Exact one- and two-particle excitation spectra of acute-angle helimagnets above their saturation magnetic field

Abstract

The two-magnon problem for the frustrated XXZ spin-$1∕2$ Heisenberg Hamiltonian and external magnetic fields exceeding the saturation field ${B}_{s}$ is considered. We show that the problem can be exactly mapped onto an effective tight-binding impurity problem. It allows to obtain explicit exact expressions for the two-magnon Green's functions for arbitrary dimension and number of interactions. We apply this theory to a quasi-one-dimensional helimagnet with ferromagnetic nearest-neighbor ${J}_{1}<0$ and antiferromagnetic next-nearest neighbor ${J}_{2}>0$ interactions. An outstanding feature of the excitation spectrum is the existence of two-magnon bound states. This leads to deviations of the saturation field ${B}_{s}$ from its classical value ${B}_{s}^{\mathrm{cl}}$ which coincides with the one-magnon instability. For the refined frustration ratio $\ensuremath{\mid}{J}_{2}∕{J}_{1}\ensuremath{\mid}>0.374\phantom{\rule{0.2em}{0ex}}661$ the minimum of the two-magnon spectrum occurs at the boundary of the Brillouin zone. Based on the two-magnon approach, we propose general analytic expressions for the saturation field ${B}_{s}$, confirming known previous results for one-dimensional isotropic systems, but explore also the role of interchain and long-ranged intrachain interactions as well as of the exchange anisotropy.