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.
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 ground-state and low-lying excited states of the S = 1 antiferromagnetic Heisenberg chain with exchange anisotropy, single-ion anisotropy and biquadratic exchange and with free boundary conditions. We present an approximate mapping of the S = 1 model to an effective S = 1/2 chain in an external magnetic field, which can be solved exactly in several limiting cases. We obtain results for the phase diagram, correlation functions and the excitation spectrum. The transition to the Haldane phase is related to the condensation of the lowest soliton modes and the hidden order parameter of the Haldane phase is identified as transverse long-range order in the effective S = 1/2 chain.
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.
In this dissertation we present the results of a theoretical investigation of spin models on two-dimensional and quasi one-dimensional lattices, all unified under the concept of quantum frustrated antiferromagnetism, and all discussing various aspects of the antiferromagnetic Heisenberg model on the kagomé lattice. In the Introduction (Chapter 1), we discuss at some length such concepts as frustration and superexchange, among others, which are of common relevance in the rest of the chapters. In Chapter 2, we study the effect of Dzyaloshinskii--Moriya (DM) interactions on the zero-temperature magnetic susceptibility of systems whose low energy can be described by short-range valence bond states. Our work shows that this treatment is consistent with the experimentally observed non-vanishing susceptibility -- in the specified temperature limit -- of the spin-1/2 kagomé antiferromagnetic compound ZnCu3(OH)6Cl2, also known as herbertsmithite. Although the objective of this work is explaining the aforementioned characteristic of the experimental system, our methods are more general and we apply them to the checkerboard and Shastry-Sutherland lattices as well. In Chapter 3, we discuss our findings in the study of ghost-mediated domain wall interactions in the diamondback ladder. These domain walls are the the spin excitations -- the kinks and the antikinks -- separating the ground states along one chain of the ladder. While as individual entities an antikink is energy costly and a kink energy free, our study finds that both interact via the ghosts that they produce in the opposite side of the ladder from where they are located. Through the study of these ghosts, we find that domain walls proliferate in the system above a critical value of the system's coupling constants. It is this proliferation that makes their treatment as free, non-interacting particles impossible, so we study here their interactions both quantitatively and qualitatively, in a region where the latter are yet not very strong, namely below the critical point. Based on the calculated two-body interaction potential, domain walls interact attractively (repulsively) when separated at even (odd) distances, with a strength that decays as 1/sp, where s is their separation and p<1. We also consider higher-order interactions. In the last chapter, Chapter 4, we present our study of the spin-1 kagomé Heisenberg antiferromagnet. Our approach is to first consider an SU(2)-symmetric parent Hamiltonian with known ground states on the S=1 kagomé lattice, in which nearest-neighbor Heisenberg interactions are already present. We then enhance these interactions by an additional Heisenberg term added perturbatively in order to move the system closer to a pure Heisenberg antiferromagnet. The results of this enhancement is obtaining a description of the system in terms of an effective Hamiltonian, namely a transverse field Ising AF on the triangular lattice. Based on the particular values of this effective Hamiltonian, our system is found to be in the order-by-disorder phase.
In this thesis we study spin systems in low dimension at zero temperature, analyzing their quantum phase transitions. In particular, we used a variational method and a linear spin wave theory for obtaining the approximate ground state. Firstly, we studied the order-disorder transitions of the quasi-one-dimensional isotropic antiferromagneticHeisenberg model of spin-1=2, dened on a square lattice with exchange interactions between nearest neighbors, J and J0. The latter interaction is responsible for onedimensional or two-dimensional behavior of the model namely, when it is zero and when it has a nonzero value, respectively. We obtained the staggered magnetizations of the antiferromagnetic and collinear antiferromagnetic phases. In the absence of exchange interactions between second neighbors, the collinear antiferromagnetic phase is obtained for J0 < 0. This model in the one-dimensional limit has a critical value for the ratio J0=J, where we have a phase transition of the order-disorder type. We founda critical value J0=J = 0, which corresponds to the value physically expected, and compare our results with the linear spin wave theory, that predicts a nonzero critical value. The variational method is also used in the study of the Heisenberg anisotropic model with frustration induced by the inclusion of exchange interactions between second neighbors J2. We renamed the exchange interactions of the rst neighbors J1 e J0 1 , so this model is known as J1 J0 1 J2. Furthermore, we include in this modelthe exchange anisotropy which follows the x and y components of the spin operators. Thus, the more general Hamiltonian presents the isotropic Heisenberg and Ising model limits for certain values of the anisotropy. With the goal to analyze the real role played by this anisotropy, we distinguish between rst and seconds neighbors by the labels 1and 2, respectively. We analyzed the staggered magnetization and the energies for the antiferromagnetic, collinear antiferromagnetic and quantum paramagnetic phase. From these analyzes, we observed quantum phase transitions for certain values of the exchange anisotropy and frustration parameter, the latter one given by the ratio J2=J1. The global phase diagram was also obtained for the cases where the rst and second neighbor exchange anisotropies are equal and where they dier from each other. In the rst case, we can observe the ordered phases, Néel and collinear antiferromagnetic, and between them a magnetically disordered phase. We checked the existence of a critical endpoint separating the three phases. In the second case, we note that the exchange anisotropy inuences the region of the disordered phase, aecting even the existence of the critical endpoint. Besides presenting, at certain intervals of the anisotropy, a behavior that was not expected a priori. We also analyze the quasi-one-dimensional behavior J01 ! 0, where the shape of the phase diagram is unchanged. We explore the degree of dimer formation via the dimerization parameter for some values of exchange anisotropy and obtain a higher degree of dimerization for the disordered phase in comparison with the ordered phases, as expected. We end this work with a generalization of the rst study without frustration in function only of the exchange anisotropy. As a challenge to the variational method, we study the more general J1 J0 1 J2 modelfor integer spin-1 without the exchange anisotropy and avoiding the unidimensional case. Dierent from what happens in the semi-integer spins case, we did not nd the disordered phase for integer spin S= 1. We note only a rst-order phase transition between the ordered phases. Finally, we return to the frustrated model with exchange anisotropies 1 and 2 to study the model from a perspective of linear spin wavetheory, in order to compare the results with those obtained via the variational method. A detailed analysis was performed based on the comparison between the two methods, which agreed qualitatively in all explored boundaries and even those behaviors that were not expected from the variational method were conrmed by the spin wave theory.This consistency of the results allowed us to get a dierent view of the role played by the exchange anisotropy when dierentiated for rst and second neighbors. If the anisotropies are considered equal they work only as a parameter that carries the model from the isotopic Heisenberg limit to the Ising model. However, when dierent, they promote disorder in the system.
The ground-state phase diagram of S=2 antiferromagnetic Heisenberg chains with coexisting uniform and alternating single-site anisotropies is investigated by the numerical exact diagonalization and density matrix renormalization group methods. We find the Haldane, large-D, Néel, period-doubled Néel, gapless spin fluid, quantized and partial ferrimagnetic phases. The Haldane phase is limited to the close neighborhood of the isotropic point. Within numerical accuracy, the transition from the gapless spin-fluid phase to the period-doubled Néel phase is a direct transition. Nevertheless, the presence of a narrow spin-gap phase between these two phases is suggested on the basis of the low-energy effective theory. The ferrimagnetic ground state is present in a wide parameter range. This suggests the realization of magnetized single-chain magnets with a uniform spin magnitude by controlling the environment of each magnetic ion without introducing ferromagnetic interactions.
This thesis investigates exotic phases of matter in magnetic materials that present anisotropic properties in two dimensions. Using the microscopic spin Hamiltonians, the study addresses two problems: (i) the appearance of a novel class of magnetic excitations in the spectra of anisotropic magnets at zero temperature, and (ii) the quantum-to-classical correspondence of spins for the description of magnetic solids in the disordered phase above the critical temperature. Different analytical and numerical means are used to this purpose.In the first part, motivated by experimental observations of multipolar excitations in afamily of strongly-anisotropic van der Waals antiferromagnets, we have developed ageneral theory for these excitations as a distinct type of quasi-particles. We called themlongitudinal magnons. They appear in easy-axis magnetic materials, and are characterized by the quantum number Sz = ±2S. They can exist as coherent long-lived excitations for D ≥ |J|, where D and J are the typical single-ion and exchange coupling constants, respectively. Starting with the strong-coupling limit, this work presented different approaches to compute the dispersion of longitudinal magnons. First, a mapping to an effective spin-1/2 model was performed and illustrated for several lattice spin models: the honeycomb-lattice zigzag antiferromagnet and the square-lattice ferro- and antiferromagnet. The excitation energy was derived for different spin values S = 1, 3/2, 2and 5/2. Second, a thorough study explored the S = 1 anisotropic square-lattice, wherethe analytical results of the effective model are compared to the multi-boson spin-wavetheory, the linked-cluster calculations in antiferromagnetic coupling J > 0, and the exactsolution of the two-magnon problem in ferromagnetic coupling J < 0. In the limit of largeJ/D, the longitudinal magnons start to overlap with the multi-magnon continuum, andtheir finite lifetime can be computed using the multi-boson theory, which is specificallyshown for two-magnon decay processes in S = 1. Moreover, critical values of J/D forthe stability of longitudinal magnons are obtained along high-symmetry lines of the Brillouin zone. Third, we investigated the appearance and condensation of the longitudinal magnons in triangular S = 1 antiferromagnets under high magnetic field. As suggested by recent experiments on a candidate material, a spin nematic phase may appear upon condensation of the multipolar excitations, opening the way for realizations of new phases of matter in classically-ordered magnets.In the second part, the correspondence between quantum and classical spins is investigated for spin Hamiltonians at finite temperature. Given a model of quantum spins, it is useful to determine the classical model that best describes its thermodynamics, which is of interest for semi-classical analyses, e.g. by classical Monte Carlo methods. Based on high-temperature series expansions, we established the scaling laws of the different microscopic constants in a quantum spin model, involving quadratic and biquadratic exchange interactions, single-ion anisotropy and the Zeeman energy. We showed that quantum corrections to the thermodynamic quantities are power series in S(S + 1). Subsequently, the findings were applied to numerically determine the phase transition temperature in several ferro- and antiferromagnetic compounds, using classical Monte Carlo simulations. To this aim, the lattice spin models were constructed with microscopic coupling constants from the literature, and the transition temperatures were compared to the experimental values.This illuminates complex collective magnetic behaviors in different models and materials.