Monday, November 30, 2009

What are deconfined spinons?

They are the spin-1/2 quasi-particle excitations associated with a spin liquid ground state of a quantum antiferromagnet. Perhaps the easiest way to understand them is in contrast to the low-lying excitations in an antiferromagnet with an ordered ground state. Spontaneously broken symmetry is the key concept behind understanding the nature of these excitations. Specifically, for infinite systems the ground state is usually degenerate and is not invariant under the samesymmetries as the system Hamiltonian. This family of ground states is described by an ``order parameter'' which describes the extent of the symmetry breaking. For example, quantum antiferromagnets can be described by a Heisenberg model Hamiltonian which describes a lattice of spins which interact with their nearest (and sometimes next-nearest) neigbours on the lattice. The model Hamiltonian is invariant under rotations of all the spins and under lattice translations. However, both these symmetries can be broken by the ground states.

The figure (a) above shows a cartoon picture of a line of alternating spins in the common antiferromagnetic ground state. This symmetry breaking can be seen by the presence of new Bragg peaks in elastic neutron scattering. In contrast, a spin liquid can be defined as a ground state in which there is no broken spin or translational symmetries.

When there is a continuous symmetry that is broken, the lowest lying excitations are weakly interacting bosons, known as Goldstone modes, and are associated with small
long wave length ``rotations'' of the order parameter. (see Figure (a) above). For an antiferromagnet, these modes have spin-1, and are also referred to as magnons. They can also be viewed as the propagation of a spin flip through the lattice.

The Heisenberg antiferromagnet in one spatial dimension has distinctly different properties from in three dimensions. In one dimension, there is no symmetry breaking or long range magnetic order; the ground state is a spin liquid. The low-lying excitations have spin 1/2, and can be viewed as domains walls or solitons in the background fluctuating magnetic order (see Figure (b) above. Haldane showed these excitations obey fractional statistics (semions which are intermediate between fermions and bosons) in contrast to the spin 1 magnons which are bosons, in three dimensions. Scattering of neutrons creates spin 1 excitations which are composed of pairs of spinons with different momenta. These spinons are deconfined, i.e., they can propagate independently of one another. (See figure above).

It is possible to directly ``see'' the quasi-particles, and measure their energy and lifetimes using inelastic neutron scattering. One scatters a beam of neutrons off a magnetic material and measures the momentum and energy of the scattered neutrons.
If there are well-defined quasi-particles they will have a particular energy for each wavevector q. The neutron scattering cross section is proportional to the dynamical spin structure factor S(E,q) will then show well-defined peaks when E=h ω(q) (see Figure above).

In an actual material the deconfinement of spinons was first seen in the compound KCuF3 which is composed of linear chains of spin-1/2 copper ions. The experimental signature of deconfined spinons was the presence of substantial spectral weight at energies above the magnon dispersion (the lower dashed line in the Figure above) one would see in a semi-classical antiferromagnet.

The Figures above are taken from a review, Mapping atomic motions in materials, by Toby Perring (ISIS) in Materials Today. The spectrometers for inelastic neutron scattering that Toby has built at ISIS have produced much of the beautiful data (such as that shown above) which is keeping theorists such as myself very busy.

Whether or not spinons exist in any real two-dimensional material is controversial. A News and Views piece I wrote for Nature Physics several years ago briefly reviews some of the issues.

Something I am still not clear on is whether a spin liquid ground state is a necessary and/or a sufficient condition for the existence of spinons.

Any ideas?

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