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.
The bound state problems associated with two magnons in a linear chain of spin 1/2 with nearest and next nearest neighbor exchange integrals, J and α J , respectively, are studied. The Hamiltonian is rewritten in terms of the Fermi operators in place of spin operators, and an integral equation to determine the bound state energies is obtained. In the cases when the total wave number K of two magnons is π or π/2, analytical solutions are easily obtained. A part of the present results disagrees with Majumdar's ones. Computer studies on the bound states are performed for a finite chain composed of spins up to twenty. They are consistent with the analytical results.
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 calculate the quantum phase diagram of an XXZ chain with nearest-neighbor (NN)
J1 and next-NN exchange
J2 with anisotropies ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2},$ respectively. In particular we consider the case ${\ensuremath{\Delta}}_{1}=\ensuremath{-}{\ensuremath{\Delta}}_{2},$ to interpolate between the XX chain $({\ensuremath{\Delta}}_{i}=0)$ and the isotropic model with ferromagnetic phase
J2. For ${\ensuremath{\Delta}}_{1}<\ensuremath{-}1,$ a ferromagnetic phase and two antiferromagnetic phases exist. For $|{\ensuremath{\Delta}}_{i}|<1,$ the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large
J2/J1, this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for ${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}=1$ suggests that a small gap is present.
A study of two-magnon bound states in the next-nearest-neighbour (NNN) ferromagnetic Heisenberg chain begun by Majumdar in 1969 is extended by finding the bound states and continuum resonances for all pair wave-vectors for eta =J 2 /J 1 in the ferromagnetic range of eta =-1/4 to infinity. An exact Green function formalism is combined with the exact analytic results recently obtained by the authors for the lattice Green functions of the system to afford a definitive analysis. The Heisenberg case exhibits a modified nearest-neighbour (NN) type bound state throughout the range of eta studied and for total pair wave-vector (K=k 1 +k 2 ) throughout the zone, in contrast to Majumdar who concluded that this was only the case for positive eta . The continuum also possesses a rich structure which is displayed via a shaded (grey-level) intensity plot in the ( omega , K) plane. An unexpected result is the presence of a second (anti-)bound state just above the continuum for negative eta . The evolution of the bound state and resonance structure is also examined for the transition to dominant NNN interactions. A general formulation is given which includes Ising and uniaxial anisotropies as well as biquadratic exchange for arbitrary range, dimension and Bravais lattice.
The phase diagram of antiferromagnetic spin-
S chain with
XY-type anisotropy and frustrating next-nearest-neighbor interaction is studied in the limit of large integer S with the help of a field-theoretical approach. It is shown that the existence of gapless and gapped chiral phases found in recent numerical studies [M. Kaburagi et al., J. Phys. Soc. Jpn. 68, 3185 (1999); T. Hikihara et al., J. Phys. Soc. Jpn. 69, 259 (2000)] is not specific for
S=1, but is rather a generic large-
S feature. Estimates for the corresponding transition boundaries are obtained, and a sketch of the typical phase diagram is presented. It is also shown that frustration stabilizes the Haldane phase against the variation of the anisotropy.