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.
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.
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.
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 nearest-neighbor quantum antiferromagnetic (AF) Heisenberg model for spin-
1∕2 on a two-dimensional square lattice is studied in the auxiliary-fermion representation. Expressing spin operators by canonical fermionic particles requires a constraint on the fermion charge
Qi=1 on each lattice site
i, which is imposed approximately through the thermal average. The resulting interacting fermion system is first treated in mean-field theory (MFT), which yields an AF ordered ground state and spin waves in quantitative agreement with conventional spin-wave theory. At finite temperature a self-consistent approximation beyond mean field is required in order to fulfill the Mermin-Wagner theorem. We first discuss a fully self-consistent approximation, where fermions are renormalized due to fluctuations of their spin density, in close analogy to FLEX. While static properties like the correlation length, $\ensuremath{\xi}(T)\ensuremath{\propto}\mathrm{exp}(aJ∕T)$, come out correctly, the dynamical response lacks the magnon-like peaks which would reflect the appearance of short-range order at low
T. This drawback, which is caused by overdamping, is overcome in a ``minimal self-consistent approximation'' (MSCA), which we derive from the equations of motion. The MSCA features dynamical scaling at small energy and temperature and is qualitatively correct both in the regime of order-parameter relaxation at long wavelengths $\ensuremath{\lambda}>\ensuremath{\xi}$ and in the short-range-order regime at $\ensuremath{\lambda}<\ensuremath{\xi}$. We also discuss the impact of vertex corrections and the problem of pseudo-gap formation in the single-particle density of states due to long-range fluctuations. Finally we show that the (short-range) magnetic order in MFT and MSCA helps to fulfill the constraint on the local fermion occupancy.
Despite nearly a century of study of the
S=1/2 Heisenberg model on the square lattice, there is still disagreement on the nature of its high-energy excitations. By tuning toward the Heisenberg model from the exactly soluble Ising limit, we find that the strongly attractive magnon interactions of the latter naturally account for a number of spectral features of the Heisenberg model. This claim is backed up both numerically and analytically. Using the density matrix renormalization group method, we obtain the dynamical structure factor for a cylindrical geometry, allowing us to continuously connect both limits. Remarkably, a semiquantitative description of certain observed features arises already at the lowest nontrivial order in perturbation theory around the Ising limit. Moreover, our analysis uncovers that high-energy magnons are localized on a single sublattice, which is related to the entanglement properties of the ground state.