长程Heisenberg链中的分数动力学和调制不稳定性
Fractional Dynamics and Modulational Instability in Long-Range Heisenberg Chains
摘要：
研究了长程相互作用下铁磁自旋链的有效动力学。我们考虑一维海森堡哈密顿量，其中自旋通过指数为$α$的幂律长程交换相互作用耦合。我们在哈密顿量中加入了$z$方向的各向异性。在半经典方法的框架下，我们利用Holstein-Primakoff变换导出了一个有效的长程离散非线性薛定谔方程.然后，我们进行连续极限，我们得到一个分数阶非线性薛定谔方程。最后，我们研究了连续极限下平面波的调制不稳定性，证明了与短程情形不同，平面波对$α< 3$是调制不稳定的。我们还研究了调制不稳定性增长率和临界波数对哈密顿量参数和指数$α$的依赖关系。
英文摘要：
We study the effective dynamics of ferromagnetic spin chains in presence of long-range interactions. We consider the Heisenberg Hamiltonian in one dimension for which the spins are coupled through power-law long-range exchange interactions with exponent $α$. We add to the Hamiltonian an anisotropy in the $z$-direction. In the framework of a semiclassical approach, we use the Holstein-Primakoff transformation to derive an effective long-range discrete nonlinear Schrödinger equation. We then perform the continuum limit and we obtain a fractional nonlinear Schrödinger-like equation. Finally, we study the modulational instability of plane-waves in the continuum limit and we prove that, at variance with the short-range case, plane waves are modulationally unstable for $α< 3$. We also study the dependence of the modulation instability growth rate and critical wave-number on the parameters of the Hamiltonian and on the exponent $α$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
