The idea of the spectrum generating algebra (SGA) is used to study the low-lying excitations in various localised-spin models of ferromagnetic, ferrimagnetic and antiferromagnetic solids. The noncompact algebra SO(2,1) approximately SU(1,1) is found to generate the spin wave or the magnon spectrum for ferromagnetic, ferrimagnetic and antiferromagnetic Bravais lattices with nearest-neighbour exchange interactions (Heisenberg model). The ferromagnetic case is treated including dipolar interactions in the model. For a quadratic antiferromagnet with easy-plane anisotropy the SGA is found to be the direct product SO(2,1)(X)SO(2,1) approximately SO(2,2). For the XY and the Heisenberg-Ising models of an antiferromagnetic linear chain the SGA is the compact counterpart of SO(2,1), i.e. SO(3) approximately SU(2). It is concluded that if the magnons are quantised as bosons then the SGA is the noncompact algebra SO(2,1), and if they are quantised as fermions then the SGA is the compact angular momentum algebra SO(3).
In the present work low energy excitations in various antiferromagnetic systems are investigated.The first part is devoted to the antiferromagnetic Heisenberg chain with two different alternating magnetic ions having spins S A and S B .The system can be characterised as a ferrimagnet with a long range ordered ground state.Linear spin wave theory (LSWT) leads to a qualitatively correct description of two excitation branches.The low energy spectrum contains ferromagnetic magnons which are discussed in the introductory chapter.The LSWT results are compared with the numerical data from exact diagonalisation for the case S A =1 and S B =1/2.In the second part of the thesis, the quantum phase transitions from the Néel phase to the Heisenberg phase and from the Néel phase to the Haldane phase for half-odd-integer spins S =1 /2, 3/2 and integer spins S =1 , 2, respectively, are studied in context of the Haldane conjecture.The models are analysed by means of high-order series expansions for the ground state properties and excitation energies.The expansion parameter is the anisotropy λ where the unperturbed case λ =0corresponds to the Ising model.For the integer-S systems the following scenario is obtained: The phase transitions from the Néel phase to the Haldane phase are determined by the condensation of the so-called spin zero defects (SZD) at finite anisotropy.These domain wall type excitations can be characterised as S z =0states embedded in the antiferromagnetic order.The critical exponents β of the order parameter and η of the correlation function are those of the two-dimensional Ising model.In contrast to the above scenario, the half-odd-integer S =1/2 chain undergoes the Kosterlitz-Thouless phase transition at the isotropic point.In the case of the S =3/2 chain the critical behaviour of the static structure factor S (zz) (π) is consistent with the exponentially divergent correlation length at the isotropic point.Finally, kink excitations of the S =1 /2 Heisenberg antiferromagnet chain with nearest and next nearest neighbour interactions (the zigzag chain) are investigated.Here a kink is defined as a free spin S =1 /2 mediating two dimerised ground states.The phase diagram of the zigzag chain contains the Majumdar-Ghosh model as well as the S =1/2 spin ladder and the S =1Heisenberg antiferromagnet.Starting from the Majumdar-Ghosh model, a two-kink variational wave function with S tot =1is constructed.The variational parameter is the characteristic distance between the two kinks.The variational wave function describes the low energy triplet excitation of the gapped zigzag chain in a qualitatively correct way.Within this approach, the elementary excitation of the spin ladder as well as the Haldane triplet of the S =1chain appears as a bound state of two kinks.The phase diagram of the zigzag chain contains the incommensurate phase where the wave vector of the lowest mode k 0 runs from k 0 =0to k 0 = π.At the qualitative level, this behaviour is correctly described by the variational approach.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone
Γ(q=0) and at the magnetic ordering wavevectors
±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.
The anisotropic
XXZ spin-
21 Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated throughout the entire temperature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Ising-like ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction
sixsjx+siysjy induces an antiferromagnetic correlation in the out-of-plane
siz component, at higher temperatures in the antiferromagnetic
XXZ chain, dominantly at low temperatures in the ferromagnetic
XXZ chain, and, in-between, at all temperatures in the
XY chain. We find that the converse effect also occurs in the antiferromagnetic
XXZ chain: an antiferromagnetic
sizsjz interaction induces a correlation in the
sixy component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or
XY) of anisotropy; and (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy.
The spectra of both two- and three-magnon excitations in Heisenberg spin chains with next-nearest-neighbour (NNN) interactions are studied using scaling methods and the recursion method respectively. Both two-spin and three-spin couplings are considered for general spin S. In the three-magnon case, the asymptotic behaviour of the recursion coefficients can be used to directly identify the presence of bound states in the spectrum. Both integrable and non-integrable models can be studied, and the integrable models display special features in the bound-state spectra. Our results for the three-magnon bound states of S = 1 chains differ appreciably from those obtained in previous studies based upon an integral equation approach.
We give a full description of a recently developed efficient Monte Carlo Approach to low-lying excitations of one-dimensional quantum spin systems. The idea is in a word expressed as extracting the lower edge of the excitation spectrum from imaginary-time quantum Monte Carlo data at a sufficiently low temperature. First, the method is applied to the antiferromagnetic Heisenberg chains of S=1/2, 1, 3/2, and 2. In the cases of S=1/2 and S=1, comparing the present results with the previous findings, we discuss the reliability of the method. The spectra for S=3/2 and S=2 turn out to be massless and massive, respectively. In order to demonstrate that our method is very good at treating long chains, we calculate the S=2 chain with length up to 512 spins and give a precise estimate of the Haldane gap. Second, we show its fruitful use in studying quantum critical phenomena of bond-alternating spin chains. Using the conformal invariance of the system as well, we calculate the central charge of the critical S=1 chain, which results in the Gaussian universality class. Third, we study an alternating-spin system composed of two kinds of spins S=1 and 1/2, which shows the ferrimagnetic behavior. We find a quadratic dispersion relation in the small-momentum region. The numerical findings are qualitatively explained well in terms of the spin-wave theory. Finally, we argue a possibility of applying the method to the higher excitations, where we again deal with the S=1 Heisenberg antiferromagnet and inquire further into its unique low-energy structure. All the applications demonstrate the wide applicability of the method and its own advantages.
The natural explanation of the excitation spectrum of the spin-1 antiferromagnetic Heisenberg chain is given from the viewpoint of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain. The energy spectrum of the latter is calculated with fixed momentum
k by numerical diagonalization of finite size systems. It consists of a branch of propagating triplet pair (triplet wave) and the continuum of multiple triplet waves for weak ferromagnetic coupling. As the ferromagnetic coupling increases, the triplet wave branch is absorbed in the continuum for small
k, reproducing the characteristics of the spin-1 antiferromagnetic Heisenberg chain.