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arXiv:cond-mat/0402445 · Physical Review B

Spin waves in antiferromagnetic spin chains with long-range interactions

Eddy Yusuf, Anuvrat Joshi, Kun Yang

Published Apr 9, 2004• 45 citations
arXivHTMLPDF

Abstract

We study antiferromagnetic spin chains with unfrustrated long-range interactions that decay as power laws with exponent $\ensuremath{\beta},$ using the spin-wave approximation. We find for sufficiently large spin S that the Neel order is stable at $T=0$ for $\ensuremath{\beta}<3,$ and survives up to a finite Neel temperature for $\ensuremath{\beta}<2,$ validating the spin-wave approach in these regimes. We estimate the critical values of S and T for the Neel order to be stable. The spin-wave spectra are found to be gapless but have nonlinear momentum dependence at long wavelength, which is responsible for the suppression of quantum and thermal fluctuations and stabilizing the Neel state. We also show that for $\ensuremath{\beta}<~1$ and for a large but finite-size system size L, the excitation gap of the system approaches zero slower than ${L}^{\ensuremath{-}1},$ a behavior that is in contrast to the Lieb-Schultz-Mattis theorem.

DOI
10.1103/physrevb.69.144412

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The antiferromagnetic spin-1∕21∕21∕2 chain with power-law decaying interactions $\ensuremath{\mid}{J}_{n}\ensuremath{\mid}\ensuremath{\propto}1∕{n}^{p}$ is studied by a Green's function method. The dispersion relation ${\ensuremath{\omega}}_{k}$ and an expression for the critical temperature TN{T}_{N}TN​ as a function of ppp are presented. By analyzing the small-kkk behavior of ${\ensuremath{\omega}}_{k}$, it is shown that there will be no long-range order at finite temperature when $p\ensuremath{\geqslant}2$, which is in good agreement with Bruno's rigorous proof. Our work refurbishes a previous work also by Green's function method, and validates the Tyablikov decoupling approximation to long-range interactions with 1<p<21<p<21<p<2.
We investigate the ground-state properties of the spin-1/21/21/2 XXZ model with power-law-decaying ($1/{r}^{\ensuremath{\alpha}}$) interactions, which describe spins interacting with long-range transverse (XX) ferromagnetic interactions and longitudinal (Z) antiferromagnetic interactions, or hard-core bosons with long-range repulsion and hopping. The long-range nature of the couplings allows us to quantitatively study the spectral, correlation, and entanglement properties of the system by making use of linear spin-wave theory, supplemented with density-matrix renormalization group in one-dimensional systems. Our most important prediction is the existence of three distinct coupling regimes, depending on the decay exponent $\ensuremath{\alpha}$ and number of dimensions ddd: (1) a short-range regime for $\ensuremath{\alpha}>d+{\ensuremath{\sigma}}_{c}$ (where ${\ensuremath{\sigma}}_{c}=1$ in the gapped N\'eel antiferromagnetic phase exhibited by the XXZ model, and ${\ensuremath{\sigma}}_{c}=2$ in the gapless XY ferromagnetic phase), sharing the same properties as those of finite-range interactions ($\ensuremath{\alpha}=\ensuremath{\infty}$); (2) a long-range regime $\ensuremath{\alpha}<d$, sharing the same properties as those of the infinite-range interactions ($\ensuremath{\alpha}=0$) in the thermodynamic limit; and (3) a most intriguing medium-range regime for $d<\ensuremath{\alpha}<d+{\ensuremath{\sigma}}_{c}$, continuously interpolating between the finite-range and the infinite-range behavior. The latter regime is characterized by elementary excitations with a long-wavelength dispersion relation $\ensuremath{\omega}\ensuremath{\approx}{\mathrm{\ensuremath{\Delta}}}_{g}+c{k}^{z}$ in the gapped phase, and $\ensuremath{\omega}\ensuremath{\sim}{k}^{z}$ in the gapless phase, exhibiting a continuously varying dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}d)/{\ensuremath{\sigma}}_{c}$. In the gapless phase of the model the zzz exponent is found to control the scaling of fluctuations, the decay of correlations, and a universal subdominant term in the entanglement entropy, leading to a very rich palette of behaviors for ground-state quantum correlations beyond what is known for finite-range interactions.
We study a generalized quantum spin ladder with staggered long range interactions that decay as a power-law with exponent α\alphaα. Using the density matrix renormalization group (DMRG) method and exact diagonalization, we show that this model undergoes a transition from a rung-dimer phase characterized by a non-local string order parameter, to a symmetry broken Neel phase at αc∼2.1\alpha_c\sim 2.1αc​∼2.1. We find evidence that the transition is second order with a dynamic critical exponent z=1z=1z=1 and ν≈1.2\nu\approx 1.2ν≈1.2. In the magnetically ordered phase, the spectrum exhibits gapless modes, while excitations in the gapped phase are well described in terms of triplons -- bound states of spinons across the legs. We obtained the momentum resolved spin dynamic structure factor numerically and found that the triplon band is well defined at high energies and adiabatically connected to the magnon dispersion. However, at low energies it emerges as the lower edge of continuum of excitations that shifts to high energies across the transition. We further discuss the possibility of deconfined criticality in this model.
We study the phase diagram of a spin-1 antiferromagnetic chain with isotropic antiferromagnetic interactions decaying with a power law $\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}1)$ accompanied by modulated single-ion anisotropy. Employing the techniques of the density-matrix renormalization group, the effects of long-range interactions and single-ion anisotropy on a variety of correlations are investigated. In order to check the consistency, the fidelity susceptibilities are evaluated across quantum phase transitions. The quantum critical points are faithfully detected and orders of phase transitions are determined. The correlation-length critical exponent is extracted from scaling functions of the fidelity susceptibility. The presence of long-range interactions leads to a quantitative change of the phase boundaries and reduces the order of the phase transition under certain conditions. A direct first-order transition between the periodic N\'eel phase and the large-DDD phase occurs for slowly decaying antiferromagnetic interactions.