THE WONDER BOOK OF MAGNETISM
CONTENTS PART I BASIC CONCEPTS IN FRUSTRATED MAGNETISM GEOMETRICALLY FRUSTRATED ANTIFERROMAGNETS: STATISTICAL MECHANICS AND DYNAMICS INTRODUCTION MODELS SOME EXPERIMENTAL FACTS CLASSICAL GROUND STATE DEGENERACY ORDER BY DISORDER GROUND STATE CORRELATIONS DYNAMICS FINAL REMARKS REFERENCES INTRODUCTION TO QUANTUM SPIN LIQUIDS BASIC BUILDING BLOCKS OF VBC AND RVB PHYSICS: THE VALENCE BONDS VALENCE-BOND CRYSTALS ZEROTH-ORDER VBC WAVE FUNCTION QUANTUM FLUCTUATIONS IN VBCS VBC EXCITATIONS RESONATING-VALENCE-BOND SPIN LIQUIDS VBCS OR RVB SPIN LIQUIDS ON KAGOMÉ AND PYROCHLORE LATTICES? CONCLUSION PART II PROBING FRUSTRATED MAGNETS NEUTRON SCATTERING AND HIGHLY FRUSTRATED MAGNETISM WHAT NEUTRON SCATTERING MEASURES SCATTERING TRIANGLE PARTIAL DIFFERENTIAL CROSS SECTION RELATION TO SAMPLE PROPERTIES SCATTERING FROM ATOMIC MAGNETIC MOMENTS ORIENTATION FACTOR AND FORM FACTOR GENERAL EXPRESSION FOR THE NEUTRON SCATTERING REAL EXPERIMENTS POWDER AVERAGING STATIC APPROXIMATION WAVEVECTOR DEPENDENT MAGNETIC MOMENT AND SUSCEPTIBILITY FULLY ORDERED MAGNET MAGNET WITH FULL OR PARTIAL DISORDER VALIDITY OF THE STATIC APPROXIMATION GENERALISED SUSCEPTIBILITY NEUTRON SPECTROSCOPY TYPICAL NEUTRON SCATTERING PATTERNS SCATTERING PLANE FREE ENERGY IDEAL PARAMAGNET CONVENTIONAL MAGNET ABOVE TC CONVENTIONAL MAGNET BELOW TC COOPERATIVE PARAMAGNET ABSENT PINCH POINTS DYNAMICAL SIGNATURE OF COOPERATIVE PARAMAGNETISM EXPERIMENTAL RESULTS COOPERATIVE PARAMAGNET STATES ORDERED STATES EXCITED STATES CONCLUSIONS NMR AND SR IN HIGHLY FRUSTRATED MAGNETS BASIC ASPECTS OF NMR AND SR TECHNIQUES LINE SHIFT AND LINE WIDTH NUCLEAR AND MUON SPIN-LATTICE RELAXATION RATE 1=T1 SR: THE STATIC CASE SR: THE DYNAMIC CASE FROM ZERO- TO THREE-DIMENSIONAL FRUSTRATED MAGNETS MOLECULAR MAGNETS ANTIFERROMAGNETS ON A SQUARE LATTICE WITH COMPETING INTERACTIONS: THE J1-J2 MODEL MAGNETIC FRUSTRATION ON A TRIANGULAR LATTICE SR AND NMR IN THE SPIN-1/2 KAGOMÉ LATTICE ZNCU3(OH)6CL2 THE PROBLEM OF C RELAXATION IN SOME KAGOMÉ LATTICES PERSISTENT DYNAMICS AND LATTICE DISTORTIONS IN THE PYROCHLORE LATTICE OPTICAL TECHNIQUES FOR SYSTEMS WITH COMPETING INTERACTIONS INELASTIC LIGHT-SCATTERING INELASTIC PHONON LIGHT-SCATTERING INELASTIC MAGNETIC, QUASIELASTIC, AND ELECTRONIC LIGHT SCATTERING THE IR EXPERIMENT SPINS, PHONONS, AND LIGHT SPIN–PHONON INTERACTION IN CR SPINELS EXCITON–MAGNON ABSORPTION IN KCUF3 PART III FRUSTRATED SYSTEMS THE GEOMETRIES OF TRIANGULAR MAGNETIC LATTICES TWO-DIMENSIONAL STRUCTURES PLANES OF EDGE-SHARING TRIANGLES PLANES OF CORNER-SHARING TRIANGLES THREE-DIMENSIONAL STRUCTURES NOTE ON SYNTHESIS OF THE COMPOUNDS HIGHLY FRUSTRATED MAGNETISM IN SPINELS SPINEL STRUCTURE BASIC ELECTRONIC CON?GURATION UNIQUENESS OF THE SPINEL AS A FRUSTRATED MAGNET MATERIALS OVERVIEW OF SPINELS FRUSTRATION IN SELECTED SPINELS PYROCHLORE ANTIFERROMAGNETS IN SPINEL OXIDES – B-SITE FRUSTRATION FRUSTRATED SPINS ON SPINEL A SITES FRUSTRATED MAGNETS BASED ON CATION- ORDERED SPINELS: THE HYPER-KAGOMÉ LATTICE OF NA4IR3O8 CHARGE FRUSTRATION IN MIXED-VALENT SPINELS SUMMARY EXPERIMENTAL STUDIES OF PYROCHLORE ANTIFERROMAGNETS THE CUBIC PYROCHLORES THE SPIN LIQUID GROUND STATE IN TB2TI2O7 ORDERED GROUND STATES IN TB2TI2O7 STRUCTURAL FLUCTUATIONS IN THE SPIN LIQUID STATE OF TB2TI2O7 MAGNETIC ORDER AND FLUCTUATIONS IN TB2SN2O7 PHASE TRANSITIONS AND FLUCTUATIONS IN GD2TI2O7 AND GD2SN2O7 KAGOMÉ ANTIFERROMAGNETS: MATERIALS VS SPIN LIQUID BEHAVIORS A SHORT THEORETICAL SURVEY: WHAT WOULD BE THE IDEAL KAGOMÉ ANTIFERROMAGNET? THE JAROSITES SYNTHESIS AND THE JAROSITE CRYSTAL STRUCTURE: IDEALIZED AND DISORDERED FE JAROSITES: S D 2 KAGOMÉ ANTIFERROMAGNETS CR JAROSITES- S D PYROCHLORE SLABS SYNTHESIS MAGNETIC NETWORK GENERIC PHYSICS NON-MAGNETIC DEFECTS CONCLUDING REMARKS TOWARDS S D 1/2 IDEAL COMPOUNDS VOLBORTHITE HERBERTSMITHITE: “AN END TO THE DROUGHT OF QUANTUM SPIN LIQUIDS [100]” OTHER COMPOUNDS ORGANIC MATERIALS Y0:5CA0:5BACO4O7 LANGASITES PART IV SPECI?C EFFECTS IN FRUSTRATED MAGNETS MAGNETIZATION PLATEAUS MECHANISMS FOR FORMATION OF MAGNETIZATION PLATEAUS SPIN GAP QUANTIZED PLATEAUS SUPER?UID-INSULATOR TRANSITION ‘QUANTUM’ PLATEAUS HIGH-ORDER PLATEAUS TRANSITION INTO PLATEAUS EXPERIMENTAL OBSERVATION OF MAGNETIZATION PLATEAUS ‘CLASSICAL’ PLATEAUS IN TRIANGULAR AND PYROCHLORE LATTICES SRCU2(BO3)2 AND THE SHASTRY–SUTHERLAND MODEL ‘QUANTUM’ PLATEAUX AND SPIN SUPERSTRUCTURE IN SRCU2(BO3)2 PHASE DIAGRAM OF SRCU2(BO3)2 RB4: A NEW FAMILY OF SHASTRY–SUTHERLAND SYSTEM SPIN-LATTICE COUPLING IN FRUSTRATED ANTIFERROMAGNETS SPIN-DRIVEN JAHN–TELLER EFFECT IN A TETRAHEDRON GENERALIZED COORDINATES AND FORCES FOUR S D 1=2 SPINS ON A TETRAHEDRON FOUR CLASSICAL SPINS ON A TETRAHEDRON COLOR NOTATION AND OTHER USEFUL ANALOGIES SPIN–JAHN–TELLER EFFECT ON A TRIANGLE MODELS WITH LOCAL PHONON MODES SPINELS COLLECTIVE SPIN–JAHN–TELLER EFFECT ON THE PYROCHLORE LATTICE COLLECTIVE JAHN–TELLER EFFECT IN CDCR2O4 SPIRAL MAGNETIC ORDER IN CDCR2O4 THEORY OF SPIRAL MAGNETIC ORDER SUMMARY AND OPEN QUESTIONS SPIN ICE WATER ICE, PAULING ENTROPY, AND ANDERSON MODEL WATER ICE AND PAULING MODEL CATION ORDERING IN INVERSE SPINELS AND ANTIFERROMAGNETIC PYROCHLORE ISING MODEL DISCOVERY OF SPIN ICE RARE-EARTH PYROCHLORE OXIDES: GENERALITIES MICROSCOPIC HAMILTONIAN: TOWARDS AN EFFECTIVE ISING MODEL DISCOVERY OF SPIN ICE IN HO2TI2O7 NEAREST-NEIGHBOR FERROMAGNETIC H111I ISING MODEL AND PAULING’S ENTROPY RESIDUAL ENTROPY OF DY2TI2O7 AND HO2TI2O7 DIPOLAR SPIN-ICE MODEL COMPETING INTERACTIONS IN THE DIPOLAR SPIN-ICE MODEL MEAN-FIELD THEORY LOOP MONTE CARLO SIMULATIONS AND PHASE DIAGRAM OF DIPOLAR SPIN ICE ORIGIN OF ICE RULES IN DIPOLAR SPIN ICE CURRENT RESEARCH TOPICS IN SPIN ICES AND RELATED MATERIALS MAGNETIC-FIELD EFFECTS DYNAMICAL PROPERTIES AND ROLE OF DISORDER BEYOND THE DIPOLAR SPIN-ICE MODEL METALLIC SPIN ICE ARTI?CIAL SPIN ICE STUFFED SPIN ICE QUANTUM MECHANICS, DYNAMICS, AND ORDER IN SPIN ICES COULOMB PHASE, MONOPOLES AND DIRAC STRINGS IN SPIN ICES SPIN NEMATIC PHASES IN QUANTUM SPIN SYSTEMS INTRODUCTION AND MATERIALS MULTIPOLAR STATES OF A SINGLE SPIN COMPETITION BETWEEN DIPOLES AND QUADRUPOLES THE BILINEAR–BIQUADRATIC MODEL ENERGY SPECTRA OF SMALL CLUSTERS QUADRUPOLAR ORDERING IN S D 1 SYSTEMS VARIATIONAL PHASE DIAGRAM ONE- AND TWO-MAGNON INSTABILITY OF THE FULLY POLARIZED STATE SPIN-WAVE THEORY FOR THE FERROQUADRUPOLAR PHASE NUMERICAL APPROACH FROM CHAINS TO THE SQUARE LATTICE NEMATIC ORDERING IN S D 1=2 SYSTEMS PART V ADVANCED THEORETICAL METHODS AND CONCEPTS IN FRUSTRATED MAGNETISM SCHWINGER BOSONS APPROACHES TO QUANTUM ANTIFERROMAGNETISM SUN / HEISENBERG MODELS SCHWINGER REPRESENTATION OF SUN / ANTIFERROMAGNETS BIPARTITE ANTIFERROMAGNET NON-BIPARTITE (FRUSTRATED) ANTIFERROMAGNETS MEAN FIELD HAMILTONIAN MEAN FIELD EQUATIONS THE MEAN FIELD ANTIFERROMAGNETIC GROUND STATE STAGGERED MAGNETIZATION IN THE LAYERED ANTIFERROMAGNET VARIATIONAL WAVE FUNCTIONS FOR FRUSTRATED MAGNETIC MODELS SYMMETRIES OF THE WAVE FUNCTION: GENERAL PROPERTIES SYMMETRIES IN THE TWO-DIMENSIONAL CASE THE MARSHALL–PEIERLS SIGN RULE SPIN CORRELATIONS CONNECTION WITH THE BOSONIC REPRESENTATION ANTIFERROMAGNETIC ORDER NUMERICAL RESULTS ONE-DIMENSIONAL LATTICE TWO-DIMENSIONAL LATTICE OTHER FRUSTRATED LATTICES QUANTUM SPIN LIQUIDS AND FRACTIONALIZATION WHAT IS A SPIN LIQUID? ABSENCE OF MAGNETIC LONG-RANGE ORDER (DE?NITION 1) ABSENCE OF SPONTANEOUSLY BROKEN SYMMETRY (DE?NITION 2) FRACTIONAL EXCITATIONS (DE?NITION 3) HALF-ODD-INTEGER SPINS AND THE LIEB-SCHULTZ-MATTIS-HASTINGS THEOREM MEAN FIELDS AND GAUGE FIELDS FERMIONIC REPRESENTATION OF HEISENBERG MODELS LOCAL SU2/ GAUGE INVARIANCE MEAN-?ELD (SPIN-LIQUID) STATES GAUGE FLUCTUATIONS Z2 SPIN LIQUIDS SHORT-RANGE RVB DESCRIPTION Z2 GAUGE THEORY, SPINON DECON?NEMENT, AND VISONS EXAMPLES HOW TO DETECT A GAPPED Z2 LIQUID GAPLESS (ALGEBRAIC) LIQUIDS OTHER SPIN LIQUIDS QUANTUM DIMER MODELS HOW QUANTUM DIMER MODELS ARISE LINK VARIABLES AND HARD CONSTRAINTS THE ORIGIN OF CONSTRAINTS TUNABLE CONSTRAINTS ADDING QUANTUM DYNAMICS THE QUANTUM DIMER MODEL HILBERT SPACE TOPOLOGICAL INVARIANTS TOPOLOGICAL ORDER FRACTIONALISATION QDM PHASE DIAGRAMS GENERAL STRUCTURE OF PHASE DIAGRAMS Z2 RVB LIQUID PHASE U(1) RVB LIQUID PHASE DECON?NED CRITICAL POINTS VALENCE BOND CRYSTALS SUMMARY OF PHASE DIAGRAMS THE ROKHSAR–KIVELSON POINT GROUND-STATE WAVEFUNCTION FRACTIONALISATION AND DECON?NEMENT SPATIAL CORRELATIONS A SPECIAL LIQUID POINT OR PART OF A LIQUID PHASE? RESONONS, PHOTONS, AND PIONS: EXCITATIONS IN THE SINGLE MODE APPROXIMATION DUALITIES AND GAUGE THEORIES EMERGENCE OF THE QDM CONTINUUM LIMIT OF THE GAUGE THEORY HEIGHT REPRESENTATION NUMERICAL METHODS DIMER PHASES IN SU(2) INVARIANT MODELS OVERLAP EXPANSION DECORATION LARGE-N KLEIN MODELS: SU(2) INVARIANT SPIN LIQUIDS OUTLOOK HOPPING FERMIONS : : : : : : AND MUCH MORE NUMERICAL SIMULATIONS OF FRUSTRATED SYSTEMS OVERVIEW OF METHODS CLASSICAL MONTE CARLO QUANTUM MONTE CARLO STOCHASTIC SERIES EXPANSION (SSE) GREEN-FUNCTION MONTE CARLO SERIES EXPANSIONS HIGH-TEMPERATURE SERIES T D 0 PERTURBATIVE EXPANSIONS FOR GROUND- AND EXCITED-STATE PROPERTIES DENSITY-MATRIX RENORMALIZATION GROUP (DMRG) FINITE T DYNAMICAL RESPONSE FUNCTIONS DMRG IN TWO AND MORE DIMENSIONS EXACT DIAGONALIZATION (ED) BASIS CONSTRUCTION CODING OF BASIS STATES SYMMETRIZED BASIS STATES HAMILTONIAN EIGENSOLVERS IMPLEMENTATION DETAILS AND PERFORMANCE ASPECTS OBSERVABLES TIME EVOLUTION FINITE TEMPERATURES MISCELLANEOUS FURTHER METHODS CLASSICAL SPIN DYNAMICS (MOLECULAR DYNAMICS) COUPLED-CLUSTER METHOD DYNAMICAL MEAN-FIELD THEORY (DMFT) CONTRACTOR RENORMALIZATION (CORE) SR-RVB CALCULATIONS SOURCE CODE AVAILABILITY EXACT RESULTS IN FRUSTRATED QUANTUM MAGNETISM DIMER MODEL EXACT RESULTS IN SPIN-1/2 HEISENBERG MODELS EXACT GROUND STATES IN COUPLED TRIANGULAR CLUSTER MODELS EXACT GROUND STATES IN COUPLED TETRAHEDRAL CLUSTER MODELS REALIZATION OF EXACT GROUND STATES EXACT RESULTS IN FRUSTRATED SPIN-1/2 MODELS WITH FOUR-SPIN INTERACTIONS GENERAL LADDER MODEL WITH FOUR-SPIN INTERACTIONS TWO-DIMENSIONAL MODEL WITH FOUR-SPIN INTERACTIONS STRONG-COUPLING EXPANSION AND EFFECTIVE HAMILTONIANS STRONG-COUPLING EXPANSION SECOND-ORDER PERTURBATION THEORY HIGH-ORDER PERTURBATION THEORY ALTERNATIVE APPROACHES YIELDING EFFECTIVE HAMILTONIANS CANONICAL TRANSFORMATION CONTINUOUS UNITARY TRANSFORMATION CONTRACTOR RENORMALIZATION PART VI FRUSTRATION, CHARGE CARRIERS AND ORBITAL DEGENERACY MOBILE HOLES IN FRUSTRATED QUANTUM MAGNETS AND ITINERANT FERMIONS ON FRUSTRATED GEOMETRIES DOPING HOLES IN FRUSTRATED QUANTUM MAGNETS THE HOLON–SPINON DECON?NEMENT SCENARIO SINGLE HOLE DOPED IN FRUSTRATED MOTT INSULATORS HOLE PAIRING AND SUPERCONDUCTIVITY DOPED QUANTUM DIMER MODEL ORIGIN OF THE QUANTUM DIMER MODEL PHASE DIAGRAMS AT ZERO DOPING CONNECTION TO THE XXZ MAGNET ON THE CHECKERBOARD LATTICE BOSONIC DOPED QUANTUM DIMER MODEL NON-FROBENIUS DOPED QUANTUM DIMER MODEL ON THE SQUARE LATTICE MOTT TRANSITION ON THE TRIANGULAR LATTICE FRUSTRATION IN ITINERANT ELECTRON SYSTEMS MOTT TRANSITION IN ORGANIC COMPOUNDS WITH TRIANGULAR GEOMETRY MOTT TRANSITION IN THE TRIANGULAR-LATTICE HUBBARD MODEL ORDERING PHENOMENA AT COMMENSURATE FERMION DENSITIES ON FRUSTRATED GEOMETRIES BOND ORDER WAVES FROM NESTING PROPERTIES OF THE FERMI SURFACE METAL–INSULATOR TRANSITIONS AND FRUSTRATED CHARGE ORDER AWAY FROM COMMENSURABILITY: DOPING THE RESONATING-SINGLET-PAIR CRYSTAL METALLIC AND SUPERCONDUCTING MATERIALS WITH FRUSTRATED LATTICES MATERIALS OVERVIEW PYROCHLORE LATTICE TRIANGULAR AND KAGOMÉ LATTICES ORGANIC CONDUCTORS WITH TRIANGULAR LATTICE THEORETICAL BACKGROUND RVB SPIN STATE AND RVB SUPERCONDUCTIVITY TRIANGULAR-LATTICE HUBBARD MODEL EXTENDED HUBBARD MODEL FOR ORGANIC CONDUCTORS SUPERCONDUCTING COMPOUNDS PYROCHLORE LATTICE: CD2RE2O7 AND AOS2O6 TRIANGULAR LATTICE: NAXCOO2 AND ITS HYDRATE ANISOTROPIC TRIANGULAR LATTICE: ORGANIC SUPERCONDUCTIVITY FRUSTRATION IN SYSTEMS WITH ORBITAL DEGREES OF FREEDOM ORBITAL DEGREES OF FREEDOM ORBITALS AND THEIR ENERGY SCALES COMPARING ORBITAL AND SPIN DEGREES OF FREEDOM ORBITAL INTERACTIONS AND ORBITAL MODELS CRYSTAL-FIELD SPLITTING OF ORBITALS JAHN–TELLER DEFORMATION JAHN–TELLER-MEDIATED ORBITAL–ORBITAL INTERACTIONS SUPEREXCHANGE-MEDIATED ORBITAL–ORBITAL INTERACTIONS SYMMETRY AND SYMMETRY-BREAKING IN ORBITAL MODELS TYPES OF SYMMETRY IN ORBITAL MODELS EXAMPLES OF INTERMEDIATE SYMMETRIES IN ORBITAL SYSTEMS A THEOREM ON DIMENSIONAL REDUCTION CONSEQUENCES OF THE THEOREM FOR ORBITAL (AND SPIN) ORDERS AND EXCITATIONS ORDER BY DISORDER IN CLASSICAL ORBITAL MODELS CONNECTION WITH QUANTUM COMPUTATION KITAEV’S HONEYCOMB MODEL KITAEV’S TORIC CODE MODEL RECENT DISCUSSIONS OF QUANTUM COMPUTING REALIZATIONS SPIN-ORBITAL FRUSTRATION GENERAL STRUCTURE OF SPIN-ORBITAL SUPEREXCHANGE MODELS SPIN-ORBITAL MODELS FOR EG PEROVSKITES SPIN-ORBITAL SUPEREXCHANGE FOR T2G PEROVSKITES SPIN-ORBITAL FRUSTRATION ON A TRIANGULAR LATTICE SPIN-ORBITAL FRUSTRATION IN SPINELS SPIN-ORBITAL ENTANGLEMENT