to the electron properties. 1 the electron’s spin and orbital. 2 Pauli exclusion principle (statistics). 3 the electron’s kinetic (delocalization) energy, for simplicity t.3 4 Coulomb repulsion between electrons, for simplicity U .3 3 Those. quantity will be defined later in the Hubbard model (see Eq. (2.10)).. 8.

(22) 2.2 Magnetism in condensed matter. Those are the four basic ingredients of quantum magnetism which induce rich sets of magnetic phases. In localized magnetism, many microscopic spin models are realizable for different effective spin sizes (from the smallest spin-1/2 up to maximal 7 spin- 15 2 in rare earths ). The different range and intensity (negative or positive) of the spin coupling defines the dimensionality of the effective spin lattice which comes with topological properties and various invariants.. 2.2.1. Electrons in a solid. Let’s start with the single electron description in a periodic ion potential Vion (x) (see Chap. I of Ref. [4]). In agreement with Bloch theorem 1 (periodicity of the wave function φnk,s (x) in real space of the nth electron s ∈ {↑, ↓}), the single electrons possess a sharp band ξk dispersion in the reciprocal space 2 2 ~ ∇ H0n φnk,s (x) = − + Vion (x) φnk,s (x) = ξk φnk,s (x) 2me where the Hamiltonian term is made of the electron kinetic term and the ion potential. Without distinguishingPeach electron bands, Ne electrons are introduced4 and the Ne system becomes H0 = n=1 H0n . Once one adds interaction between electrons HMB = H0 +. 1X Ve-e (|xi − xj |) 2 i,j. (2.3). the separability of the Hamiltonian is spoiled due to the Coulomb interaction e2 Ve-e (r) = 4π . Some parts of the interacting term can be injected in the single parrr ticle theory (process called as “renormalization” H0 → Hef f where the irrelevant interacting terms vanish under this transformation, generally due to an initial scale invariance structure 49 ). The remaining relevant non diagonal expressions generate multiple problems – welcome to the challenging Many-Body physics. 2.2.1.1. Particle statistics. An appropriate basis for describing microscopic many-body models is the Fock space (see the recent book number III of Ref [1]). The single body or two body operator are defined in this formalism as X H(1) = (α| E ef f |γ) Cα† Cγ (2.4) αγ (2) Hint. 1 X = (α, β| V ef f |γ, δ) Cα† Cβ† Cδ Cγ 2 αβγδ. 4 They. form a Slater determinant wave function due to the Pauli exclusion principle.. 9.

(23) 2. BRIEF INTRODUCTION TO QUANTUM MAGNETISM. where the |· · · ) forms a complete basis of one and two body states. E ef f and V ef f are respectively the one and two body operators describing the kinetic energy and the interaction between two particles (compare with Eq. 2.3). In the second quantization calculations, the combinatorial analysis of this creation Cγ† and destruction Cγ operator over the Fock space lead to different statistics. For P a quadratic Hamiltonian H = k ξk Ck† Ck in the particle numbers, two statistics occurs (see additionally the spin-statistics theorem 1 ). • Fermions Cα† ≡ fα† pick up a minus sign when commuted. n o n o † † fα , fγ† = δαγ fα , fγ = fα , fγ = 0. (2.5). For a single band dispersion ξk , they distribute in the reciprocal space according to the Fermi-Dirac statistics h i D E Tr e−βH fk† fk 1 fF (ξk ) = fk† fk = = βξ (2.6) Tr[e−βH ] e k +1 β ◦ Bosons Cα† ≡ b†α remains untouched when commuted. h i i † † h bα , b†γ = δαγ bα , bγ = bα , bγ = 0. (2.7). For a dispersion ξq , their distribution follows the Bose-Einstein statistics in the reciprocal space h i D E Tr e−βH b†q bq 1 = βξq (2.8) fB (ξq ) = b†q bq = −βH Tr[e ] e −1 β The anti-commutator is denoted by the braces {·, ·} while the commutator is denoted by the square brackets [·, ·]. The average h· · · iβ at equilibrium is given at a temperature T = 1/β where the Boltzmann constant of statistical mechanics is set to one kB ≡ 1 in this thesis. Comparing the angular momentum algebra 2.1 with the one of fermions 2.5 and bosons 2.7, one deduces that spins actually anti-commute on the same site and commute on different locations. Spins are much more bosonic-like since they commute everywhere but on-site. However it is possible to map them to fermions as well for spin-1/2 chains. The Jordan-Wigner transformation maps the spins to fermions with a very non-local string operator sign corrector (see App. A). The Holstein-Primakoff (originally Schwinger) transformation maps the spins to constrained finite number of bosons (see App. B).. 10.

(24) 2.2 Magnetism in condensed matter. 2.2.2. Many-body models. The many-body interaction (see Eq. (2.4)) can be rewritten for spinfull electrons † Cr,σ ≡ c†ri σi in the second quantization formalism. Z 1 (2) Hint = dr1 dr2 Ve-e (r1 − r2 )c†r1 σ1 c†r2 σ2 cr2 σ2 cr1 σ1 (2.9) 2 This expression generates different possible terms. Until now, we worked with the Coulomb interaction, but we can now introduce another type of interaction Ve-e (r) = U δ(r) such as a local contact repulsion U . Let us illustrate this many-body formalism by introducing two fundamental microscopic models. In the following chapters, we will additionally introduce external potentials or Zeeman magnetic fields in order to play with even richer phase diagrams (see Chap. 3). 2.2.2.1. The Hubbard model. One illustration of this local repulsion U > 0 interaction is the Hubbard model (2.10) based on spinfull fermions c†i,σ describing interacting electrons in a solid. X † X HH = −t ci,σ cj,σ + h.c. + U ρi,↑ ρi,↓ (2.10) i. hi,ji,σ. Keep in mind that the energy magnitude of local interaction U and hopping t in materials are of the order of thousand of eV while magnetic couplings are more of the order of the meV coupling. Despite his outside simplicity, this model remains yet unsolved but in dimension one (d = 1) and infinity (d = ∞). It describes in a more general way the transitions from insulating state to metallic phases. 2.2.2.2. The Heisenberg model. Among many spin models (see Chap. 3), the Heisenberg model is the most famous occurrence of localized spin systems. X HH = JH Si · Sj (2.11) hi,ji. It conserves totally the SU (2) symmetry of the spin and the associated global quantum numbers |s, sz i. We illustrate this model on a chain geometry to avoid any frustration. 50,51 The degenerate ground state is aligned if the coupling is negative tot |stot , stot i ≡ |· · · ↑↑ · · ·i ≡ |· · · ↓↓ · · ·i while in a Néel state5 |stot , stot z = ±s z = 0i if the coupling is positive. 5 Note. the difference with the classical Néel state . . . ↑↓↑↓↑ . . .. 11.

(25) 2. BRIEF INTRODUCTION TO QUANTUM MAGNETISM. In the following chapters, we will introduce an external magnetic field to such models which will favor a direction of quantization and reduce the symmetry to a phase in the transverse in-plane (⊥ ẑ) perpendicular to the quantification axis. For antiferromagnets, one can polarize the spin system with large external magnetic fields (see Sec. 4.3). For ferromagnets, the spins will point in the direction of the external applied field (see App. B).. 2.2.3. Magnetic exchanges. We now briefly illustrate simple mechanisms that induce positive and negative spin coupling within the Hamiltonian that generates respectively anti- and ferromagnetic ground state respectively. More details and generalities can be found in Chap. II of Ref. [4]. 2.2.3.1. Ferromagnetism. Suppose that we have two spinfull electron c†`,σ orbitals ` = 1, 2 localized on an atom P2 r, i.e., ψσ† (r) = `=1 φ∗` (r)c†`σ and that the ground state population prefers to have two electrons on different orbitals ` (see Fig. 2.2) due to a local contact repulsion Ve-e (r) = U δ(r) (see Eq. (2.9)). The adequate picture generated in the effective spin sector of the electron is the Heisenberg model Heff ≈ JF S1 · S2 with 1 † S` = c 2 `,↑. c†`,↓. h↑| σ |↑i h↓| σ |↑i. ! c`,↑ h↑| σ |↓i h↓| σ |↓i c`,↓. where the effective exchange coupling JF for such problem (2.9) is negative. Z 2 2 JF ∝ −U d3 r |φ1 (r)| |φ2 (r)| (2.12) The Pauli matrices σ are defined in Sec. 2.1.1. Since the wave function of the electrons have to be antisymmetric – which they are in the orbital sector – the spin sectors are now aligned (see Fig. 2.2). The ferromagnetic exchange strength depends on the overlap of the orbitals. The alignment of the spins thus reduces the effect of the Coulomb interaction in agreement with first Hund’s rule (see Sec. 2.1.2). The overlap of the orbital vanishes where the Coulomb potential is generally maximal Ve-e (r → 0). 2.2.3.2. Antiferromagnetism. Antiferromagnetism appears for instance in the Hubbard model (2.10). We will quantitatively illustrate the mechanism below. The effective spin model of two. 12.

(26) 2.2 Magnetism in condensed matter. neighboring sites can be fixed in a second order expansion of the hopping term (two hopping processes, see Ref. [4]). We immediately see that aligned spins are freezed by Pauli’s principle for fermions, but anti-aligned spins can temporary share sites with a cost of U to gain some energy of order t2 . This effect of virtual double occupancy U. |↑, ↓i. −t. |↑↓, ·i. −t. |↓, ↑i. |↑, ↑i. − t . |↑, ↑i . |↓, ↑i. −t. |·, ↑↓i. −t. |↑, ↓i. |↓, ↓i. − t . |↓, ↓i . Figure 2.1: Illustration of the superexchange mechanism. Even though there is a barrier U , the spins can exchange by virtually occupying the same site. This process is not allowed for polarized particles thus favorising antiferromagnetism. Picture adapted from Ref. [4]. allowed for the spins is called superexchange. The effective parent Hamiltonian in 2 such 4tU expansion mechanism gives a term Hef f ≈ JAF S1 · S2 2. with positive coupling JAF = 4tU > 0 and lead to favor anti-alignement of the spins. In Fig. 2.3, we show an illustration of two separated electrons creating some antiferromagnetic effective exchange. With this overview of emerging quantum magnetism, we now move to the main introduction to the research area by introducing in the next Chap. 3 the low-dimensional models under study. In the following, we will focus on antiferromagnetic properties at large external magnetic field of effective strongly correlated quasi-one dimensional theories.. 13.

(27) 2. BRIEF INTRODUCTION TO QUANTUM MAGNETISM. Figure 2.2: Ferromagnetism takes place when relevant spin orbitals are orthogonal but share a common location in space. The electron is antisymmetric in the wave function sector and symmetric in the spin sector. Illustration of those mechanism can be found in compound such as iron or cobalt (see Sec. 2.1.2). Picture adapted from Ref. [4].. Figure 2.3: Antiferromagnetism takes place when orbitals are not orthogonal but spatially separated in space. The coupling of the spins occurs with a virtual superexchange process. In this thesis, all compounds under study have antiferromagnetic exchanges. Picture adapted from Ref. [4].. 14.

(28) CHAPTER. 3. Low-dimensional antiferromagnetic models. In this chapter, we present spin-1/2 models which interact strongly on an effective low-dimensional geometry by antiferromagnetic exchange (see Sec. 2.2.3.2). We discuss the different phase diagram of each model in Sec. 3.1 to help the reader understand the observed experimental measurement in the subsequent sections. In Sec. 3.2, we list good compounds candidates for such theoretical models which drew attention in the low-dimensional quantum magnetism community.. 3.1. Spin-1/2 models. In this thesis, we will predominantly focus on spin-1/2 systems only. But most of the techniques could be used for spin-1 system as well (see Chap. 4 and App. B). One important low-dimensional difference between integer and half integer spin chains is known as the Haldane “conjecture” 8 which exhibits different gapped or gapless ground state depending on the spin size. In this thesis, we mainly study gapless excitations in low-dimensional spin-1/2 systems.. 3.1.1. The XXZ spin chain. In Sec. 2.2.2, we introduced the Heisenberg model (2.11). Here we complete this model with additional terms starting from the overall previous spin exchange J. We add an external magnetic field hz and an anisotropy ∆ originated from spin orbit. 15.

(29) 3. LOW-DIMENSIONAL ANTIFERROMAGNETIC MODELS. coupling terms. This model is commonly referred as the anisotropic XXZ model. HXXZ = J. N −1 X. N X y x z S`x S`+1 + S`y S`+1 + ∆S`z S`+1 − hz S`z. `=1. =. Jxy 2. N −1 X. `=1 − (S`+ S`+1 + h.c.) + Jz. `=1. X `. N X z S`z S`+1 − hz S`z. (3.1). `=1. We will always use the convention hz = gµB H z , where H z is the large experimental applied Zeeman field in units of Tesla. This magnetic term reduces the total spin symmetry (s is not a good quantum number anymore) to the xy-plane (phase symmetry U (1) which conserves the total stot implies that the z spin number). ThisP ground state of such model has always the total magnetization [ ` S`z , H] = 0 as conserved quantity.. 1. FP. TLL. -1. Néel 1. Figure 3.1: Zero temperature phase diagram of the XXZ model (3.1). The phase where the excitations are gapless (TLL) delimits two types of order, namely the Fully Polarized order at large Zeeman field and the quantum Néel order at large anisotropy (see Sec. 2.2.2.2). The energy for flipping one spin down from the polarized state delimits the critical magnetic field hzc = J(1 + ∆) = Jxy + Jz . The animation follows the excitations along the vertical axis of this phase diagram.Picture adapted from Ref. [22]. One can get rid of the units kB , ~, µB and a (the lattice spacing of the sites) of the microscopic model by expressing all energy scale in terms of the overall coupling J. The spin symmetries (2.2) connect the phase diagram (see Fig. 3.1) to different models with staggered magnetic field. The positive J > 0 without magnetic field hz = 0 with a negative anisotropy ∆ = −1 therefore corresponds to the ferromagnetic Heisenberg point ( JF ≡ −J = −Jxy = Jz < 0). While the antiferromagnetic. 16.

(30) 3.1 Spin-1/2 models. Heisenberg corresponds to the case ∆ = 1, i.e., JAF ≡ J = Jxy = Jz > 0 (see Eq. (2.11)). Now, tuning the magnetic field, one gets the phase diagram in Fig. 3.1. The particular ∆ = 0 point corresponds to dispersive spins without interaction ωk = Jxy cos(ka) − hz in the xy-plane. Exact results exists for correlations in the XY model (A.2) as computed in App. A (see the animation below). In this work, we will only meet the two particular cases ∆ = 1 in the critical region near saturation hzc (see Chap. 5) and ∆ = 12 in the middle of the gapless phase hz = 0 (see Chap. 7) every time with units fixed to J = 1. The gapless modes will be described in more details in Chap. 4 introducing the bosonization formalism (see Sec. 4.1.1).. 3.1.2. The dimerized model(s). In this thesis, we studied two dimerized models. The coupling between the spin-1/2 could altern with different intensities thus creating a pattern along the lattice (see Fig. 3.2). In the compounds of interests (see below), a two unit cell of strongly connected spin-1/2 emerges and creates dimers which become the starting points of our dimerized study. In the strong dimerization limit (isolated dimers), the two spin-1/2 combine fully and form an effective Hilbert space made of a singlet |s = 0, sz = 0i 1 |si = √ (|↑↓i − |↓↑i) 2 and a degenerate triplet |s = 1, sz ∈ {−1, 0, +1}i, where the degeneracy is lifted under an external perturbation such as an external applied magnetic field (see Fig. 3.2). The three different triplets are described by t+ = |↑↑i ,. 1 t0 = √ (|↑↓i + |↓↑i) , 2. t− = |↓↓i .. The way the two-cell dimers connect can lead to different border effects. We didn’t pay too much attention on the finite size effects and focus instead on the bulk properties of the two following dimerized models – the strong-rung two-leg ladder and the dimerized chain (see Fig. 3.2). We then briefly illustrate some low-dimensional synthetized compounds which provides such effective theories (see Sec. 3.2). 3.1.2.1. Strong-rung two-leg ladder L. We consider a two-leg ladder system with spin-1/2 coupled by antiferromagnetic Heisenberg couplings (see Fig. 3.2) on rungs ` and legs η ∈ {1, 2} X X X z S`,η (3.2) HL = Jk S`,η · S`+1,η + J⊥ S`,1 · S`,2 − hz `,η. `. 17. `,η.

(31) 3. LOW-DIMENSIONAL ANTIFERROMAGNETIC MODELS. The ladder geometry 2. 2. 2. 2. 2. 2. 2. 1. 1. 1. 1. 1. 1. 1. The dimerized chain on the ladder geometry 2. 1. 2. 1. 2. 1. 2. 1. 2. 1. 2. 1. 2. 1. Figure 3.2: Weakly coupled ladder (3.2) and dimer (3.3) representation. The index η corresponds for the ladder to the bottom or upper leg. For the dimer, η corresponds to the left or right strong bond cell, thus the labels shuffle when mapped on the ladder geometry. We add an arrow on the middle cell to visualize the symmetry when we inverse the dimerized chain. Picture adapted from Ref. [40].. Such effective realization are provided by materials for both the weak-rung limit 52–54 and the strong-rung limit 55 (see Table (3.1)). In the following, we will always remain in the weak-leg limit and use a unitless framework in general expressed in the J⊥ . Such strong-rung model exhibits a phase diagram described below and in Figs. 3.3 and 7.1.. 3.1.2.2. Dimerized chain D. If we remove alternatively the weak bonds along the ladder (see Fig. 3.2) and map the model to a chain we get a dimerized chain of alternative bonds. 22,56 For an even number of sites N , we always have N2 strong bonds Js and N2 − 1 weak bonds Jw . The model is thus HD =. N −1 X `=1. `. z. (J − (−1) δJ)S` · S`+1 − h. N X `=1. 18. S`z. ( Jw = J − δJ where Js = J + δJ. (3.3).

(32) 3.1 Spin-1/2 models. starting with a strong bond at each border Js = J + δJ and alternating with the weak bonds Jw = J − δJ along the chain. Starting the opposite way (we avoid that) induces topological protected states at each border. 57 The coupling values are given in Eq. (6.2) and expressed theoretically in the Js units. Experimental realization exists as described below (See Sec. 3.2 and Table 3.1). Naively, the dimerized chain (without protected border state, see Fig. 3.2) is similar to the ladder case with half of the weak coupling absent. Therefore, the phase diagram can be described similarly with Js ↔ J⊥ and Jw ↔ 2Jk (see Fig. 3.3). However, the observables are quite reshuffled due to the new labeling of the sites when moving from a chain geometry to a ladder – we will come back to this issue in Chap. 7. The phase diagram of such dimerized systems induces three different phases depending on the external magnetic field strength 1. The spin liquid or quantum disordered (QD) phase is ordered by a spinsinglet configuration in the isolated dimer picture (see Fig. 3.3). When applying the external magnetic field, the gap of the spin-liquid state reduces until reaching the gapless phase. This phase is delimited by a first critical magnetic value |hz | < hzc1 where the system is made of singlets ground state. 2. The gapless or Tomonaga-Luttinger Liquid (TLL) phase takes place when the triplet |t+ i and the singlet |si interplay creates zero-energy excitation. Those modes lie with a linear spectrum on specific (in-) commensuration points depending on the magnetization as described in the XY spin model (see animation below). The low lying excitations will be well described by the bosonization formalism in Sec. 4.1.1. This phase occurs between the two critical fields hzc1 < |hz | < hzc2 . In this regime, one can identify the two low lying states with a spin-chain behavior for the low energy physics (see Sec. 3.1.2.3). 3. The fully polarized (FP) phase is gapped again with a fully polarized ground state (made of the |t+ i state). This phase occurs when the exchange energy inside the system cannot compensate the large Zeeman term hzc2 < |hz | that gap the system. The critical value for two-leg ladders is hzc2 = J⊥ + 2Jk and hzc2 = Js + Jw for the dimerized chain. 4. The quantum critical point (QCP) phases occur exactly at the quantum transitions hzc1 and hzc2 . A zero-energy mode with a quadratic dispersion occurs where the correlation function fall into universal scaling laws with zero scale-factor expressions. 37 The only expected difference at hzc1 and hzc2 is the energy location of the triplet |t− i and the asymmetry of the magnetization curve evolution when raising the temperature.. 19.

(33) 3. LOW-DIMENSIONAL ANTIFERROMAGNETIC MODELS. Figure 3.3: Phase diagram of the strong-rung ladder (3.2). The coupling within the two spin-1/2 unit cell is J⊥ while the dimers are coupled by Jk . The external magnetic field splits the triplets, moving the ground state from a spin-liquid phase (singlet |si) towards a fully polarized state |t+ i crossing a TLL region arising from their interplay. The four levels are located at energies Et± = J⊥ /4 ∓ hz , Et0 = J⊥ /4 and Es = −3J⊥ /4 and disperse along the leg with Jk exchange. The dimerized chain (see Eq. 3.3) has an analog phase diagram with Js playing the role of J⊥ and Jw playing the 2Jk role. Note however that due to the relabeling of the sites (see Fig. 3.2), both models remains somehow differents. Picture adapted from Ref. [22].. 3.1.2.3. Spin-Chain mapping - ∆ =. 1 2. XXZ chain C. _. Both previous models possess an intermediate phase at low temperature between the QD and the FP phase (see Fig. 3.3). There, the singlet |si and triplet |t+ i creates interplaying gapless modes similar to the ones encountered in spin-1/2 chains (see Sec. 3.1.1). In fact, if the magnetic field and temperature are such that we can neglect the triplet t0 and |t− i population, one can identify a spin-chain behavior between the two |t+ i and |si states. For this proposal, we introduce the pseudo-spin-1/2 ~S in the basis | i ≡ |t+ i, | i ≡ |si mapped by Sη± ≡ √η2 S± and Sηz ≡ 14 (1 + 2Sz ) for the singlet-triplet crossing. _. 20.

(34) 3.2 Synthetized compounds. region. 58–60 The mapping leads to a spin-1/2 XXZ chain with ∆ = HC = J. N −1 X `=1. 1 Sx` Sx`+1 + Sy` Sy`+1 + Sz` Sz`+1 2. . − hzef f. N X. 1 2. anisotropy. Sz`. (3.4). `=1. The spin-chain mapping fixes the following microscopic parameters for the XXZ model (3.1) : 1. ladder : J ≡ Jk and hzef f = hz − J⊥ − 2. dimer : J ≡. −Jw 2. Jk 2 ,. and hzef f = hz − Js −. Jw 4 .. Although these models can be studied independently we will consider them in the regime where their low-energy properties are roughly equivalent (see Chap. 7). In this thesis, we will not discuss the correction of the constant and the boundary terms and we note for future works that the value Jw should rather be compared to 2Jk . We will discuss how the spin-chain mapping manifests differently in Chap. 6.. 3.2. Synthetized compounds. As presented in Sec. 2.1.2, the Copper Cu is a prime candidate for localized magnetism with his outermost 4s1 orbital, giving an effective isotropic spin-1/2 effective shell. The two compounds under consideration in this thesis (see Chap. 5 and Chap. 7 for further details) are both insulators with spins localized on Cu2+ ions, namely the K2 CuSO4 Cl2 [39] known as the natural mineral chlorothionite 38 and (C5 H12 N)2 CuCl4 [42,61] known as the bis-piperidinium copper tetra-chloride (BPCC) as can be seen in Fig. 3.4. Large variety of compounds exists in the literature (see Ref. [62] for a non-exhaustive list). As part of them, let us mention the most relevant candidates for models (3.1, 3.2 and 3.3) in Table 3.1. Compounds KCuF3 K2 CuSO4 Cl2 (C5 H12 N)2 CuCl4 (C5 H12 N)2 CuBr4 (C7 H10 N)2 CuBr4 Cu(NO3 )2 · 2.5D2 O NiCl2 -4SC(NH2 )2. Abbrev.. models. BPCC BPCB DIMPY. quasi-1D chain (5.2) quasi-1D chain (5.2) weak-leg ladder (3.2) weak-leg ladder (3.2) strong-leg ladder (3.2) dimerized chain (3.3) spin-1 single ion anisotropy (3.5). DTN. weak-coupling ∼ = −18.56 K ∼ 0.45 K = ∼ 1.34 K = ∼ 3.6 K = ∼ 4.34 K = ∼ 1.474 K = ∼ 2.2 K =. JF ⊥ J⊥ Jk Jk J⊥ Jw JΞ. strong-coupling Jk ∼ = 394.4 K Jk ∼ = 2.94 K J⊥ ∼ = 3.42 K J⊥ ∼ = 12.8 K Jk ∼ = 8.42 K Js ∼ = 5.28 K ∼ D 8.9 K =. Refs. [63,64] [38] [42,61] [12,42,61] [65] [41,66] [12]. Table 3.1: Short list of effective low-dimensional antiferromagnetic models founds in real compounds. We took 1 meV ≈ 11.6 K when the coupling where expressed in eV in the references. The (C5 H12 N)2 is sometimes abbreviated as Hpip. 45 A broader list can be found in Ref. [62]. 21.

(35) 3. LOW-DIMENSIONAL ANTIFERROMAGNETIC MODELS. The synthetized compounds found in Table 3.1 are challenging to create and study. In the table, we added a spin-1 compound based on Ni Nickel with a relevant spin orbit coupling leading to a strong spin-1 single ion anisotropy. 67 HDTN = JΞ. N −1 X. S` · S`+1 + D. `=1. N X l=1. 2. (S`z ) − hz. N X. S`z. (3.5). `=1. This model with single ion anisotropy D met in the DTN compound in the large D phase 67 could have a similar behavior to previous dimerized models. Indeed, the single ion anisotropy splits by an energy D the spin-1 and separates the |s = 1, sz = 0i which now play the role of the previous singlet |si. The two other states |s = 1, sz = ±1i are then playing the role of the two polarized triplets |t± i. We thus get an equivalent phase diagram (see Fig. 3.3) without the zero triplet t0 with the single ion playing the role of the strong coupling D ↔ J⊥ and the spin-1 coupling JΞ playing the role of the leg exchange JΞ ↔ Jk . Such model will not appear within this work but it remains a motivation for the following works since many open questions remains in the literature (see Ref. [12]). Similar studies to the one in this thesis can be found in such spin-1 model in Ref. [24]. Other compounds 62 have recently attracted attention in the low-dimensional magnetism community. The LiCuSbO4 68 provides a good candidate for the study of frustrated low-dimensional effects. Other compounds such as BaCo2 V2 O8 69 and YbAlO3 70 probed recently spinon confinement excitation in a more complicated geometry-like XXZ models than the one encountered in this thesis. Materials offer a rich set of phases which are challenging to study in details. In the next Chap. 4, we present the theoretical techniques we used in order to compare with the previous mentioned theoretical models and compounds.. 22.

(36) 3.2 Synthetized compounds. Figure 3.4: Single crystals of (C5 H12 N)2 CuCl4 , refered as BPCC, grown at University of Bern. Two crystals are shown in Fig 3.4 compared to a 2 CHF Swiss coin (8.8 g). Picture taken from Ref. [48].. 23.

(37) 3. LOW-DIMENSIONAL ANTIFERROMAGNETIC MODELS. 24.

(38) CHAPTER. 4. Methods. In this chapter, three methods are presented to study low-dimensional antiferromagnetic models. The first method in Sec. 4.1 is a field theory, commonly referred as bosonization 5 which consists of describing low-dimensional systems at low-energy with bosonic collective mode excitations. In opposition with higher dimensional Fermi liquid theory, 71 materials described by such theory are referred as TomonagaLuttinger Liquid (TLL). 5 The second method in Sec. 4.2 is a numerical tool called the Density Matrix Renormalization Group (DMRG). 13–19 The third method in Sec. 4.3 is an experimental method consisting in an Inelastic Neutron Scattering (INS) 10 process. In addition to the neutron scattering, other experimental methods such as Nuclear Magnetic Resonance (NMR), 11,12 Muon Spin Rotation (µSR) 72 or Electron Spin Resonance (ESR) 73,74 are used to analyze the properties of quantum magnets. Some of the related literature were important motivation for the following work (see Ref. [12]).. 4.1. Tomonaga-Luttinger Liquid. The Fermi liquid theory with Landau single electron-like quasi-particle excitations 71 is no longer an optimal description for low-dimensional systems 75 . The relevance of the interaction and the geometrical constrain favor collective behavior. From there on, the bosonization procedure 5 consists of moving from initial spin, electron or boson on a lattice to a bosonic field theory description (see Eq. (4.3)). The systems which enter this low-dimensional low-energy description are referred to as. 25.

(39) 4. METHODS. Tomonaga-Luttinger Liquid (TLL). A naive picture can be viewed as follow. In the absence of large energy scale compared to the band structure, the particle-hole excitations are confined to the Fermi surface on a linear branch (see Fig. 4.1). The new effective excitation is now made of collective particle-hole excitation with linear spectrum of bosonic nature.. ξ(k ). q∼2 k F. −v F k. E k (q). +v F k. q∼0. 0. k. 2kF. q. Figure 4.1: Particle-hole excitation mechanism and spectrum in one dimensional system as presented for fermions in Ref. [5]. The low particle-hole excitation has a well-defined momentum q and energy Ek (q) = ξ(k + q) − ξ(k) confined to the Fermi surface. The effective physics is now described by collective excitations with linear dispersion ±vF |k|. Picture adapted from Ref. [5]. The whole microscopic lattice can then be described by an effective bosonic field theory. In the context of a TLL description, two massless bosonic fields (φ and θ) are introduced forming the following quadratic Hamiltonian. 5 Z ~ uK u 2 2 HTLL = dx (∇θ(x)) + (∇φ(x)) (4.1) 2π ~2 K The φ and θ are canonical conjugate field (πΠ(x) = ∇θ(x)) [φ(x1 ), Π(x2 )] = iδ(x1 − x2 ). (4.2). and can be interpreted as a particle-hole density term ∇φ(x) and a phase eiθ(x) in the fermionic picture. The connection of fermions with spins is done in App. A. All physical characteristics of the microscopic model (hopping, interaction,...) are now fixing the Luttinger parameters K and u. The K dimensionless parameter. 26.

(40) 4.1 Tomonaga-Luttinger Liquid. controls the decay of the correlation functions (powerlaw-like in 1D, see Eq. (4.5)) while u is the velocity propagation of the elementary excitation. The underlying structure of such Hamiltonian (4.1) is embedded in a larger conformal field theory (CFT). 76 In those theories, the conformal invariance fixes primary fields which are defined by an operator product expansion and a central charge structure. 76 Some microscopic models can then be described in the continuum with those conformal invariant theories. However, not all microscopic models enters in the TLL Hamiltonian (4.1) description. In the following, we discuss the bosonization prediction of the XXZ spin-1/2 chain (3.1) in more details.. 4.1.1. Bosonization of the spin-1/2 chain. The microscopic spin operator6 can be described in term of the two previously introduced bosonic fields (according to Ref. [5]). Now, φ and θ plays classically the role of longitudinal and transversal angles on the unit-sphere on which the spin lives (see App. B). S z (r) = mz +. −1 2 ∇φ(r) + cos(2φ(r) − π(1 + 2mz )x) π 2πα. e∓iθ(r) (cos(πx) + cos(2φ(r) − 2πmz x)) S ± (r) = √ 2πα. (4.3). where r ∼ = (x, uτ ), x being the space, τ = it being the imaginary time7 and mz is the magnetization. α is a short distance cutoff. This mapping can be used in order to study the XXZ model (3.1). The Zeeman z term can be seen as a shift in the field ∇φ = ∇φ̃ + Kh uπ and the free exchange term Jxy as a quadratic TLL (4.1). However, the interacting longitudinal part brings further complication. The XXZ model (3.1) with observables (4.3) leads to a sine-Gordon 5 Hamiltonian Z 2g3 HsG = HTLL − dx cos(4φ(x)) (4.4) (2πα)2 where the cosinusoidal term is a remnant of the interacting anisotropic term g3 = a J ∆ (see the Umklapp phenomenon 5 ). For this model, the renormalization group flows show different phases. • For K > 1/2, the coupling becomes irrelevant, and we remain with a quadratic TLL system with fixed parameter K ∗ and g3 → 0. This phase contains the XY model (A.2) which corresponds to K = 1 (see App. A) and delimits repulsive from attractive interactions. 6 The. lattice spacing a and the continuum short distance cutoff α are introduced to map spins to √ the continuum (S z (x) ↔ Siz /a and S ± (x) ↔ Si± / a). 7 Up to a cutoff in the Wick rotation.. 27.

(41) 4. METHODS. • when K < 1/2, the flow of the renormalization equation generates large g3 and the system enters the strong coupling regime. The cosinus term can be expanded near a semi-classical solution (ordering of the φ field) that minimize the energy (see App. B). • The particular case Ks = 1/2 is on the separatrix and the cosine becomes marginal; without entering into the details, logarithmic corrections 77,78 can appear in some correlations. This particular point corresponds to the onedimensional isotropic antiferromagnetic Heisenberg model (2.11).. 4.1.2. Correlations in TLL theories. From this microscopic to field theory identification (see Eq. (4.3)), one predicts how the direct space correlation functions at T = 0 decays for the XXZ model (3.1) using a quadratic TLL formalism up to renormalization procedures. 2K K −1 1 hS (x)S (0)i = + 2 2 + Az cos (π(1 + 2mz )x) (4.5) 2π x x 1/(2K) 2K+1/(2K) 1 1 + − S (x)S (0) = Ax cos (πx) − Bx cos (2πmz x) x x z. z. m2z. Those results (see Ref. [5,77]) are found using the path integral formalism. The non-universal Ax , Az and Bx amplitudes are strongly dependent on the microscopic model under study (see Sec. 4.1.2.1). This expression predicts a dependence of the modes according to the filling mz (see the film below). The longitudinal hS zz i mode remains fixed at q ∼ 0 but moves at q ∼ π(1 ± 2mz ) with the external magnetic field. The transverse hS xx i mode remains fixed at q ∼ π and moves with the filling q ∼ ±2πmz . At finite time, the excitation remains within a light-cone where u takes the role of a Lieb-Robinson velocity 79 that delimits the borders. A more general expression for dynamical correlations at finite temperature in the TLL regime can be derived from the following retarded susceptibilities in the reciprocal space 5,80,81 by a deformation of the conformal symmetry in the CFT theory 76 sin(πκ)α2 χκ (q̆, ω) = − u. . 2κ−2. κ β(ω − uq̆ + iε) B −i ,1 − κ 2 4π κ β(ω + uq̆ + iε) B −i ,1 − κ 2 4π. 2πα βu. . (4.6). where β = 1/T is the inverse temperature and ε → 0+ . The q̆ is the momentum centered according to the field-dependence as described in Eq. (4.5) and the κ. 28.

(42) 4.2 Density Matrix Renormalization Group. parameter is a function of K that depends on the precise correlation function under consideration. This last expression starts decaying with a power-law and end with an exponential suppression beyond a thermal length. 4.1.2.1. Extraction of TLL parameters. From the previous correlations (4.5), one can extract the TLL parameters. For the XXZ model (3.1), exact Bethe-Ansatz solutions 5 relates the microscopic parameters to the TLL prediction π K= (4.7) 2 arccos(−Jz /Jxy ) 1 Jxy 1 u= (4.8) 1 sin(π(1 − 2K )) 2 1 − 2K where repulsive (resp. attractive) interaction corresponds to 1/2 < K < 1 (resp. K > 1). In particular, the XY model (A.2) corresponds to K = 1, and the isotropic ∆ = 1 antiferromagnetic Heisenberg model (2.11) is on the separatrix K = 1/2 (see above). If one wants to relate different microscopic models to the TLL theory, one can extract the TLL parameters using numerical tools. For instance, with the K z magnetization curves, we extract the compressibility πu = ∂m ∂hz . On the other hand, the ratio. Ku π. =L. d2 hψ0 |H|ψ0 i dΦ2. can be extracted from the ground state variation Φ=0. under twisted boundaries or flux current Φ as previously done in Ref [82]. Another method is extracting the K parameter as well as the non-universal amplitudes directly from the T = 0 static correlations Eq. (4.5) as previously done in Refs. [20,21,40]. We present now the most appropriate numerical tools to date for such extraction.. 4.2. Density Matrix Renormalization Group. The Density Matrix Renormalization Group (DMRG) was first invented by S. R. White [13,14] and generated a large set of numerical tools. 15–19 The underlying matrix product structure revealed there (see Ref. [19]) is the main reason of the success of the method.8. 4.2.1. Numerics. 4.2.1.1. A numerical complexity problem. The main problem in quantum many-body systems is the exponential growth of the Hilbert space inaccessible to classical computer for large system sizes N & 40 8 This. can be found under various names in the literature such as tensor train, tensor network, left/right blocks, etc.. 29.

(43) 4. METHODS. – mainly due to the intrinsic inseparability of degrees of freedom in quantum mechanics. Indeed, N interacting spins-s of d = 2s + 1 local degree of freedom {σl } on a 1D lattice l ∈ {1, . . . , N } are described according to quantum mechanics without loss of generality with the basis X |ψi = cσ1 ...σN |σ1 . . . σN i (4.9) σ1 ...σN. where cσ1 ...σN contains all dN complex C-numbers which describes the quantum system fully. The quantity of numbers is exponentially big in the chain length and therefore unusable for simulations with large number of sites. Of course, many values could be zeros and quantum number conservation laws could reduce the dimension of the problem furthermore (see App. C). Luckily, we can take advantage of existing numerical tools. 4.2.1.2. Some numerical analysis tools. Let us first introduce some numerical analysis tools that will be useful for the algorithm that we will present in Sec. 4.2.2. This section provides a short complexity discussion of standard routines used within the DMRG framework. ã SVD - low rank approximation ã Suppose M ∈ Kn×m being a random matrix, the best low rank approximation to this matrix M (in the Froebenius/matrix norm) is given by the matrix M̃ of rank r where M =US V† and M̃ = Ũ S̃ Ṽ † (4.10) U, V, Ũ and Ṽ are unitary matrices (U ·U † =U † ·U =1) and the S and S̃ matrices share the same diagonal with singular values λj = Sj,j with j ∈ {1, . . . , min(n, m)} restricted to the rank r for the other one. Note that M · M † or M † · M have eigenvalues corresponding to the squares of the singular values λ2j . The complexity of such low rank approximation is naively given by O(min(n2 · m, n · m2 )) and the precision of such method is bounded by the highest discarded singular value max {λj |j > r} j. (automatically sorted). Matrix multiplication Suppose A ∈ Kn×k and B ∈ Kk×m being two random matrices, the matrix multiplication complexity A · B has complexity of the order O(n · k · m). Let’s now suppose that we multiply both matrices and then SVD the results. One immediately sees that the complexity is dominated by the low rank approximation in case k < min(n, m). Note that we will always be in that situation in the following – up to border terms which are irrelevant complexity discussions.. 30.

(44) 4.2 Density Matrix Renormalization Group. Therefore the reduced rank r is the key feature concerning not only the precision but the complexity encounter in the matrix product and SVD! In the following, the rank r will be referred as the bond dimension χ. All the above complexity techniques could actually be improved by working with different type of matrices. One such type are the sparse matrices which will be implemented using quantum number conservation technique (see App. C). 4.2.1.3. A renormalization procedure. Let us use previous numerical tools to translate the problem (4.9) into an available computation framework. The general state |ψi can be rewritten in a product of matrices by naively truncating to a dimension χ all singular values encountered. As a result, one expects the following outer product of matrices P. cσ1 ...σ ≡c(σ. 1 ),(σ2 ...σN ). N. P. =. a1. =. P. 1 ...σN −1 ),(σN. X. =. Uσ1 ,a1 Sa1 ,a1 (V † ). σ1 A1,a U Sa2 ,a2 1 (a1 σ2 ),a2. cσ1 ...σ ≡c(σ N. a1. = ). aN −1. σ. P. a1 ,(σ2 ...σN ). =. P. †. (V )a2 ,(σ3 ...σN ) =. a1 ,a2. a1. 1 c̃ A1,a a 1. U(σ1 ...σN −1 ),aN −1 SaN −1 ,aN −1 (V † ). c̃(σ1 ...σN −1 ,aN −1 ) BaσNN−1 ,1. 1 ,(σ2 ...σN ). σ1 σ A1,a A 2 c̃ =... 1 a1 ,a2 a2 ,(σ3 ...σN ). aN −1 ,σN. = .... aN −1. to fully describe the many-body system (4.9). The above procedure, turning a matrix in a left (respectively right) normalized A (respectively B) matrix is commonly called a sweep step. With this procedure, we have transformed a general state |ψi σl l unusable by a computer into an outer product of matrices Aσal−1 ,al and/or Bal−1 ,al which can be stored efficiently in a computer if one agrees beforehand renormalizing the contained information to the bond χ dimension. The resulting object arising from such transformation is called a Matrix Product State (MPS). X X |ψi = Maσ01,a1 Maσ12,a2 . . . MaσNN−1 ,aN |σ1 . . . σN i (4.11) σ1 ...σN a0 ...aN. =. X. X. σl σN l−1 Aσa01,a1 . . . Aaσl−1 ,al Sal ,al Bal ,al+1 . . . BaN −1 ,aN |σ1 . . . σN i (4.12). σ1 ...σN a0 ...aN. In case the information can be stored in this MPS form, the cσ1 ...σN coefficients are stored in the outer product of smaller matrices, where unimportant information have been renormalized by fixing an overall bond dimension χ which focus on the most relevant data. In fact, this procedure transforms an exponential problem O(dN ) to a linear problem of N matrices of size dχ2 (respectively the SVD will be cubic in χ). This limitation is sometimes referred to as the “area law” problem where theoretical work explains 83 why this method works in the particular case of low-dimensional models.. 31.