Quantum Magnetism
Introduction
The most common model of insulating magnets is of a Néel ordered ground state and spin-wave excitations. In many cases this semi-classical picture gives a fairly accurate account of the observed behaviour. Discrepancies arise when the effects of quantum mechanics start to dominate the behaviour. In particular this occurs in materials which have magnetic ions with small spin values which are coupled by antiferromagnetic exchange interactions where these interactions are quasi-one-dimensional, (couple the spins in one crystallographic direction only) or frustrated (favour contradictory spin alignments). Our measurements of quantum magnets reveal the novel behaviour found in these systems and the highlights are given below
Background
An example of a quantum magnet is the ideal one-dimensional spin-½ Heisenberg antiferromagnet with no interchain coupling. This system does not develop Néel ordered even at the lowest temperatures and its excitations are spinons not spin-waves.
Spinons can be regarded as twists in a chain and they are fractional particles which possess a spin moment if S=½ in contrast to spin-waves or magnons which possess a spin moment of S=1. Because in quantum mechanics the spin moment can only change by one unit, even numbers of spinons are created in each neutron scattering event so that their total angular momentum adds up to S=1, this leads to the observation of a continuum of scattering in inelastic neutron measurements.
An alternative way of approaching the one-dimensional, spin-½, Heisenberg antiferromagnet is through the theory of the quantum critical points. A quantum critical point is a phase transition that is suppressed to T=0, by quantum fluctuations. This is exactly the situation occurring in the ideal one-dimensional Spin=½ Heisenberg antiferromagnet where long-range Neel order is suppressed by quantum fluctuation and this system is at the Luttinger Liquid quantum critical point at T=0. The concept of quantum criticality was introduced to unify the physics of different quantum regions, since it was discovered that quantum critical points in different systems have similar features. One of these is energy/temperature scaling where the behaviour of a system in the vicinity of a quantum critical point depends on the ratio of energy divided by temperature rather than on these two quantities independently. Scaling in the spin-½, Heisenberg antiferromagnet is implied by Shulz’s formula for the magnetic excitations of the system
The correlation function S(q,w) multiplied by temperature (T) at wavevector q=ω depends only on the ratio of energy (ω) to temperature. While the spinon picture has been confirmed experimentally, less in known about the validity of the quantum critical point model. In addition the ideal one-dimensional, spin-½, Heisenberg antiferromagnet is not realised in the real world, inevitable there is always some interchain coupling or dimerisation, so the question arises as to how accurate these models are in describing real materials.
Figure 1: A spin-½, Ising, one-dimensional, antiferromagnet is illustrated on the top line and represents the ground state. The second, third and fourth lines represents excitations where one, three and five consecutive spins are reversed from their ground state direction. Unlike the case of a three-dimensional antiferromagnet, these excitations are all degenerate with energy J and must be combined and diagonalised to find the eigenstates. In an alternative picture we can think of domain walls excitations (surrounded by the black boxes) since this is where energy and spin is gained these are the analogons to the spinons.
Figure 2: The upper panel shows the dispersion of a single spinon as a function of wavevector and energy. However in the neutron scattering process pairs of spinons are created and this gives rise to the multispinon continuum represented by the blue shading in the lower panel. For example the two spinons represented by the solid circles in the upper panel give rise to neutron signal at the circle in the lower panel, and similarly for the square symbols.
Quantum Criticality in a Quasi-One-dimensional, Spin-½, Heisenberg Antiferromagnet
Nature Materials 4 329-334 (2005)
The compound KCuF3 is a quasi-one-dimensional, Heisenberg insulating antiferromagnet. The copper ions possess spin-½ moments and the dominant antiferromagnetic exchange interactions couple them into chains along the crystallographic c direction. However there are weaker ferromagnetic exchange interaction in the a and b direction that act to couple these chains together. Although the ratio of interchain to intrachain coupling is only ~3%, the interchain coupling is sufficient to give rise to long-range antiferromagnetic order below a Néel temperature of 39K. Nevertheless long-range antiferromagnetic order is achieved only 50% of each spin moment points in the ordering direction even at the lowest temperatures, the remaining 50% continues to fluctuate. This system then clearly has some aspects of both one-dimensional and three-dimensional spin-½ magnetism, and the question arises as to what extent it can be said to be a one-dimensional system.
The full excitation spectrum of KCuF3 was collected as a function of wavevector and energy for a range of temperature using neutron scattering and the results are shown in figure 3. The data shows the presence of continuum-like excitations (contained between the superimposed dashed lines) similar to the predictions for a one-dimensional spin-½, Heisenberg antiferromagnet. This data was used to test for one-dimensionality by testing for quantum criticality as implied by universal energy/temperature scaling. From this is was possible to conclude that KCuF3 demonstrates energy/temperature scaling over a large range of energies and temperatures implying one-dimensional behaviour in spite of the interchain coupling present in this material. Other regimes of behaviour were also deduced and a magnetic crossover diagram was constructed.
Figure 4 shows the magnetic crossover diagram. One-dimensional behaviour as characterised by Luttinger Liquid quantum critical scaling dominates the behaviour over a large range of energies and temperatures (blue region). However at low temperatures (below the Néel temperature) and low energies, spin-wave type excitations are found, characteristic of a three-dimensional magnet (red region) and these phases are separated by another region where the crossover from one-dimensional to three-dimensional behaviour occurs.
The energy ranges of the various phases are also indicated in figure 5, superimposed on the neutron scattering spectrum at T=6K.
Figure 3: Inelastic neutron scattering data for KCuF3. The data is plotted as a function of E and q parallel to the chains for the temperatures a, T = 6 K, b, T = 50 K, c, T = 150 K and d, T = 300 K. The colours indicate the size of the neutron scattering cross-section S(q,E) and the superimposed black dashed lines indicate the region where the multi-spinon continuum is predicted at T = 0 K. The data was collected using the MAPS time-of-flight spectrometer at ISIS, Rutherford Appleton Laboratory, UK.
Figure 4: Magnetic crossover diagram showing the different physical regimes of KCuF3 as a function of energy and temperature. The regimes are labelled on the diagram and the dashed lines mark the places where one phase crosses over into another.
Figure 5: The data collected at T=6K with the energy ranges of the magnetic regimes indicted by the horizontal white lines.
Dimensional Crossover and the Longitudinal Mode in KCuF3
Physical Review Letters 85 832 (2000) and Phys. Rev. B 71 134412 (2005). A detailed investigation of the crossover region revealed some new and unexpected phenomena. Data collected at T=6K, q~π and energies between 8 and 23meV is shown in figure 6a. The red V-shaped feature represents the spin-waves found at low energies and temperatures which are indicative of three-dimensional magnetism. An additional feature arises at E=16meV between the spin-wave branches, this is the first measurement of the longitudinal mode which is a characteristic of the crossover phase.
The longitudinal mode is in a quantum feature and arises in a system where there is long-range order co-existing with strong quantum fluctuations. The feature exists below the Néel temperature and is gapped with a gap size related to the size of the interchain coupling. The longitudinal mode is not found in ordinary three-dimensional magnets because its gap size becomes larger and its intensity weaker with increasing three-dimensionality. In theory the longitudinal mode in KCuF3 is predicted at 17meV and in the experiment it is observed at 16meV, the mode is also predicted to be sharp whereas in the experiment it has a width of 5meV. Figure 6b shows a simulation of theoretically predicted longitudinal mode in KCuF3, the mode is given an energy gap of 16meV and a width of 5meV, all other theoretical aspects of the mode are maintained including the ratio of longitudinal to spin-wave intensities.
Figure. 6: (a) Energy-wave vector contour map of the magnetic
signal collected at 10 K; the colors indicate the relative scattering
intensities. (b) Simulation of the magnetic signal over the same reciprocal
space region using the theoretical dispersions for the transverse and
longitudinal modes convolved with the resolution function.
