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.
In this thesis the elementary excitations of the spin-1=2 Heisenberg antiferromagnet on the square lattice are studied. This fundamental model is a paradigmatic example for long range ordered quantum phases and the spontaneous breaking of a continuous symmetry in condensed matter. The elementary excitations at long wavelengths are well understood by quantized spin waves, i.e., magnons, which are the gapless Goldstone bosons of the broken SU(2) symmetry. However, recent findings reveal an anomalous energy dip at short wavelengths, which cannot be explained by the conventional spin wave theory. The nature of spin waves at short wavelengths remains unclear to this day. A central aim of this thesis is the derivation of an effective magnon description for the square lattice antiferromagnet which is quantitatively valid for all length scales. For this purpose, the method of continuous unitary transformations (CUT) is extended to its use for gapless excitations. CUT methods provide a powerful tool for the derivation of effective quasi-particle descriptions by means of flow equations. However, typical approaches are truncated in real space restricting their scope of application to gapped quantum phases. In order to circumvent these limitations a CUT approach in momentum space is developed. The scaling dimension of operator terms is established as a suitable truncation criterion for gapless quasi-particles. Moreover, a diagrammatic representation of operator terms is introduced which simplifies the derivation and interpretation of flow equations in momentum space. The developed approach is used to transform the Heisenberg Hamiltonian into an effective Hamiltonian which conserves the number of magnons. In addition, effective observables required for the theoretical description of inelastic neutron scattering are derived. The initial magnon operators before the transformation are defined by the Dyson-Maleev representation which leads to a superficially non-hermitian Hamiltonian. As a result, the transformation constitutes a continuous similarity transformation (CST). This formalism enables the formulation of spin wave interactions by means of quartic boson operators. A crucial step of the transformation is the proper renormalization of the spin wave interaction which is achieved by a selfsimilar CST for scaling dimension d = 2. The resulting effective magnon description is consistent at all length scales. In particular, the magnon dispersion exhibits an energy dip at short wavelengths which is in quantitative agreement with recent numerical and experimental data. Further analysis of the spectral properties indicates a strong attraction between spin waves giving rise to a resonance in the longitudinal two-magnon channel. This resonance can be interpreted as the amplitude or Higgs mode of the continuously broken SU(2) symmetry with finite lifetime. The origin of the anomalous dispersion is ascribed to a scattering between the Higgs resonance and single magnon states. The theoretical results are directly compared with experimental data obtained by inelastic neutron scattering and various numerical results which support this view. The quantitative agreement between the effective spin wave approach and numerous numerical and experimental findings shows that magnons are a valid description of the elementary excitations at all length scales. Therefore, this thesis provides a final settlement of the question about the nature of the elementary excitations in the square lattice quantum antiferromagnet.
We apply continuous similarity transformations (CSTs) to the easy-axis antiferromagnetic XXZ model on the square lattice. The CST flow equations are truncated in momentum space by the scaling dimension
d so that all contributions with $d\ensuremath{\le}2$ are taken into account. The resulting quartic magnon-conserving effective Hamiltonian is analyzed in the zero-, one-, and two-magnon sector. In this way, a quantitative description of the ground-state energy, the one-magnon dispersion, and its gap as well as of two-magnon bound states is gained for anisotropies ranging from the gapped Ising model to the gapless Heisenberg model. We discuss the critical properties of the gap closing as well as the evolution of the one-magnon roton mininum. The excitation energies of two-magnon bound states are calculated, and their decay into the two-magnon continuum is determined via the inverse participation ratio.
A quantitative description of magnons in long-range ordered quantum antiferromagnets is presented which is consistent from low to high energies. It is illustrated for the generic S=1/2 Heisenberg model on the square lattice. The approach is based on a continuous similarity transformation in momentum space using the scaling dimension as the truncation criterion. Evidence is found for significant magnon-magnon attraction inducing a Higgs resonance. The high-energy roton minimum in the magnon dispersion appears to be induced by strong magnon-Higgs scattering.
A key feature of the newly discovered altermagnet is that its spin degeneracy is lifted, although it has an antiferromagnetic order and zero net magnetization. In this work, we investigate a frustrated spin-1/2
J1-
J2-
δ Heisenberg model on the square lattice by the tensor network methodin combination with the linear spin-wave theory, with our focus on both the magnon excitations and longitudinal excitations.For a small
J2 and a finite range of
δ we demonstrate that such a model hosts an altermagnetic ground state. Its magnon spectrum is split into two branches and the largest splitting occurs at
(±π/2,±π/2) in the Brillouin zone. The magnitudes of splitting in the two magnon modes are equal with respect to the case of
δ=0. Dynamical spin structure factors show that the low-energy peak in the longitudinal spectral weight around
(π/2,π/2) is also split, and thus the relative positions of the magnon modes and longitudinal modes in energy may change in the presence of a finite
δ. These findings demonstrate that the altermagnets harbor more complex quantum dynamics than the conventional collinear antiferromagnets.
Altermagnets, a newly classified type of collinear magnet, exhibit zero magnetization, akin to antiferromagnets, but have spin-split electronic bands, akin to ferromagnets. Their low-energy excitations are predicted to be magnons, but with linear energy-momentum dispersion, typical of antiferromagnets, and additional chiral splitting analogous to those of electronic bands. However, such predictions are made using linear spin wave theory, which considers infinitely long-lived noninteracting magnons. This study employs nonperturbative quantum many-body calculations, via tensor network algorithms, to find how a chiral magnon will acquire finite lifetime and how the bands will shift due to magnon-magnon interaction. Such corrections to noninteracting magnon bands could be detected by Raman scattering experiments, for which the recipe is provided. Although the same interaction effects are often described in magnetism by nonlinearity of the Landau-Lifshitz-Gilbert equation, direct comparison using altermagnetic magnons as the testbed, shows how such classical methods cannot fully reproduce the results of quantum many-body calculations.