An Introduction to Quantum Spin Systems
The topic of lattice quantum spin systems is a fascinating and by now well-established branch of theoretical physics. However, many important questions remain to be answered. Their intrinsically quantum mechanical nature and the large (usually effectively
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Quantum Magnetism
Abstract Many quantum spin systems of spin quantum number s > 1/2 or in spatial dimensions greater than 1D cannot be solved exactly. One source of this “lack of integrability” comes from the competition between different bonds, as in quantum frustration. We begin this final chapter by considering the application of approximate methods to one-dimensional models such as the spin-half J1 –J2 model, and the spin-one Heisenberg and biquadratic models. The properties of the spin-half Heisenberg model for Archimedean lattices such as the square (unfrustrated) lattice, and the triangular and Kagomé (frustrated) lattices are listed for a variety of approximate techniques. The phenomenon of “order-from-disorder” is investigated in the context of the triangular lattice antiferromagnet in the presence of an external field. Finally, the properties of the square-lattice J1 –J2 model and the Shastry-Sutherland model are studied. The chapter shows how the application of a range of approximate techniques, in addition to the few isolated exact results, can provide a comprehensive and compelling description of the ground-state properties of a wide range of quantum spin systems.
11.1 Introduction In previous chapters we saw that the spin-half one-dimensional Heisenberg model may be solved exactly by using the Bethe Ansatz and that the Néel order of the classical ground state is removed by quantum fluctuations. However, there are other ways in which the classical ordering may be replaced by states of quantum order that have no classical counterpart or by quantum disorder. One possible situation in which this can happen is in frustrated systems [1]. The term ‘frustration’ indicates that different bonds in the Hamiltonian compete against each other. A simple example is a triangle of spins with antiferromagnetic exchange interactions. The spins wish to align antiparallel to their neighbours but this is not possible for all bonds (connected pairs of atoms) simultaneously. Classically, this means that the energies for these different bonds are higher than for their corresponding unfrustrated counterparts. Bipartite lattices (such as the square lattice) with nearest neighbour antiferromagnetic exchange interactions are unfrustrated since in the classical Néel ground state Parkinson, J.B., Farnell, D.J.J.: Quantum Magnetism. Lect. Notes Phys. 816, 135–152 (2010) c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-13290-2_11
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11 Quantum Magnetism
each spin is aligned antiparallel to all its neighbours, and the energy associated with each bond is −J S 2 . (Bipartite lattices are those lattices that may be divided into two interpenetrating sublattices in which nearest neighbours to sites on one sublattice are always on the other sublattice and vice versa.) By contrast, the classical ground state of the spin-half Heisenberg model on the frustrated triangular lattice has spins at an angle of 120◦ to each other. The classical energy of a single antiferromagnetic bond in the Hamiltonian is now − 12 J S 2 , i.e
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