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arXiv:2401.14101 · Physical review. B./Physical review. B

Few-magnon excitations in a frustrated spin- S ferromagnetic chain with single-ion anisotropy

Jiawei Li, Ye Cao, Ning Wu

Published May 2, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.109.174403

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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].
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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 study the dependence of the magnetization M with magnetic field B at zero temperature in the spin-1/2 XXZ\mathrm{XXZ}XXZ chain with nearest-neighbor (NN) J1{J}_{1}J1​ and next-NN J2{J}_{2}J2​ exchange interactions, with anisotropies ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2},$ respectively. The region of parameters for which a jump in M(B)M(B)M(B) exists is studied using numerical diagonalization, and analytical results for two magnons on a ferromagnetic background in the thermodynamic limit. We find a line in the parameter space ${(J}_{2}{/J}_{1},{\ensuremath{\Delta}}_{1}{/J}_{1},{\ensuremath{\Delta}}_{2}{/J}_{1})$ (determined by two simple equations) at which the ground state is highly degenerate. M(B)M(B)M(B) has a jump near this line, and at or near the isotropic case with ferromagnetic J1{J}_{1}J1​ and antiferromagnetic J2,{J}_{2},J2​, with $|{J}_{2}{/J}_{1}|\ensuremath{\sim}1/4.$ These results are relevant for some systems containing CuO chains with edge-sharing CuO4{\mathrm{CuO}}_{4}CuO4​ units.
We consider analytically and numerically an anisotropic $\text{spin}\ensuremath{-}\frac{1}{2}$ delta chain (sawtooth chain) with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions. For certain values of the interactions a lowest one-particle band becomes flat and there is a class of localized-magnon eigenstates which form a ground state with a macroscopic degeneracy. In this case the model depends on a single parameter which can be chosen as the anisotropy of the exchange interactions. When this parameter changes from zero to infinity the model interpolates between the one-dimensional isotropic ferromagnet and the frustrated Ising model on the delta chain. It is shown that the low-temperature thermodynamic properties in these limiting cases are governed by the specific structure of the excitation spectrum. In particular, the specific heat has one or infinite number of low-temperature maxima for the small or the large anisotropy parameter, correspondingly. Qualitative features of such behavior survive when the interaction parameters deviate from the relations providing the local magnon ground state.
We numerically investigate the magnetization process of a frustrated S =1 chain with antiferromagnetic nearest-neighbor (nn) and next-nearest-neighbor (nnn) exchange interactions, which compete with each other, and also with uniaxial single-ion-type anisotropy energies. Special attention is paid to the half magnetization plateau which appears at the half of the saturation magnetization in the ground-state magnetization curve. We determine the parameter region where this plateau appears by employing the level spectroscopy method, and find that the plateau appears even in the absence of the single-ion-type anisotropy, if the ratio of the nnn to the nn interaction constant is in a specified region.