Systematic study of multi-magnon binding energies in the FM-AFM $J_1$-$J_2$ chain

We present a systematic study of multi-magnon bound states (MBSs) in the spin-$\tfrac{1}{2}$ FM-AFM $J_1$-$J_2$ chain under magnetic fields using the density-matrix renormalization group method. As a quantitative measure of stability, we compute the magnon binding energy $E_{\rm b}(M,p)$ for bound clusters of size $p$ over wide ranges of the frustration ratio $J_2/|J_1|$ and the normalized magnetization $M/M_{\rm s}$. Near saturation, we benchmark our data against the analytic two-magnon result and map out a clear hierarchy of $p$-magnon states, whose phase boundaries follow an empirical scaling $J_{2,{\rm c}}(p;p\!+\!1)/|J_1|\!\approx\!0.34\,p^{-2.3}$ for large $p$. We further quantify the relation between the most stable $p$ and the zero-field pitch angle $\theta$, verifying the conjectured inequality $1/p>\theta/\pi>1/(p+1)$ up to $p \lesssim 9$. The binding energy shows pronounced suppression as $J_2/|J_1|\!\to\!1/4^+$ and, for some frustration values, attains a maximum below full saturation, indicating that partial depolarization enhances bound-magnon mobility. Close to the FM instability, $E_{\rm b}(M_{\rm s},p)$ exhibits an empirical power-law vanishing consistent with a quantum-Lifshitz scenario. Our results provide a comprehensive, experimentally relevant map of MBS stability across field and frustration, offering concrete guidance for inelastic probes in quasi-one-dimensional magnets.