Dynamical properties of frustrated XXZ models in the Ising limit on spatially anisotropic two-dimensional lattices are investigated by a weak-coupling approach from the one-dimensional limit. Restricting the Hilbert space to that spanned by exact eigenstates of XXZ chains in the Ising limit, we calculate dynamical structure factor 5 -+ (k,ω) in spatially anisotropic two-dimensional systems. As in the case of frustrated Heisenberg models [M.Kohno, et al. , Nat. Phys. 3, 790 (2007)], spinons form bound states through hopping process between chains. We calculate the density-density correlation function of spinons in the Ising limit, and show that spinons in the bound state hop together as a pair, although those in a continuous spectrum behave almost independently. We discuss possible implications of the present results on fractionalization of bosonic and fermionic particles in spatially anisotropic frustrated systems with strong repulsion.
Ground-state properties of antiferromagnetic quantum XXZ models on triangular and square lattices are investigated within a framework of conventional perturbative spin-wave theory using the Dyson-Maleev transformation. To lowest order in the perturbation expansion, the ground-state energy, where dynamical interactions are taken into account, the sublattice magnetization, spin-spin correlation functions, and susceptibilities are calculated. Higher-order calculations are made for the correlation functions and for the sublattice magnetization. For the triangular Heisenberg antiferromagnet, divergences in the perturbation expansion of the sublattice magnetization, which means the conventional spin-wave theory based on the sublattice order is not applicable to that system, are shown to exist. Explicit asymptotic behaviors of the correlation functions are also given.
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.
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 investigate the dynamics of the quantum Ising model on two-dimensional square lattices up to 16 $\ifmmode\times\else\texttimes\fi{}$ 16 spins. In the ordered phase, the model exhibits dynamically constrained dynamics, leading to confinement of elementary excitations and slow thermalization. After demonstrating the signatures of confinement, we probe the dynamics of interfaces in the constrained regime through sudden quenches of product states with domains of opposite magnetization. We find that the nature of excitations can be captured by perturbation theory throughout the confining regime and identify the crossover to the deconfining regime. We systematically explore the effect of the transverse field on the modes propagating along flat interfaces and investigate the crossover from resonant to diffusive melting of a square of flipped spins embedded in a larger lattice.
The quantitative description of long-range order remains a challenge in quantum many-body physics. We provide zero-temperature results from two complementary methods for the ground-state energy per site, the sublattice magnetization, the spin gap, and the transverse spin correlation length for the spin-1/2 quantum XXZ antiferromagnet on the square lattice. On the one hand, we use exact, large-scale quantum Monte Carlo (QMC) simulations. On the other hand, we use the semianalytic approach based on continuous similarity transformations in terms of elementary magnon excitations. Our findings confirm the applicability and quantitative validity of both approaches along the full parameter axis from the Ising point to the symmetry-restoring phase transition at the Heisenberg point and further provide quantitative reference results in the thermodynamic limit. In addition, we analytically derive the relation between the dispersion and the correlation length at zero temperature in arbitrary dimension, and discuss improved second-moment QMC estimators.