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

TL;DR

The paper investigates multi-magnon bound states in the spin-$\tfrac{1}{2}$ FM-AFM $J_1$-$J_2$ chain under magnetic field using DMRG to map magnon binding energies $E_{\rm b}(M,p)$ across a broad space of frustration $\alpha$ and magnetization $M/M_{\rm s}$. It reveals a hierarchical sequence of bound states near saturation with phase boundaries that follow an empirical scaling and establishes a quantitative link between the most stable bound-magnon size $p$ and the zero-field pitch angle $\theta$, consistent with the inequality $\tfrac{1}{p}>\tfrac{\theta}{\pi}>\tfrac{1}{p+1}$ up to $p\lesssim 9$. The study also finds strong suppression of binding energy as $\alpha \to 1/4^+$ and, for some frustrations, a maximum below full saturation, indicating enhanced bound-magnon mobility upon partial depolarization; near the FM instability, $E_{\rm b}(M_{\rm s},p)$ vanishes with a quantum-Lifshitz-like power law. These results provide a detailed, experimentally relevant map of MBS stability across field and frustration and offer concrete benchmarks for inelastic probes in quasi-one-dimensional magnets.

Abstract

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 $θ$, verifying the conjectured inequality $1/p>θ/π>1/(p+1)$ 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.

Figures (7)

• Figure 1: Lattice structure of the FM-AFM $J_1$-$J_2$ chain, where each circle denotes a spin-$\tfrac{1}{2}$ site.
• Figure 2: (a) DMRG results for the pitch angle $\theta/\pi$ as a function of the frustration ratio $\alpha$. The classical result $\theta/\pi=\cos^{-1}(|J_1|/4J_2)/\pi$ is shown for comparison. Inset: Difference between the quantum and classical values of $\theta/\pi$. (b) Log–log plot of $\theta/\pi$ versus $\alpha-\tfrac{1}{4}$, comparing the classical and DMRG scaling near the FM critical point.
• Figure 3: (a) DMRG results for the binding energy $E_{\rm b}(M,p)$ near full saturation ($M/M_{\rm s}=1$) as a function of $\alpha$. The most stable bound-magnon size $p$ is indicated for each region. (b) Critical coupling $\alpha_{\rm c}(p;p+1)-1/4$ between the $p$- and $(p\!+\!1)$-MBS phases plotted as a function of $1/p$. The solid line shows a fit to $0.34\,p^{-2.3}$.
• Figure 4: (a) Pitch angle $\theta/\pi$ (black circles) at zero field as a function of $\alpha-1/4$, together with the corresponding most energetically stable (MES) bound-magnon number $p$ at saturation. The dashed line denotes the quantum critical scaling $\theta/\pi=(\pi/4)(\alpha-1/4)^{1/\pi}$. (b) Magnon binding energy $E_{\rm b}(M_{\rm s},p)$ corresponding to $p$ in the MES at saturation. The dotted line represents an empirical power-law fit $E_{\rm b}(M_{\rm s},p) \propto (\alpha-1/4)^{\pi/2}$.
• Figure 5: (a) Representative cross sections of the binding energy $E_{\rm b}(M,p)$ as a function of $M/M_{\rm s}$ for selected frustration ratios $\alpha=0.50$, $0.34$, and $0.28$, which correspond to the nematic ($p=2$), triatic ($p=3$), and quartic ($p=4$) states, respectively. (b) Color map of $E_{\rm b}(M,p)$ obtained by DMRG as a function of $1/\alpha$ ($\alpha$) and $M/M_{\rm s}$.