Few-magnon bound states in quantum spin chains have been long studied and attracted much recent attention. For a higher-spin ferromagnetic XXZ chain with single-ion anisotropy, several features regarding the evolution of the low-lying two-magnon bound states with varying wave number were observed in the literature. However, most of these observations are only qualitatively understood due to the lack of analytical tools. By combining a set of exact two-magnon Bloch states and a plane-wave ansatz, we achieve a complete solution of the two-magnon problem in such a system. We identify parameter regions that support different types of two-magnon bound states, with the boundaries defined by algebraic equations. We discover a narrow region in which two single-ion bound states coexist. We show that the phase diagrams for distinct wave numbers are similar to each other, which enables us to map the evolution of the bound states to the rectilinear movement of a representative point for given parameters in a rescaled phase diagram. This dynamic picture provides quantitative interpretations of the observed features.
We solve few-magnon problems for a finite-size spin-
S periodic Heisenberg XXZ chain with single-ion anisotropy through constructing sets of exact Bloch states achieving block diagonalization of the system. Concretely, the two-magnon (three-magnon) problem is converted to a single-particle one on a one-dimensional (two-dimensional) effective lattice whose size depends linearly (quadratically) on the total number of sites. For parameters lying within certain ranges, various types of multimagnon bound states are manifested and shown to correspond to edge states on the effective lattices. In the absence of the single-ion anisotropy, we reveal the condition under which exact zero-energy states emerge. As applications of the formalism, we calculate the transverse dynamic structure factor for a higher-spin chain near saturation magnetization and find signatures of the multimagnon bound states. We also calculate the real-time three-magnon dynamics from certain localized states, which are relevant to cold-atom quantum simulations, by simulating single-particle quantum walks on the effective lattices. This provides a physically transparent interpretation of the observed dynamics in terms of propagation of bound state excitations. Our method can be directly applied to more general spin or itinerant particle systems possessing translational symmetry.
Few-magnon excitations in Heisenberg-like models play an important role in understanding magnetism and have long been studied by various approaches. However, the quantum dynamics of magnon excitations in a finite-size spin-
S XXZ chain with single-ion anisotropy remains unsolved. Here, we exactly solve the two-magnon (three-magnon) problem in the spin-
S XXZ chain by reducing the few-magnons to a fictitious single particle on a one-dimensional (two-dimensional) effective lattice. Such a mapping allows us to obtain both the static and dynamical properties of the model explicitly. The zero-energy-excitation states and various types of multimagnon bound states are manifested, with the latter being interpreted as edge states on the effective lattices. Moreover, we study the real-time multimagnon dynamics by simulating single-particle quantum walks on the effective lattices.
We study few-magnon excitations in a finite-size spin-
S chain with ferromagnetic nearest-neighbor (NN) interaction
J>0 and antiferromagnetic next-nearest-neighbor (NNN) interaction ${J}^{\ensuremath{'}}<0$, in the presence of the single-ion (SI) anisotropy
D. We first reveal the condition for the emergence of zero-excitation-energy states. In the isotropic case with $\mathrm{\ensuremath{\Delta}}={\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}=1$ ($\mathrm{\ensuremath{\Delta}}$ and ${\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}$ are the corresponding anisotropy parameters), a threshold of $J/|{J}^{\ensuremath{'}}|$, above which the ground state is ferromagnetic, is determined by exact diagonalization for short chains up to 12 sites. Using a set of exact two-magnon Bloch states, we then map the two-magnon problem to a single-particle one on an effective open chain with both NN and NNN hoppings. The whole two-magnon excitation spectrum is calculated for large systems, and the commensurate-incommensurate transition in the lowest-lying mode is found to exhibit different behaviors between
S=1/2 and higher spins due to the interplay of the SI anisotropy and the NNN interaction. For the commensurate momentum $k=\ensuremath{-}\ensuremath{\pi}$, the effective lattice is decoupled into two NN open chains that can be exactly solved via a plane-wave ansatz. Based on this, we analytically identify in the ${\mathrm{\ensuremath{\Delta}}}^{\ensuremath{'}}\ensuremath{-}D/|{J}^{\ensuremath{'}}|$ plane the regions supporting the SI or NNN exchange two-magnon bound states near the edge of the band. In particular, we prove that there always exists a lower-lying NN exchange two-magnon bound state near the band edge for arbitrary $S\ensuremath{\ge}1/2$. Finally, we numerically calculate the
n-magnon spectra for
S=1/2 with $n\ensuremath{\le}5$ by using a spin-operator matrix element method. The corresponding
n-magnon commensurate instability regions are determined for finite chains, and consistent results with prior literature are observed.
Abstract The two-magnon problem in the k = − π /2 sector of a finite-size spin-1/2 chain with ferromagnetic nearest-neighbor (NN) interaction ( J 1 > 0) and antiferromagnetic next-nearest-neighbor (NNN) interaction ( J 2 < 0) and anisotropy parameters Δ 1 and Δ 2 is solved exactly by combining a set of exact two-magnon Bloch states and a plane-wave ansatz. Two types of two-magnon bound states (BSs), i.e., the NN and NNN exchange BSs, are revealed. We establish a phase diagram in the J 1 /(∣ J 2 ∣Δ 2 )-Δ 1 plane where regions supporting different types of BSs are analytically identified. It is found that no BSs exist (the two types of BSs coexist) when both Δ 1 and Δ 2 are small (large) enough. Our results for the isotropic case Δ 1 = Δ 2 = 1 are consistent with an early work [Ono I, Mikado S and Oguchi T 1971 J. Phys. Soc. Japan 30 358].
The quantum scattering of magnon bound states in the anisotropic Heisenberg spin chain is shown to display features similar to the scattering of solitons in classical exactly solvable models. Localized colliding Gaussian wave packets of bound magnons are constructed from string solutions of the Bethe equations and subsequently evolved in time, relying on an algebraic Bethe ansatz based framework for the computation of local expectation values in real space-time. The local magnetization profile shows the trajectories of colliding wave packets of bound magnons, which obtain a spatial displacement upon scattering. Analytic predictions on the displacements for various values of anisotropy and string lengths are derived from scattering theory and Bethe ansatz phase shifts, matching time-evolution fits on the displacements. The time-evolved block decimation algorithm allows for the study of scattering displacements from spin-block states, showing similar scattering displacement features.
Long-lived spin-helix states facilitate the study of non-equilibrium dynamics in quantum magnets. We consider the decay of transverse spin-helices in antiferromagnetic spin-
S XXZ chains with single-ion anisostropy. The spin-helix decay is observable in the time evolution of the local magnetization that we calculate numerically for the system in the thermodynamic limit using infinite time-evolving block decimation simulations. Although the single-ion anisotropy prevents helix states from being eigenstates of the Hamiltonian, they still can be long-lived for appropriately chosen wave numbers. In case of an easy-axis exchange anisotropy the single-ion anisotropy may even stabilize the helices. Within a spin-wave approximation, we obtain a condition giving an estimate for the most stable wave number
Q that agrees qualitatively with our numerical results.