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Abstract
This article discusses at a qualitative level a number of issues at the forefront of current understanding and developments in frustrated quantum magnetism. The focal point of the presentation is the spin liquid, which is introduced in terms of (un)broken spin and lattice symmetries. An overview of the full spectrum of research activity in the field is obtained by considering selected examples from experimental approaches to realising spin-liquid states, from theoretical efforts which seek both to classify spin liquids according to their physical properties and to broaden the search for spin-liquid behaviour, and from numerical techniques which offer the prospect of qualitatively new insight into frustrated spin systems.
Acknowledgements
The author would like to thank C. Batista, F. Becca, F. Mila, G. Misguich, T.-K. Ng, Z. Nussinov, A.M. Oleś and T. Xiang for their input to the scientific content of this article, and is grateful to M. Sigrist and X. Wang for their hospitality during its preparation.
Notes
1. It is assumed here, and throughout this article, that spin space is decoupled from real space, i.e. that only the relative moment directions are important. This is not the case in the presence of crystalline anisotropies in real materials, or for certain types of interaction, including the familiar dipolar coupling, where the bond direction appears in the Hamiltonian.
2. The AF interaction is unfrustrated on a square lattice, or any other bipartite system; while the focus here is on the simplest interaction types, in principle the definition of frustration depends on the interaction and may need to be extended for more complex situations.
3. The author is grateful to F. Mila for this classification.
4. Note that ‘broken lattice symmetry’ remains a spontaneous phenomenon concerning the spin degrees of freedom of the system, and has no connection to the degrees of freedom of the lattice itself. While such a connection may exist (‘magnetoelastic coupling’, ‘Jahn-Teller coupling’), and indeed is a determining factor in the electronic and magnetic properties of many real materials, it is not the origin of any of the physics on which this article concentrates.
5. Here, however, rather more exceptions with weaker interactions do exist.
6. To date almost all consideration of interacting atoms in optical lattices has involved systems where the atoms are in their lowest, s -wave states. However, maintaining all the atoms in higher, p - or even d -wave states is possible, and leads to some of the directional specificity of interaction and kinetic parameters found in real materials. One example may be found in Citation27.
7. These statements may be modified depending on the nature of the splitting of the ground manifold shown in . They also apply only in the limit of dilute spinons: when the spinons meet each other they will interact.
8. The spin states may in principle be represented using a variational wavefunction composed of either bosonic or fermionic quasiparticles. Because a bosonic system has many more basis states, it is in general significantly more difficult to work with. However, for certain frustrated quantum magnets the bosonic representation can deliver a better variational treatment, one example of which may be found in Citation58.
9. Although Citation63,Citation64 illustrate the application of the TRG method only in 2d, the formulation is in principle applicable in any dimension.