Introduction In the previous chapters we showed how series expansion methods can be used to study ground state properties and elementary excitations in simple (and not so simple) antiferromagnets. In the present chapter we extend this to a consideration of more complex properties. We begin by discussing various spin-spin correlation functions, or ‘correlators’. These play an important role in characterizing the nature of the ground state and, as we shall show, can be used to locate quantum phase transition points. The correlators can be combined to form the dynamical twospin structure factors S( k , ω), which are measured in inelastic neutron scattering experiments. We will discuss how these can be computed via series methods, and show some results for the integrated (static) forms. While one-particle excited states are usually the dominant excitations in quantum systems, there are situations in which two-particle excitations, including bound states, and more general multi-particle excitations also play an important role. There are a number of experimental probes which are beginning to show features associated with multi-particle continuum and bound states, including two-magnon Raman spectroscopy, photo-emission and inelastic neutron scattering. For example, Tennant et al . (2003) have measured two-magnon states in copper nitrate, a quasi one-dimensional antiferromagnet (see Section 6.4.4). Thus one can hope to build up a detailed picture of the dynamics of these quasiparticle excitations, and to construct an effective Hamiltonian to describe them. In the last part of the chapter we outline how one may calculate series expansions for the spectrum of multi-particle excitations in a quantum lattice model at T = 0.
Introduction In the previous chapters we showed how series expansion methods can be used to study ground state properties and elementary excitations in simple (and not so simple) antiferromagnets. In the present chapter we extend this to a consideration of more complex properties. We begin by discussing various spin-spin correlation functions, or ‘correlators’. These play an important role in characterizing the nature of the ground state and, as we shall show, can be used to locate quantum phase transition points. The correlators can be combined to form the dynamical twospin structure factors S( k , ω), which are measured in inelastic neutron scattering experiments. We will discuss how these can be computed via series methods, and show some results for the integrated (static) forms. While one-particle excited states are usually the dominant excitations in quantum systems, there are situations in which two-particle excitations, including bound states, and more general multi-particle excitations also play an important role. There are a number of experimental probes which are beginning to show features associated with multi-particle continuum and bound states, including two-magnon Raman spectroscopy, photo-emission and inelastic neutron scattering. For example, Tennant et al . (2003) have measured two-magnon states in copper nitrate, a quasi one-dimensional antiferromagnet (see Section 6.4.4). Thus one can hope to build up a detailed picture of the dynamics of these quasiparticle excitations, and to construct an effective Hamiltonian to describe them. In the last part of the chapter we outline how one may calculate series expansions for the spectrum of multi-particle excitations in a quantum lattice model at T = 0.
Inelastic neutron scattering experiments are commonly used to unveil how excitations on Heisenberg spin models play a role in dynamical correlations functions. For a certain class of materials, like CsCoCl3 or CsCoBr3 salts, it turns out that their magnetic properties are fairly well approximated by quasi-one dimensional XXZ models, which enjoy the property of quantum integrability. In these instances, one can in principle use their underlying algebraic structure to describe very precisely how excitations, the so-called spinons, participate in dynamical correlations functions. Even though the available theories (either algebraic Bethe ansatz or quantum group approach) provide all the needed physical quantities such as form factors, complete set of eigenstates and spectrum, it is typically a rather daunting task, however, to obtain sufficiently simple analytical expressions for computing Dynamical Structure Factors (DSFs), valuable, e.g., for parameters estimation based on experimental data. This is particularly the case for the longitudinal DSF of the XXZ model, which has eluded a formal mathematical treatment thus far. Using the quantum group approach, we present here an exact and simple expression of the 2-spinon longitudinal DSF and show our results to be consistent with the expected sum rules and the isotropic and Ising antiferromagnet limiting cases.
Interactions between elementary excitations in quasi-one-dimensional antiferromagnets are of experimental relevance and their quantitative theoretical treatment has been a theoretical challenge for many years. Using matrix product states, one can explicitly determine the wave functions of the one- and two-particle excitations, and, consequently, the contributions to dynamical correlations. We apply this framework to the (nonintegrable) frustrated dimerized spin-1/2 chain, a model for generic spin-Peierls systems, where low-energy quasiparticle excitations are bound states of topological solitons. The spin structure factor involving two quasiparticle scattering states is obtained in the thermodynamic limit with full momentum and frequency resolution. This allows very subtle features in the two-particle spectral function to be revealed which, we argue, could be seen, e.g., in inelastic neutron scattering of spin-Peierls compounds under a change of the external pressure.
We have developed strong-coupling series expansion methods to study the two-particle spectra in quantum lattice models. The properties of bound states and multiparticle excitations can reveal important information about the dynamics of a given model. At the heart of this method lies the calculation of an effective Hamiltonian in the two-particle subspace. We use an orthogonal transformation to perform this block diagonalising, and find that maintaining orthogonality is crucial for cases where the ground state and the two-particle subspace have identical quantum numbers. The two-particle Schrödinger equation is solved by using a finite lattice approach in coordinate space or an integral equation in momentum space. These methods allow us to determine precisely the low-lying excitation spectra and dispersion relations for the two-particle bound states. The method has been tested for the (1+1) D transverse Ising model, and applied to the two-leg spin-1/2 Heisenberg ladder. We study the coherence lengths of the bound states, and how they merge with the two-particle continuum. Finally, these techniques are applied to the frustrated alternating Heisenberg chain, which has been of considerable recent interest due to its relevance to spin-Peierls systems such as CuGeO 3. Starting from a limit corresponding to weakly-coupled dimers, we develop high-order series expansions for the effective Hamiltonian in the two-particle subspace. In the regime of strong dimerisation, various properties of the singlet and triplet bound states, and the quintet antibound states, can be accurately calculated. We also study the behaviour as the external bond alternation vanishes, and the way in which the bound states of triplet dimer excitations make the transition to a soliton-antisoliton continuum.
Introduction: simple antiferromagnets In the previous chapter we showed how to derive perturbation expansions for quantum spin models at T = 0, for both ground state bulk properties and for excitations. We used, as an example, the transverse field Ising model. This is arguably the simplest model to show a quantum phase transition. Moreover the model has a natural perturbation parameter, Γ/ J or J /Γ, and the series are relatively straightforward to derive and interpret. In the present chapter we turn to quantum antiferromagnets. These are systems in which the exchange interaction favours anti-alignment of neighbouring spins. The physics will turn out to be much richer, and we will use series methods to explore not only simple antiferromagnets on bipartite lattices but also the effects of frustration, competing interactions, destruction of antiferromagnetism due to singlet formation on dimers or plaquettes, and other related topics. With some minor modifications the techniques are the same as in Chapter 4. In this chapter, however, we will provide fewer details and concentrate more on results. The Ising expansion Let us start by considering a spin-½ system on a bipartite lattice, with nearest-neighbour antiferromagnetic exchange. A bipartite lattice is one which consists of two interpenetrating sub-lattices A, B such that the nearest neighbours of A spins all lie on sub-lattice B and vice versa. Examples are the linear chain, the square and honeycomb lattices in two dimensions, the simple cubic and body-centred cubic lattices in three dimensions.
Many phenomena in magnetism can be described using classical models, yet magnetism is a quantum mechanical effect. In addition to thermal fluctuations, quantum fluctuations arise due to the quantum uncertainty of the ground state. These fluctuations are enhanced by low dimensionality and low spin. Spin chains show interesting excitations associated with quasiparticles. Model materials for spin chains can be realized in compounds where the interchain interactions are weak enough. This thesis presents neutron scattering studies of two model materials for spin chains with different spin symmetry. The spin2 Heisenberg antiferromagnetic chain CuSO4*5D2O has a main exchange of J ∼ 3 K, which permits to study its excitation spectrum up to temperatures of ten times the exchange. The spin2 Heisenberg magnet has been studied widely both theoretically and experimentally. The excitation spectrum of the spin2 Heisenberg antiferromagnet consists of quasiparticles called spinons, and calculations for it exist for T = 0 and as a function of temperature. A quantitative comparison between experimental data and theoretical models across a range of temperatures is presented. The excitation spectrum of CuSO4*5D2O is excellently described by the excitation spectrum of a spin2 Heisenberg antiferromagnetic chain at finite temperatures. Energy-temperature scaling is observed at temperatures up to 2J . For these temperatures the Luttinger liquid description of the spin2 Heisenberg antiferromagnetic chain is valid. A spinon continuum is confirmed at all measured temperatures. Therefore, even at ten times the main exchange (T ≈ 10 J), CuSO4*5D2O shows the behavior of a quantum magnet, and not of a classical system at infinite temperatures, for which one would expect single spin flips. The Ising chain is one of the most studied models in magnetism, one model compound for it is RbCoCl3. Optical spectroscopy measurements suggest a lower gap than for similar compounds, which makes it better suited for high resolution neutron spectroscopy. Magnetic correlations along the chains arise in RbCoCl3 below 80 K. A gapped continuum is observed, which arises from isolated antiferromagnetic Ising chains. Below the first magnetic ordering temperature TN1 ≈ 28 K bound states are observed, which arise due to correlations which form between the chains. Strong diffuse scattering leads to the conclusion that disorder is still present. At the second magnetic ordering transition, at TN2 ≈ 14 K, the bound states split as the correlations between the chains change. Fits indicate that the correlations change continuously as a function of temperature. Above TN2 the excitation spectrum of RbCoCl3 can be described by a spin1 2 Ising chain with an XY-component which is treated as a perturbation. Below TN2 the main features are reproduced correctly, but deviations from the details indicate that additional terms in the Hamiltonian may be necessary. This thesis was carried out at the Institut Laue-Langevin, the Laboratory for Neutron Scattering and Imaging at the Paul Scherrer Institute and ETH Zurich.
The research presented here focuses on the theoretical investigation of spinons in quasi one-dimensional quantum spin chains from the perspective of neutron scattering, with emphasis placed on the calculation of dynamic spin structure factors for both unpolarised and polarised incident neutrons. Two anisotropic versions of the Heisenberg spin chain are considered for such an examination: the spin-1/2 antiferromagnetic XXZ model and the spin-1/2 ferromagnetic XYZ model.The XYZ model supports spinon scattering (modelled at finite temperature) and is related to the so-called Villain mode. The introduction of a perturbation in the form of an external transverse magnetic field or Dzyaloshinskii-Moriya interaction (DMI) leads to the emergence of incommensurability in the system; such a result is a signature of fractional excitations. The presence of these interactions also give rise to a chiral response associated with the spinons in the polarised term of the scattering cross section. The inelastic scattering cross section is computed for both unpolarised and polarised neutrons.In relation to the antiferromagnetic XXZ model, emphasis is on spinon pair creation (modelled at T=0rmK). Motivated by the presence of a chiral response in the case of the XYZ model mentioned above, a new technique is developed in order to incorporate DMI and the transverse magnetic field so that one can determine if there is a chiral property associated with the two-spinon system. This technique is numerical in nature and is based on Green function methods. Using the results produced by the Green function approach, the inelastic cross section in the presence of an external transverse magnetic field is computed for the first time. The result is compared with experimental neutron scattering data; good agreement is demonstrated between theory and experiments on rmCsCoBr3.Finally, the Green function approach is generalised to account for the interactions between spin chains in magnetic compounds described by a staggered field. Various dynamic structure factors are computed for this scenario with results compared to experiment; good agreement between the theory and experiments on rmRbCoCl3 is shown.