Fundamentals of the Physics of Solids / 15-Elementary Excitations in Magnetic Systems

15.6 Spin Liquids

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As a first example consider a two-dimensional square lattice with nearestneighbor interactions. In this system spins are not frustrated, and quantum fluctuations cannot destroy the Néel-type antiferromagnetic order. However, in the presence of a su ciently strong (Jd ≈ 0.5J) antiferromagnetic coupling between (diagonally separated) next-nearest neighbors the ground state is disordered – although the exact nature of this disorder is not fully understood yet.

Frustration arising from the geometry (topology) of the lattice is observed in the antiferromagnetic triangular lattice. Considering first a single triangle, the three spins at the vertices are each others’ nearest neighbors. If, owing to the antiferromagnetic coupling, the spins at two vertices are oriented oppositely (quantum mechanically: if their spin projections are 1/2 and −1/2), then they act oppositely on the third spin. With its orientation undetermined, the state of the third spin is the combination of the sz = 1/2 and sz = −1/2 components with equal weights since the two spin orientations are of the same energy. As a resolution of this frustration Anderson15 suggested that the ground state of the isotropic antiferromagnetic Heisenberg model on a triangular lattice is a disordered singlet. It is a superposition of singlet states in which every spin forms a singlet pair with a nearby spin. In the language of quantum chemistry, the state is a superposition of valence bonds. The energy can be lowered if the singlet pairs break up and reform – that is, if they resonate. Hence the name for this hypothetical spin liquid: resonating valence bond spin liquid (RVB spin liquid).

It turned out that resonating singlets do not su ciently lower the energy on a triangular lattice. The true ground state is a three-sublattice Néel-type state. If, however, in addition to nearest-neighbors interactions other couplings

– e.g., multi-spin exchange processes – are also important, the ground state may be a spin-liquid state.

There exist certain lattices in nature that are topologically more strongly frustrated than the triangular lattice, e.g., the two-dimensional kagome lattice or the three-dimensional pyrochlore lattice. The kagome lattice shown in Fig. 5.4 can be described as a network of corner-sharing (interlaced) triangles. In the pyrochlore lattice corner-sharing tetrahedra form a face-centered cubic lattice. Assuming classical spins in both cases, the orientation of the spins on neighboring units is not fixed even when the sum of the spins on each triangle or tetrahedron is required to vanish, and the ground state is disordered. The quantum mechanical ground states of these models are not yet known, but they are expected to be spin liquids. An indication for this is that Cu3V2O7(OH)2·2H2O – in which the spin-1/2 copper ions form a kagome lattice – does not have any usual magnetically ordered state. Similarly, in Tb2Ti2O7 – which has a pyrochlore structure, with magnetic rare-earth ions at the vertices of the tetrahedra – no magnetic ordering has been observed down to 70 mK.

15 P. W. Anderson, 1973.

586 15 Elementary Excitations in Magnetic Systems

If the RVB state proposed by Anderson is realized in these materials, then, on account of the resonance of short-range singlet valence bonds, the two spins liberated in the excited state from the singlet pair are expected to propagate practically independently, and the true elementary excitations are gapped spin-1/2 spinons. According to the numerical results obtained for such models the spectrum of triplet excitations does, indeed, have a finite gap, however, it is filled with a continuum of singlet excitations. This indicates that there are still a lot of mysteries to be resolved about the spin-liquid state.

Further Reading

1.A. Herpin, Théorie du magnétisme, Presses Universitaires de France, Paris (1968).

2.L.-P. Lévy, Magnetism and Superconductivity, Texts and Monographs in Physics, Springer-Verlag, Berlin (2000).

3.D. C. Mattis, The Theory of Magnetism I: Thermodynamics and Statistical Mechanics, Springer-Verlag, Berlin (1988); The Theory of Magnetism

II:Statics and Dynamics, Springer-Verlag, Berlin (1985).

4.K. Yosida, Theory of Magnetism, Springer Series in Solid-State Sciences, Vol. 122, Springer-Verlag, Berlin (1996).

5.S. V. Vonsovskii, Magnetism, John Wiley & Sons, New York (1974).

6.R. M. White, Quantum Theory of Magnetism, Springer Series in SolidState Sciences, Vol 32, Springer-Verlag, Berlin (2002).