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 study the ground-state phase diagrams and properties of spin-1/2 Heisenberg models on the diamond-like-decorated square and triangular lattices. The diamond-like-decorated square (triangular) lattice is a lattice in which the bonds of a square (triangular) lattice are replaced with diamond units. The diamond unit has two types of antiferromagnetic exchange interactions, and the ratio λ of the strength of the diagonal bond to that of the other four edges determines the ground-state properties. In particular, the macroscopically degenerated tetramer-dimer states, which are equivalent to the dimer covering states of the original lattices, are stabilized for λc < λ < 2, where the value of λc depends on the lattices. To determine the phase diagrams and boundaries λc, we employ the modified spin wave (MSW) method and the quantum Monte Carlo (QMC) method to estimate the ground-state energies of the ferrimagnetic states for λ < λc, where we can consider the mixed spin-1 and spin-1/2 Lieb-lattice and triangular Lieb-lattice Heisenberg antiferromagnets instead, and obtain λc(square)=0.974 and λc(triangular)=0.988. We also calculate the long-range order (LRO) parameters using the MSW and QMC methods and find the scaling relations where the spin reductions of each sublattice are inversely proportional to the number of sublattice sites. We prove these scaling relations by applying an infinitesimal uniform magnetic field. Furthermore, by examining the calculation process in the MSW, we clarify the mathematical structure behind the scaling relations for the sublattice LROs.
Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.
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.
Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L −1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln γ ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.
Second-order spin-wave expansions are used to compute the ground-state energy and sublattice magnetizations of the quantum one-dimensional Heisenberg ferrimagnet with nearest-neighbor antiferromagnetic interactions and two types of alternating sublattice spins
S1>S2. It is found that in the extreme quantum cases
(S1,S2)=(1,1/2), (3/2,1), and (3/2,1/2), the estimates for the ground-state energy and sublattice magnetizations differ less than 0.03% for the energy and 0.2% for the sublattice magnetizations from the recently published density-matrix renormalization-group numerical calculations. The reported results strongly suggest that the quantum Heisenberg ferrimagnetic chains give another example of a low-dimensional quantum spin system where the spin-wave approach demonstrates a surprising efficiency.