By extending our recently proposed magnon-density waves to low dimensions, we investigate, using a microscopic many-body approach, the longitudinal excitations of the quasi-one-dimensional (quasi-1d) and quasi-2d Heisenberg antiferromagnetic systems on a bipartite lattice with a general spin quantum number. We obtain the full energy spectrum of the longitudinal mode as a function of the coupling constants in the original lattice Hamiltonian and find that it always has a nonzero energy gap if the ground state has a long-range order and becomes gapless for the pure isotropic 1d model. The numerical value of the minimum gap in our approximation agrees with that of a longitudinal mode observed in the quasi-1d antiferromagnetic compound KCuF3 at low temperature. It will be interesting to compare values of the energy spectrum at other momenta if their experimental results are available.
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.
We investigate 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 noncollinear 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 compare our results for the longitudinal energy gaps at the magnetic wave vectors with the experimental results for several antiferromagnetic compounds with both integer and noninteger spin quantum numbers, and we find good agreement after the higher-order contributions are included in our calculations.
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.
This thesis investigates two-dimensional quantum antiferromagnets, with a focus on the spin-
21 antiferromagnetic easy-axis XXZ model on a square and a honeycomb lattice. Describing the excitations and interactions of magnons in these models is of substantial interest, as magnons are the fundamental quasiparticles and can thus explain experimental observations. Already for the paradigmatic antiferromagnetic spin-
21 Heisenberg model on the square lattice, high-energy features of the magnon dispersion are understood via strong magnon-magnon interactions. Besides that, interesting high-energy features were recently observed in the honeycomb lattice, including a possible decay of the single-magnon mode. The goal of this thesis is to obtain effective descriptions of the aforementioned systems that capture the relevant features of the magnon excitations and their interactions. To study these features, this thesis employs continuous similarity transformations (CSTs), which enables a systematic derivation of an effective quasiparticle picture via so-called flow equations. The approach enables decoupling different magnon sectors and thus for investigating ground-state properties, single-magnon excitations, and multi-magnon bound states. The magnon description is derived from the non-Hermitian Dyson-Maleev representation, which describes fluctuations around a long-range magnetically ordered ground state. The flow equations are set up in momentum space, truncated by the scaling dimension, retaining only operators up to scaling dimension of two, and solved numerically. For the square lattice, a magnon-conserving effective Hamiltonian is obtained across the full anisotropy range by interpolating between the Ising limit and the Heisenberg limit. The ground-state energy, staggered magnetization, dispersion, and critical exponents are in excellent agreement with the literature, supporting the validity of CST also for gapped phases. Two-magnon excitations in the
Sz=0 subspace comprise four bound states, which are tracked across the anisotropy range. Their successive decay into the two-magnon continuum is determined by using the inverse participation ratio. For the honeycomb lattice, the magnon-conserving CST is reliable only up to
λ≲0.57; an adapted CST scheme that decouples only the ground state while preserving couplings between different higher-magnon sectors yields effective descriptions for all
λ and confirms stable long-range order. The low-energy properties determined in this way match the literature values. However, in the Heisenberg limit, the adapted approach fails to capture high-energy features, such as the experimentally observed decay into the multi-particle continuum at corners of the Brillouin zone. A termination of the magnon-conserving flow equations at a finite flow parameter before the divergence reveals a single-magnon decay at high energies, which is connected to a crossing of a two-magnon bound state with the single-magnon mode. The observed energy dip qualitatively agrees with Quantum Monte Carlo results, but magnon–magnon interactions are overestimated, hindering a fully quantitative characterization of the single-magnon dispersion. Analyzing multi-magnon bound states over the anisotropy range unveils that the crossing of a three-magnon bound state with the single-magnon mode accounts for the breakdown of the magnon-conserving CST at higher
λ. This work demonstrates that CST with scaling-dimension truncation provides a powerful framework for studying two-dimensional quantum magnets, capturing low-energy properties and bound states. Nevertheless, challenges remain in accurately describing high-energy features arising from a persistent overlap of magnon sectors due to strong binding effects.
Investigation of dynamical excitations is difficult but crucial to the understanding of many exotic quantum phenomena discovered in quantum materials. This is particularly true for highly frustrated quantum antiferromagnets whose dynamical properties deviate strongly from theoretical predictions made based on the spin-wave or other approximations. Here, we present a large-scale numerical calculation on the dynamical correlation functions of spin-1/2 triangular Heisenberg model using a state-of-the-art tensor network renormalization group method. The calculated results allow us to gain for the first time a comprehensive picture on the nature of spin excitation spectra in this highly frustrated quantum system. It provides a quantitative account for all the key features of the dynamical spectra disclosed by inelastic neutron scattering measurements for Ba_{3}CoSb_{2}O_{9}, revealing the importance of the interplay between low- and high-energy excitations and its renormalization effect to the low-energy magnon bands and high-energy continuums. We identify the longitudinal Higgs modes in the intermediate-energy scale and predict the energy and momentum dependence of spectral functions along the three principal axes that can be verified by polarized neutron scattering experiments. Furthermore, we find that the spin excitation spectra weakly depend on the anisotropic ratio of the antiferromagnetic interaction.