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.

Fractional dynamics and modulational instability in long-range Heisenberg chains

Trombettoni, Andrea;
2023-01-01

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/3073868
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