Frustrated Spin Systems

The following sections are included:

• PREFACE

• CONTENTS

After a short introduction on frustrated spin systems, we study in this chapter several two-dimensional frustrated Ising spin systems which can be exactly solved by using vertex models. We show that these systems contain most of the spectacular effects due to the frustration: high ground-state degeneracy, existence of several phases in the ground-state phase diagram, multiple phase transitions with increasing temperature, reentrance, disorder lines, partial disorder at equilibrium. Evidences of such effects in non solvable models are also shown and discussed.

The following sections are included:

• Introduction

• Ising model on two-dimensional frustrated lattice and on stacked frustrated lattice

• Ising model on antiferromagnetic triangular lattice

• Ising model on stacked antiferromagnetic triangular lattice

• Ising model with large S on antiferromagnetic triangular lattice

• Ising model with infinite-spin on antiferromagnetic triangular lattice

• Ising-like Heisenberg model on antiferromagnetic triangular lattice

• Ising model with infinite-spin on stacked antiferromagnetic triangular lattice

• Phase diagram in spin-magnitude versus temperature for Ising models with spin S on stacked antiferromagnetic triangular lattice

• Effect of antiferromagnetic interaction between next-nearest-neighbor spins in xy-plane

• Three-dimensional Ising paramagnet

• Concluding remarks

• Acknowledgements

• References

This chapter is devoted to the recent advances in the renormalization group (RG) approaches to the physics of classical frustrated magnets in three dimensions. We first recall the main features of the field theoretical approach to these systems including considerations on symmetries, symmetry breaking schemes, continuum limits, topological contents, etc. We then provide an overview of the phenomenological situation, stressing on the most striking aspect of the physics of frustrated magnets: the existence of nonuniversal scaling behaviors. We then review the various perturbative and nonperturbative RG approaches that have been used to investigate frustrated magnets. Finally, a large part of this chapter is devoted to a recent nonperturbative approach that has clarified the intricate physical situation of frustrated magnets.

The following sections are included:

• Introduction

• Breakdown of symmetry
  • Symmetry in the high-temperature region
  • Breakdown of symmetry for ferromagnetic systems
  • Breakdown of symmetry for frustrated systems
    • Stacked triangular antiferromagnetic lattices
    • bct helimagnets
    • Stacked J₁−J₂ square lattices
    • The simple cubic J₁−J₂ lattice
    • J₁−J₂−J₃ lattice
    • Villain lattice and fully frustrated simple cubic lattice
    • Face-centered cubic lattice (fcc)
    • Hexagonal-close-packed lattice (hcp)
    • Pyrochlores
    • Other lattices
    • STAR lattices
    • Dihedral lattices V_N,2
    • Right-handed trihedral lattices V_3,3
    • P-hedral lattices V_N,P
    • Ising and Potts-V_N,1 model
    • Ising and Potts-_N,2 model
    • Landau–Ginzburg model
    • Cubic term in Hamiltonian
    • Summary

• Phase transitions between two and four dimensions: 2 < d ≤ 4
  • O(N)/O(N − 2) breakdown of symmetry
    • Fixed points
    • MCRG and first-order transition
    • Complex fixed point or minimum in the flow
    • Experiment
    • Value of N_c
    • Phase diagram (N,d)
    • Renormalization-group expansions
    • Short historical review
    • Relations with the Potts model
  • O(N)/O(N − P) breakdown of symmetry for d = 3
  • Z₂ ⊗ SO(N)/SO(N − 1) breakdown of symmetry for d = 3
  • Z₃ ⊗ SO(N)/SO(N − 1) breakdown of symmetry for d = 3
  • Z_q ⊗ O(N)/O(N − 2) and other breakdown of symmetry in d = 3

• Conclusion

• O(N) frustrated vector spins in d = 2
  • Introduction
  • Non frustrated XY spin systems
  • Frustrated XY spin systems: Z₂ ⊗ SO(2)
  • Frustrated XY spin systems: Z₃ ⊗ SO(2)
  • Frustrated XY spin systems: Z₂ ⊗ Z₂ ⊗ SO(2) and Z₃ ⊗ Z₂ ⊗ SO(2)
  • Frustrated Heisenberg spin systems: SO(3)
  • Frustrated Heisenberg spin systems: Z₂ ⊗ SO(3), Z₃ ⊗ SO(3) …
  • Topological defects for N ≥ 4

• General conclusions

• Acknowledgments

• Appendix A: Monte Carlo simulation

• Appendix B: Renormalization group: Landau-Ginzburg theory, expansions in fixed dimension d = 3 and for d = 4 − ε and its implications for experiments

• References

The following sections are included:

• Introduction

• J₁−J₂ model on the square lattice
  • Classical ground-state and spin-wave analysis
  • Order by disorder (J₂ > J₁/2)
  • Non-magnetic region (J₂ ≃ (J₁/2)
    • Series expansions
    • Exact diagonalizations
    • Quantum Monte Carlo

• Valence-bond crystals
  • Definitions
  • One-dimensional and quasi one-dimensional examples (spin-½ systems)
  • Valence bond solids
  • Two-dimensional examples of VBC
    • Without spontaneous lattice symmetry breaking
    • With spontaneous lattice symmetry breaking
  • Methods
  • Summary of the properties of VBC phases

• Large-N methods
  • Bond variables
  • SU(N)
  • Sp(N)
    • Gauge invariance
    • Mean-field (N = ∞ limit)
    • Fluctuations about the mean-field solution
    • Topological effects - instantons and spontaneous dimerization
    • Deconfined phases

• Quantum dimer models
  • Hamiltonian
  • Relation with spin-½ models
  • Square lattice
    • Transition graphs and topological sectors
    • Staggered VBC for V/J > 1
    • Columnar crystal for V < 0
    • Plaquette phase
    • Rokhsar-Kivelson point
  • Hexagonal lattice
  • Triangular lattice
    • RVB liquid at the RK point
    • Topological order
  • Solvable QDM on the kagome lattice
    • Hamiltonian
    • RK ground-state
    • Ising pseudo-spin variables
    • Dimer-dimer correlations
    • Visons excitations
    • Spinons deconfinement
    • ℤ₂ gauge theory
  • A QDM with an extensive ground-state entropy

• Multiple-spin exchange models
  • Physical realizations of multiple-spin interactions
    • Nuclear magnetism of solid ³He
    • Wigner crystal
    • Cuprates
  • Two-leg ladders
  • MSE model on the square lattice
  • RVB phase of the triangular J₂−J₄ MSE
    • Non-planar classical ground-states
    • Absence of Néel LRO
    • Local singlet-singlet correlations - absence of lattice symmetry breaking
    • Topological degeneracy and Lieb-Schultz-Mattis theorem
    • Deconfined spinons
  • Other models with MSE interactions

• Antiferromagnets on the kagome (and related) lattices
  • Miscellaneous models on the kagome lattice
  • Spin-½ Heisenberg model on the kagome lattice: an extreme play-ground for “quantum fluctuations”
    • Ground-state energy per spin
    • Correlations
    • Spin-gap
    • An exceptional density of low lying excitations in the singlet sector
    • Absence of gap in the singlet sector
    • Anomalous density of states in other spin sectors
  • Nearest-neighbor RVB description of the spin-½ kagome antiferromagnet
  • Experiments in compounds with kagome-like lattices
  • “Haldane's conjecture”

• Conclusions

• Acknowledgments

• References

This chapter is intended as a brief overview of some of the zero-temperature quantum spin liquid phases with unbroken SU(2) spin symmetry that have been found in one dimension. The main characteristics of these phases are discussed by means of the bosonization approach. A special emphasis is laid on the interplay between frustration and quantum fluctuations in one dimension.

Pauling's model of hydrogen disorder in water ice represents the prototype of a frustrated system. Over the years it has spawned several analogous models, including Anderson's model antiferromagnet and the statistical “vertex” models. Spin Ice is a sixteen vertex model of “ferromagnetic frustration” that is approximated by real materials, most notably the rare earth pyrochlores Ho₂Ti₂O₇, Dy₂Ti₂O₇ and Ho₂Sn₂O₇. These “spin ice materials” have the Pauling zero point entropy and in all respects represent almost ideal realisations of Pauling's model. They provide experimentalists with unprecedented access to a wide variety of novel magnetic states and phase transitions that are located in different regions of the field-temperature phase diagram. They afford theoreticians the opportunity to explore many new features of the magnetic interactions and statistical mechanics of frustrated systems. This chapter is a comprehensive review of the physics - both experimental and theoretical - of spin ice. It starts with a discussion of the historic problem of water ice and its relation to spin ice and other frustrated magnets. The properties of spin ice are then discussed in three sections that deal with the zero field spin ice state, the numerous field-induced states (including the recently identified “kagomé ice”) and the magnetic dynamics. Some materials related to spin ice are briefly described and the chapter is concluded with a short summary of spin ice physics.

Much recent experimental progress has been made in the study of magnetic materials made up of antiferromagnetically-coupled magnetic moments residing on networks of corner-sharing tetrahedra. Such networks of magnetic moments display phenomena known broadly as geometric frustration. They are found in nature in a variety of cubic pyrochlore, spinel and Laves phase materials, with the magnetic moments arising from either rare earth or transition metal electrons. Typically, the rare earth moments are relatively large and are only weakly coupled by exchange, such that the competition between antiferromagnetic exchange and dipolar interactions is important. Antiferromagnetic exchange interactions are stronger in transition metal-based materials, and the manifestation of geometrical frustration is different. This review focusses on experimental progress in this area from the last ten years primarily due to neutron scattering studies.

We review recent findings on spin glass models. Both the equilibrium properties and the dynamic properties are covered. We focus on progress in theoretical, in particular numerical, studies, while its relationship to real magnetic materials is also mentioned.

The following sections are included:

• INDEX