Multimagnon and multispinon $L_3$-edge RIXS spectra of an effective $\tilde{J}_1-\tilde{J}_2-\tilde{J}_3$ square lattice Heisenberg model

TL;DR

This work tackles the origin of satellite intensity in L3-edge RIXS for a square-lattice J1–J2–J3 Heisenberg system in the Néel phase. By combining 1/S interacting spin-wave theory with Schwinger-boson mean-field theory, the authors show that conventional magnons alone cannot explain higher-order spectral weight, and that RVB-like bond fluctuations (via SBMFT) reproduce the multimagnon and multispinon features, including a Higgs-like condensate signature. They demonstrate that satellite intensity can originate from both one-to-three-magnon hybridization and condensed spinon dynamics, revealing a rich interplay of quantum fluctuations, entanglement, and gauge interactions in quantum magnets as probed by RIXS. The results provide a unified framework linking high-energy multimagnon continua to RVB physics, with implications for interpreting RIXS data in cuprates and related frustrated antiferromagnets. The study highlights the potential of L3-edge RIXS as a sensitive probe of nonlocal spin correlations and Higgs-type resonances in correlated electron systems.

Abstract

We investigate the multimagnon and the multispinon $L_3$-edge resonant inelastic x-ray scattering (RIXS) spectra of a spin-1/2 effective $\tilde{J}_1-\tilde{J}_2-\tilde{J}_3$ square lattice Heisenberg model in its Néel ordered phase. Motivated by the observation of satellite intensity peaks above the single magnon dispersion in the $L$-edge RIXS spectrum, we propose a resonating valence bond (RVB) inspired RIXS mechanism that incorporates the local site ultrashort core-hole lifetime (UCL) expansion. We compute the multimagnon and the multispinon excitations using $\mathcal{O}(1/S)$ interacting spin wave theory and Schwinger boson mean-field theory (SBMFT) formalism, respectively. We treat the x-ray scattering process up to second order in the UCL expansion. Our calculations of two-magnon, bimagnon, and three-magnon RIXS intensities reveal that interacting spin wave theory fails to fully capture all the quantum correlations in the antiferromagnetic ordered phase. However utilizing the SBMFT framework, with a ground state that combines Néel order and fluctuating RVB components, we demonstrate that a RIXS bond-flipping mechanism provides an alternative deeper physical explanation of the satellite intensities. Specifically, we find that the spin correlation spectra predicted by the fluctuating RVB mechanism aligns with higher order UCL expansion results. We further show that the satellite intensity above the single-magnon mode can originate both from a one-to-three-magnon hybridization vertex process and from condensed spinons exhibiting Higgs mechanism. These features reflect the interplay of quantum fluctuation, entanglement, and gauge interaction effects of quantum magnetism probed by RIXS.

Figures (10)

• Figure 1: (a) Spin configuration of the Néel ordered state in a square lattice. Spin wave theory was constructed out of this broken $SU(2)$ symmetry configuration. The fluctuating RVB states considered in the Schwinger boson theory are not shown. The filled blue circles represent magnetic atoms. The red arrows denote the orientations of the spins. Antiferromagnetic ordering is exhibited as an example. Exchange interactions up to the third nearest neighbor are illustrated. The effective magnetic couplings $(\tilde{J}_1,\tilde{J}_2,\tilde{J}_3) = (J_1-2J_cS^2,J_2-J_cS^2,J_3)$ are defined in terms of the original exchange interactions where the first-neighbor is $J_1$, the second-nearest neighbor is $J_2$, the third-nearest neighbor is $J_3$, and the cyclic exchange coupling is $J_c$PhysRevLett.86.5377PhysRevB.79.235130. (b) The Brillouin zone of the square lattice. Red dotted line represents boundary of the magnetic Brillouin zone (MBZ), with high-symmetry points $\Gamma(0,0)$, $M(\pi,0)$, $K(\frac{\pi}{2}$, $\frac{\pi}{2})$, and $X(\pi,\pi)$.
• Figure 2: (a) Single site $L_3$-edge RIXS process in the spin-conserving (SC) and non-spin-conserving (NSC) channels. In the initial state, an incident photon excites a single electron to the $3d_{x^2-y^2}$ orbital leaving behind a core-hole in the $2p$ orbital. The excited electron fills the empty hole in the $3d$ orbital. In the intermediate state, there are effects of the core-hole potential and spin-orbit coupling. In the final state the electron loses its energy and falls back to the $2p$ orbital. If it has the same spin orientation it belongs to the SC channel. If has the opposite spin orientation it belongs to the NSC channel. (b) $L_3$-edge bond spin-flip RIXS mechanism. The solid blue lines represent singlet bonds. The blue hatched circles represent sites where a single-spin flip mechanism is active. The wavy orange lines represent gauge interaction (the glue that holds the bond together). Incident photons can induce resonating valence bonds fluctuations. Initially photons strike an existing RVB bond (panel(b), left figure) and ruptures it. Next, the decoupled sites undergo a core-hole mediated RIXS spin-flip at each site (within the UCL scheme). After, the single-site process is completed, the photons leave the system. However, the RVB bonds can be restored in another orientation (or not) due to the local constraint $\lambda$ and due to intrinsic parity-gauge in the Schwinger-boson ground state fradkin2013fieldchandra1990quantum. If the bonds are reorientated then a RIXS photon induced RVB fluctuation is created. Such a bond spin-flip mechanism is able to capture a three-site spin correlation (described within a bond spin-flip process). Based on our RIXS intensity calculations, we show that there is a finite amplitude of such a process occurring in a quantum magnet.
• Figure 3: Renormalized single magnon dispersion $\omega_\mathbf{q}$ (the left axis) and $1/S$ -corrected $L_3$-edge (single magnon) RIXS spectral weight $W_{\mathrm{1m}}$ (the right axis). The momentum $\mathbf{q}$ path is $\mathrm{\Gamma}-\mathrm{M}-\mathrm{K}-\mathrm{X}-\mathrm{M}$. (a) $(J_2,J_3,J_c)=0$. Along the MBZ boundary path $\mathrm{M}-\mathrm{K}$ the magnon band is flat with $\omega \sim 2.3J_1$. At the $\mathrm{X}$ point, the dispersion is gapless while the single-magnon spectral weight diverges PhysRevLett.105.167404. (b) $J_2=J_3=(0,0.05,0.1)J_1$ and $J_c=0$. As the frustration parameters $(J_2, J_3)$ increase, the dispersion energy is suppressed. However, the single-magnon RIXS spectral weight $W_{\mathrm{1m}}$ is relatively insensitive to the changes. (c) $J_c=(0.3,0.5,0.7)J_1$ and $J_2=J_3=0.05J_1$. The cyclic exchange interaction $J_c$ predominantly reduces the magnon energy at the $\mathrm{K}$ point, but has minimal influence on $W_{\mathrm{1m}}$.
• Figure 4: Three-magnon DOS [Eq. \ref{['eq:d3m']}] in (a) and (b) and three-magnon intensity $I^{(0,1/S,1)}_{\mathrm{3m}}(\mathbf{q},\omega)$ [$\text{Eq.}\ref{['eq:i1s3m']}$] in (c) and (d). Panels (a) and (c) use parameters $(J_2,J_3,J_c) = 0$, whereas panels (b) and (d) include frustration effects with $(J_2,J_3,J_c) = (0.05,0.05,0)J_1$. In (a) the upper bound of the three-magnon DOS is $\approx 6.9 J_1$. With finite frustration $(J_2,J_3) = (0.05,0.05)J_1$, this upper bound is lowered to $\approx 6.2 J_1$ in panel (b). Panel (c) presents the three-magnon intensity $I^{(0,1/S,1)}_{\mathrm{3m}}(\mathbf{q},\omega)$, where a broad high-energy band feature is evident. Upon incorporating the one-to-three hybridization process, the three-magnon excitation peaks shift toward the one-magnon excitation regions. Notably, the intensity peaks vanish at the $\mathrm{X}$ point instead of diverging, indicating that these peaks do not correspond to single-magnon dispersion peaks, which are known to have gapless dispersion and always diverge at the $\mathrm{X}$ point PhysRevLett.105.167404. With finite frustration, the overall spectrum is shifted to lower energies, and the high-energy band peaks become more pronounced. Additionally, some peaks remain near $\sim 2J_1$, suggesting the presence of satellite intensity peaks associated with single-magnon dispersion peaks. In panel (d), consistent with the DOS, the inclusion of frustrations causes lower energy excitations in RIXS spectrum. The inset in panel (d) is showing the Feynman diagram corresponding to the one-to-three magnon hybridization process.
• Figure 5: Two-magnon DOS $D_{\mathrm{2m}}$ [Eq. \ref{['eq:d2m']}] (the upper panels) and RIXS intensity $I^{(1,S^2,0)}_{\mathrm{2m}}$ [Eq. \ref{['eq:i12m']}] (the bottom panels) within $\mathcal{O}(\text{UCL}[1])$. (a) Two-magnon DOS with $(J_2,J_3,J_c)=0$ shows a flat upper bound at $\approx 4.6J_1$. (b) Two-magnon DOS with $(J_2,J_3,J_c)=(0.05,0.05,0.4) J_1$. The upper bound becomes wavy with a local minimum at the $\mathrm{M}$ point. (c) Two-magnon RIXS intensity with $(J_2,J_3,J_c)=0$. (d) Two-magnon RIXS intensity with $(J_2,J_3,J_c)=(0.05,0.05,0.4)J_1$. Two-magnon RIXS intensity vanishes at $\mathrm{\Gamma}$ and $\mathrm{X}$ points in $\mathcal{O}(\text{UCL}[1])$PhysRevB.77.134428. The intensity peak at $\mathrm{M}$ point has higher energy than that at $\mathrm{K}$ point. Moving to $\mathrm{X}$ point, the energy of two-magnon intensity becomes $\approx 4.6J_1$ instead of $0$, which indicates the two-magnon intensity is not diverging at the $\mathrm{X}$ point. Comparing (c) and (d), the inclusion of $(J_2,J_3,J_c)$ leads to a downward shift in the energy range and a softening of the peaks.