Introduction
Whether studying elementary particles, nuclei, molecules, or condensed matter, physicists often encounter systems composed of a large number of particles that interact strongly with each other. The collective behaviour of the particles can lead to emergent properties that are surprising, counterintuitive and hard to predict. In quantum many-body theory, an important concept is that low-energy excited states can be described in terms of weakly interacting 'quasiparticles'. However, identifying what these quasiparticles are for a particular system is a major challenge because the quasiparticles sometimes have distinctly different quantum numbers and quantum statistics than the constituent particles. For example, the electrons (fermions with charge e and spin 1/2) in a fractional quantum Hall state can act collectively so that the quasiparticles have charge e/3 and are not fermions or bosons but 'anyons' with fractional statistics. In carbon nanotubes and linear chain compounds such as SrCuO2, the charge and spin of the electrons 'separate' into spinless charge excitations (holons) and neutral spin-1/2 excitations (spinons)1. On page 790 of this issue, Kohno, Starykh and Balents2 report the use of a sophisticated theoretical method to identify and describe the quasiparticles (spinons and triplons) of an important class of quantum antiferromagnets in two dimensions (2D). This problem has attracted significant theoretical and experimental attention over the past two decades, largely stimulated by the challenge of understanding high-temperature superconductors3, 4.
In three dimensions, quantum antiferromagnets usually form a
ground state with a net magnetic moment on each atom. Consequently, spin
rotational symmetry is broken and the lowest-lying excitations are 'Goldstone
modes' — weakly interacting bosons with magnetic spin quantum numbers,
Sz = 
1. These quasiparticles, called
magnons, are associated with long-wavelength rotations of the local magnetic
moment.
The work of Kohno, Starykh and Balents2 reveals the composite structure of spin excitations in a particular class of quantum antiferromagnets. Here, an analogy with elementary particle physics is useful. Pions are the particles that mediate the strong nuclear force that holds together the protons and neutrons in atomic nuclei. The three possible charge states of the pion form a triplet representation of the isospin symmetry. When they are probed at low energies, pions can be viewed as the Goldstone bosons associated with chiral symmetry breaking. However, pions (with integer quantum numbers) are composed of a pair of quarks (with fractional quantum numbers) that are bound together (confined) by the 'gluons' associated with the gauge field of quantum chromodynamics. Pion scattering experiments involving large momentum and energy transfers reveal this composite structure.
Quantum antiferromagnets in one spatial dimension have distinctly
different properties from those in three dimensions. In one dimension, there is
no symmetry breaking or net magnetic moment; the ground state is called a 'spin
liquid'. The low-lying excitations have spin 1/2, and obey fractional
statistics. In contrast, in three dimensions the magnons act like bosons.
Scattering of neutrons creates excitations with Sz = 1, and these single magnons
propagate through the lattice. In one dimension, the Sz = 1 excitations are composed of
pairs of spinons with different momenta. These spinons are deconfined, so they
can propagate independently of one another (see Fig.
1).
Figure 1: In a one-dimensional chain of quantum spins (spin S = 1/2) with antiferromagnetic nearest-neighbour interactions, the lowest-lying eigenstates can be described as weakly interacting particles called spinons of spin-1/2.
a, A possible ground state. A second possible state is found by flipping all the spins. b, Flipping a spin in the centre of the chain produces an excited state with a total spin of 1. c, Flipping two spins on each side of the central spin produces a state with the same energy, so the spinons can move independently within the chain. The spinons can also be viewed as domain walls (denoted by stars) between the two possible ground states. Kohno, Starykh and Balents2 show that individual spinons can only move coherently between chains by pairing up into a composite particle, a triplon. Figure derived from ref. 11.
Full size image (29 KB)It is possible to directly 'see' quasiparticles and measure their energy and lifetimes using experimental probes such as inelastic neutron scattering and angle-resolved photoemission spectroscopy. Well-defined quasiparticles appear as sharp peaks in the scattering cross-section. The first material in which deconfined spinons were clearly observed was KCuF3, which is composed of linear chains of spin-1/2 copper ions5 — a quasi-one-dimensional system.
But the search for a two-dimensional system with spin excitations with fractional spin — triggered by Anderson's theory of high-temperature superconductivity, which involves an intimate connection between spin liquids and superconductivity3 — has been less successful. Significant experimental support for the existence of deconfined spinons in two dimensions came a few years ago from beautiful inelastic neutron-scattering experiments that Radu Coldea and Alan Tennant performed on Cs2CuCl4 (ref. 6). This material can be viewed as a spatially anisotropic triangular lattice in which chains of spin-1/2 copper ions are coupled together in a frustrated manner such that not all pairs of spins can minimize their antiferromagnetic interaction energy. The experiments clearly showed high-energy scattering, reminiscent of that associated with spinons in one dimension. Furthermore, the observed dispersion of the energy of the excitations for momenta in the direction perpendicular to the chains suggested that the spinons could move coherently between the chains.
Kohno, Starykh and Balents provide an alternative interpretation of these experimental results. They make use of a theoretical method that starts from the exact quantum states in one dimension and show how spinons can exist in a two-dimensional model; they are 'descendants' of the spinons that exist within the individual chains. The dispersion perpendicular to the chains only occurs because pairs of spinons form bound states, triplons, that can move between the chains. Thus, the spinons are deconfined within individual chains but confined to a chain.
Triplons have a dispersion relation similar to that of the magnons that would exist if the ground state had a broken symmetry. However, the triplons are distinctly different physical entities as they exist in the absence of broken symmetry. Furthermore, all three triplon modes have the same dispersion relation, whereas the three magnon modes associated with a broken symmetry state (a spiral state) have different dispersion relations, depending on their polarization. It is interesting that even though in Cs2CuCl4 the interaction between spins on neighbouring chains is only three times smaller than the interaction between spins on the same chain, the authors get an excellent quantitative description treating the interchain interaction as a perturbation. This agreement confirms an observation that the combination of frustration and large quantum fluctuations make the system properties much more one-dimensional than the hamiltonian7.
So why is this work2 significant? It provides (1) concrete evidence for fractional quantum numbers in a two-dimensional model, (2) detailed quantitative agreement between quantum many-body theory and experimental data on a real material, and (3) a sprinkling of rain in the current drought of reliable theoretical methods for calculating properties of strongly correlated electron models in two dimensions. Moreover, we can see how frustration allows fractional quantum numbers to survive in two dimensions. In particular, the work is consistent with the ideas of Anderson that quasiparticles with fractional quantum numbers must combine into composite quasiparticles with integer quantum numbers to survive in a system of higher dimension.
Future studies could focus more on the connection between spin
liquids and superconductivity. In this regard, the most promising potential
material is an organic molecular crystal, .gif)
-(BEDT-TTF)2Cu2(CN)3,
which seems to have a spin-liquid ground state at ambient pressure but becomes
superconducting at high pressures. Interestingly, the relevant model hamiltonian
is closely related to that for Cs2CuCl4 (ref. 8).
Solving this problem will require extending the current technique for Heisenberg
models to Hubbard models that can describe holons and spinons in sets of chains
with frustrated interchain coupling. Also, experimentalists should search for
the predicted anti-triplon mode and do polarized neutron scattering to see if
all three triplons have the same dispersion relation.
More broadly, the approach of Kohno, Starykh and Balents illustrates how properties of the ground state (vacuum) and low-energy excited states (quasiparticles) of quantum many-body systems are emergent. That fractional quantum numbers are emergent phenomena leads some condensed-matter theorists to make radical claims that in elementary particle physics the 'vacuum' state, fundamental symmetries and even fermion statistics are also emergent phenomena9, 10. If so, what are the truly 'elementary' particles of nature?