Frustrated Spin Systems

The following sections are included:

• Preface of the Second Edition

• Preface of the First Edition

• 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:

• Abbreviations

• Introduction

• Breakdown of Symmetry
  • Symmetry in the high-temperature region
  • Breakdown of symmetry for ferromagnetic systems
  • Breakdown of symmetry for frustrated systems

• Phase transitions between two and four dimensions: 2 < d ≤ 4
  • O(N)/O(N - 2) breakdown of symmetry
  • 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

• Note added for the 2nd Edition

• Appendix A: Monte Carlo Simulation

• Appendix B: Renormalization Group: Landau–Ginzburg theory, expansions in fixed dimension d = 3 and for d = 4- ∊ and its implications for experiments

This review presents some theoretical advances in the field of quantum magnetism in two-dimensional systems, and quantum spin liquids in particular. The first version published in 2005 has been updated for the present second edition of the book: the section devoted to the kagome antiferromagnet (Sec. 7) has been completely rewritten, as well as the concluding section (Sec. 8). The other sections (Secs. 1-6) are unchanged from the first edition of the book.

This chapter is intended as a brief overview of some of the quantum spin liquid phases with unbroken SU(2) spin symmetry available 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