Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (2024)

PrabhakarSchool of Physical Sciences, National Institute of Science Education and Research, a CI of Homi Bhabha National Institute, Jatni 752050, India  Subhajyoti PalSchool of Physical Sciences, National Institute of Science Education and Research, a CI of Homi Bhabha National Institute, Jatni 752050, India  Umesh KumarDepartment of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA  Manoranjan KumarS. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India  Anamitra Mukherjeeanamitra@niser.ac.inSchool of Physical Sciences, National Institute of Science Education and Research, a CI of Homi Bhabha National Institute, Jatni 752050, India

(October 19, 2024; October 19, 2024)

Abstract

We theoretically investigate the resonant inelastic X-ray scattering (RIXS) spectra in a quasi-1D chain of weakly coupled frustrated spin-1/2 trimers, as realized in Na2Cu3Ge4O12, with Cu d9superscript𝑑9d^{9}italic_d start_POSTSUPERSCRIPT 9 end_POSTSUPERSCRIPT 1/2 spins. We compute multi-spin correlations contributing to spin-conserving (SC) and spin non-conserving (NSC) RIXS cross-sections using ultra-short core-hole lifetime expansion within the Kramer-Heisenberg formalism. These excitations involve flipping spins of up to three spin-1/2 trimers and include the inelastic neutron scattering (INS) single spin-flip excitations in the lowest order of the NSC channel. We identify the fractionalization of two coupled frustrated trimers in terms of spinons, doublons, and quartons in the spectra evaluated using exact diagonalization, complementing prior studies single spin-spin flip excitation in inelastic neutron scattering. Specifically, we uncover two new high-energy modes at ω2.4J1𝜔2.4subscript𝐽1\omega\approx 2.4J_{1}italic_ω ≈ 2.4 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 3.0J13.0subscript𝐽13.0J_{1}3.0 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in the NSC and SC channels that are accessible at the Cu K𝐾Kitalic_K-edge and L𝐿Litalic_L-edge RIXS spectra, which were missing in the INS study. This, therefore, provides pathways to uncover all the possible excitations in coupled trimers. Our work opens new opportunities for understanding the nature of fractionalization and RIXS spectra of frustrated, low-dimensional spin chains.

I Introduction

Since the breakthrough discovery of high-temperature superconductivity in cuprates Dagotto (1994a), cuprates have continued to be at the forefront of research in condensed matter physics. These materials have undergone extensive experimental and theoretical investigation, yet they continue to yield surprising behavior. Alongside their manifestation in 2D geometries SchüttlerandFedro (1992); SimónandAligia (1993); Feineretal. (1996); BelinicherandChernyshev (1994), cuprates manifest in one-dimensional (1D) chain Chenetal. (2021); Kimetal. (1996) and ladder structures Dagottoetal. (1992). The 1D geometries exhibit distinctive behaviors and serve as a platform for exploring many-body physics. Magnetism, in one dimension, is strongly influenced by quantum fluctuations and has been thoroughly investigated since the beginning of quantum mechanics. The Bethe ansatz Bethe (1931) based solution of the 1D Heisenberg antiferromagnetic chain (HAC) is a cornerstone result and has deepened our understanding of low dimensional interacting systems Karabachetal. (1997); Karbachetal. (1998). However, it is well-known that the integrability of the HAC is destroyed once longer range spin-interactions are introduced. One such system of interest is the spin-1/2 trimer chain Beraetal. (2017); HidaandAffleck (2005); Chengetal. (2022a), a 1D arrangement of three spin unit cells that can exhibit unique quantum phenomena. These trimer chains are known to exhibit intriguing properties such as fractionalized excitations and exotic magnetic states.Recently, a new family of cuprates X2Cu3Ge4-xSixO12 with X=Li, Na has been discoveredMoetal. (2006). The x=0 compound consists of [Cu3O8]-10 spin-trimers, with each Cu acting as a spin S=1/2. These trimers are periodically arranged as shown in Fig.1 (a) and are coupled by magnetic exchanges in addition to magnetic exchanges coupling spins within the trimer. Continued experimental effort Yasuietal. (2014), and detailed materials theoretic modeling has revealed an effective model of frustrated trimers coupled with magnetic exchanges shown in Fig.1 (b).

A recent study of inelastic neutron scattering (INS) experiment on Na2Cu3Ge4O12 revealed signature of fractional excitation termed as spinon, doublon and quarton modes Beraetal. (2022). However, higher-order spin excitations have yet to be reported due to the limitations in the energy range of the INS.Apart from new high energy features, current theoretical analysis has that have so far reported single spin-flip excitations only. The contributions of multi-spin excitations in the energy range of dynamical (single-spin-flip) excitations have remained open for frustrated spin-trimer chains. The latter is important in constraining sum-rules, as has been done for HAC recentlyMourigaletal. (2013). Resonant Inelastic X-ray Scattering (RIXS) has emerged as a standard probe for measuring multi-spin excitations Amentetal. (2011); den Brink (2007); vanden BrinkandvanVeenendaal (2005); Pengetal. (2017); Braicovichetal. (2010); Amentetal. (2009). We thus focus of the theory of RIXS-inspired computation of multi-spin correlation functions.

Resonant Inelastic X-ray Scattering (RIXS)Amentetal. (2011); Gel’mukhanovetal. (2021); deGrootetal. (2024); den Brink (2007); vanden BrinkandvanVeenendaal (2005); Pengetal. (2017); Braicovichetal. (2010); Amentetal. (2009) has become an invaluable, powerful spectroscopic tool for probing complex materials’ electronic structure, magnetic properties, and lattice dynamics. By tuning the incident X-ray energy to an absorption edge, RIXS provides element-specific orbital-selective insights into various excitations, ranging from charge transfer and spin waves to phonon and orbital excitations. This makes it particularly useful for studying low-dimensional systems where interactions between electronic, magnetic, and lattice degrees of freedom are often pronounced. RIXS has enabled the detection of multi-spinon excitation Klauseretal. (2011); Schlappaetal. (2018), spin-orbital separation Schlappaetal. (2012); Wohlfeldetal. (2013) in antiferromagnetic Heisenberg material Sr2CuO3 and high energy spin excitations in the quantum spin liquid candidate Zn-substituted barlowite Smahaetal. (2023). In recent years, the improved resolution in the meV scale has opened new opportunities to explore magnetism in materials such as Na2Cu3Ge4O12Zhouetal. (2022).The Kramers-Heisenberg (KH) formalism Amentetal. (2011), used to simulate the RIXS cross-section, is complex and makes interpreting RIXS data difficult. Nevertheless, significant progress has been made in exploring quantum magnets with RIXS. The ultrashort core-hole lifetime (UCL) expansion in RIXS expands the cross-section into spin-conserving (SC) and non-spin-conserving (NSC) channels for the K𝐾Kitalic_K and L𝐿Litalic_L-edge RIXS respectivelyKumaretal. (2022); Martinellietal. (2022); Jiaetal. (2016); Kumaretal. (2019); Forteetal. (2008); IgarashiandNagao (2012).For the cuprates, usually at L𝐿Litalic_L-edge, J1/Γ<1subscript𝐽1Γ1J_{1}/\Gamma<1italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / roman_Γ < 1, where ΓΓ\Gammaroman_Γ is the inverse of the core-hole life time. Due to this, one can expand the RIXS cross-section in orders of 1/Γ1Γ1/\Gamma1 / roman_Γ. Due to richer statistics in RIXS, higher-order corrections can be experimentally observedKumaretal. (2022). The NSC channel of RIXS is typically considered analogous to the INS probe for the lowest order of the UCL expansion, with higher-order contributions remaining largely unexplored in the literature. On the other hand, the SC channel successfully determines the four spinons Kumaretal. (2018); Schlappaetal. (2018) in 1D cuprates and multi-magnons Robartsetal. (2021); Paletal. (2023), multi-triplons Notbohmetal. (2007); Windtetal. (2001) in 2D cuprates which are inaccessible in INS experiments.

Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (1)

In this work, we investigate resonant inelastic X-ray scattering (RIXS) in quasi-1D spin-1/2 trimer chains, focusing on Na2Cu3Ge4O12subscriptNa2subscriptCu3subscriptGe4subscriptO12\text{Na}_{2}\text{Cu}_{3}\text{Ge}_{4}\text{O}_{12}Na start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Cu start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Ge start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT O start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT. Through Lanczos diagonalization and Fock-space Recursive Green’s function (F-RGF) technique, we uncover a richer fractionalization pattern than previously reported. We identify two new high-energy modes in both the NSC and SC channels. We provide a detailed understanding of the single and the multiple spin-flip processes in terms of spinons, doublons, and quartons. We also report weakly dispersing fractionalized modes in multi-spin correlations.

The paper is structured as follows. In Sec. II, we first introduce the spin model for the 1D antiferromagnetic Heisenberg trimer chain investigated in this paper. We then explain the response functions used for the corrections in the UCL expansion of the RIXS cross-section in SC as well as NSC channels and provide the expression for various operators in detail. We present our results in Sec. III and conclude the paper in Sec. IV.

II Hamiltonian & RIXS cross-section

II.1 Hamiltonian

Na2Cu3Ge4O12subscriptNa2subscriptCu3subscriptGe4subscriptO12\text{Na}_{2}\text{Cu}_{3}\text{Ge}_{4}\text{O}_{12}Na start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Cu start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Ge start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT O start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT is known to be the host of the frustrated spin-half antiferromagnetic Heisenberg chain of trimers Yasuietal. (2014). The effective spin HamiltonianBeraetal. (2022) shown in Fig.1 (b) is given by,

H=r=0N/31[J1(Sr,aSr,b+Sr,bSr,c)+J2Sr,cSr+1,a+J3Sr,aSr,c].𝐻superscriptsubscript𝑟0𝑁31delimited-[]subscript𝐽1subscript𝑆𝑟𝑎subscript𝑆𝑟𝑏subscript𝑆𝑟𝑏subscript𝑆𝑟𝑐subscript𝐽2subscript𝑆𝑟𝑐subscript𝑆𝑟1𝑎subscript𝐽3subscript𝑆𝑟𝑎subscript𝑆𝑟𝑐\begin{split}H=&\sum_{r=0}^{N/3-1}[J_{1}(\vec{S}_{r,a}\cdot\vec{S}_{r,b}+\vec{%S}_{r,b}\cdot\vec{S}_{r,c})\\&+J_{2}\vec{S}_{r,c}\cdot\vec{S}_{r+1,a}+J_{3}\vec{S}_{r,a}\cdot\vec{S}_{r,c}]%.\end{split}start_ROW start_CELL italic_H = end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_r = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N / 3 - 1 end_POSTSUPERSCRIPT [ italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_a end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_b end_POSTSUBSCRIPT + over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_b end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_c end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_c end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r + 1 , italic_a end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_a end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_c end_POSTSUBSCRIPT ] . end_CELL end_ROW(1)

The unit cell consists of three spins, as shown in Fig.1, and summation r𝑟ritalic_r is over the unit cell. Where Sr,μsubscript𝑆𝑟𝜇\vec{S}_{r,\mu}over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_r , italic_μ end_POSTSUBSCRIPT is the spin operator at μthsuperscript𝜇𝑡\mu^{th}italic_μ start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT(=a,b,c) site of rthsuperscript𝑟𝑡r^{th}italic_r start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT trimer. J1subscript𝐽1J_{1}italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT,J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT(=αJ1)absent𝛼subscript𝐽1(=\alpha J_{1})( = italic_α italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and J3subscript𝐽3J_{3}italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT(=βJ1)absent𝛽subscript𝐽1(=\beta J_{1})( = italic_β italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) are the nearest-neighbor intra trimer, nearest-neighbor inter trimer and next-nearest-neighbor intra-trimer interaction respectively. β𝛽\betaitalic_β is a frustration parameter for isolated trimers. From the materials analysis it is known that both α𝛼\alphaitalic_α and β𝛽\betaitalic_β are equal to 0.18 Beraetal. (2022). Thus, for the present material, we are at the limit of weak frustration and weak inter-trimer coupling.

Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (2)

II.2 RIXS cross-section

RIXS is a photon-in photon-out process. The RIXS process measures the scattering cross-section of an incoming X-ray whose energy is tuned to the energy difference between specific core atomic energy level and valence band of a material under consideration. For transition metal oxides, the incoming X-ray energy can be made resonant with the core levels of either the transition metal atom or Oxygen in a transition metal oxide. Depending on incident energy, the X-ray can cause a photo-excitation from 1s1𝑠1s1 italic_s to 4p4𝑝4p4 italic_p (K𝐾Kitalic_K-edge) or 2p2𝑝2p2 italic_p to 3d3𝑑3d3 italic_d (L𝐿Litalic_L-edge). It has been well-established that due to the spin-orbit interaction in the 2p state, the L𝐿Litalic_L-edge can probe spin angular momentum non-conserving spin excitation processes Forteetal. (2008); Amentetal. (2009), while K𝐾Kitalic_K-edge probes only spin angular momentum conserving spin-excitations. We note that the L𝐿Litalic_L-edge can also probe spin-conserving excitations as well.For an incoming photon with momentum qinsubscript𝑞inq_{\text{in}}italic_q start_POSTSUBSCRIPT in end_POSTSUBSCRIPT, and an outgoing photon with momentum qoutsubscript𝑞outq_{\text{out}}italic_q start_POSTSUBSCRIPT out end_POSTSUBSCRIPT, the experiment involves measuring the intensity of the scattered radiation as a function of energy loss and momentum transfer.

In the present work, we envisage a RIXS (L𝐿Litalic_L-edge) experiment at the Cu atoms in the spin-trimer that can probe both the NSC and SC channels. Hence, we investigate the spin Hamiltonian given by Eq.(1) within the UCL approximation of the Kramer-Heisenberg (KH) formalism. RIXS intensity decomposed into correlation functions in spin-conserving (SC) and non-spin-conserving (NSC) channels in UCL expansion. The RIXS cross-section is given by KH formalism; IRIXS=f|f|Dout𝒪Din|g|2δ(EfEgω)subscript𝐼RIXSsubscript𝑓superscriptquantum-operator-product𝑓subscript𝐷out𝒪subscript𝐷in𝑔2𝛿subscript𝐸𝑓subscript𝐸𝑔𝜔I_{\text{RIXS}}=\sum_{f}|\langle f|D_{\text{out}}\mathcal{O}D_{\text{in}}|g%\rangle|^{2}\delta(E_{f}-E_{g}-\omega)italic_I start_POSTSUBSCRIPT RIXS end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | italic_D start_POSTSUBSCRIPT out end_POSTSUBSCRIPT caligraphic_O italic_D start_POSTSUBSCRIPT in end_POSTSUBSCRIPT | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_δ ( italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT - italic_ω ). Here, |g(|f)ket𝑔ket𝑓|g\rangle(|f\rangle)| italic_g ⟩ ( | italic_f ⟩ ) are the ground (final) states from the Hamiltonian H0subscript𝐻0H_{0}italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT with energy Eg(Ef)subscript𝐸𝑔subscript𝐸𝑓E_{g}~{}(E_{f})italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ). q(=qoutqin)annotated𝑞absentsubscript𝑞outsubscript𝑞inq(=q_{\text{out}}-q_{\text{in}})italic_q ( = italic_q start_POSTSUBSCRIPT out end_POSTSUBSCRIPT - italic_q start_POSTSUBSCRIPT in end_POSTSUBSCRIPT ) and ω𝜔\omegaitalic_ω is the momentum transfer and the energy loss respectively. Din(out)subscript𝐷in(out)D_{\text{in(out)}}italic_D start_POSTSUBSCRIPT in(out) end_POSTSUBSCRIPT is the dipole operator and 𝒪𝒪\mathcal{O}caligraphic_O accounts for the evolution of the system in the presence of the core hole. We refer to recent literature for a detailed exposition of the simplification of the cross-section (See Ref.Kumaretal. (2022)). Following the literatureKumaretal. (2022); Jiaetal. (2016), we employ ultra-short core-hole lifetime approximation. This introduces a broadening factor (ΓΓ\Gammaroman_Γ), which is the inverse of the core-hole lifetime. The RIXS intensity IRIXSlχlNSC(q,ω)+lχlSC(q,ω)proportional-tosubscript𝐼RIXSsubscript𝑙superscriptsubscript𝜒𝑙NSC𝑞𝜔subscript𝑙superscriptsubscript𝜒𝑙SC𝑞𝜔I_{\text{RIXS}}\propto\sum_{l}\chi_{l}^{\text{NSC}}(q,\omega)+\sum_{l}\chi_{l}%^{\text{SC}}(q,\omega)italic_I start_POSTSUBSCRIPT RIXS end_POSTSUBSCRIPT ∝ ∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT NSC end_POSTSUPERSCRIPT ( italic_q , italic_ω ) + ∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SC end_POSTSUPERSCRIPT ( italic_q , italic_ω ), the label ‘l𝑙litalic_l’ refers to the order of expansion in the inverse of the core-hole lifetime, as discussed below. The proportionality constant involves polarization-dependent matrix elements emerging from the dipole operators and differs for the NSC and SC channels Amentetal. (2009). We provide the detailed derivation RIXS cross-section in Appendix A.

II.2.1 Non spin conserving (NSC) channel

The lthsuperscript𝑙𝑡l^{th}italic_l start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT order terms in the UCL expansion in the non-spin-conserving channel is,

χlNSC(q,ω)=1Γ2l+2f|f|1Ni=1NeιqriOi,lNSC|g|2×δ(EfEgω)superscriptsubscript𝜒𝑙𝑁𝑆𝐶𝑞𝜔1superscriptΓ2𝑙2subscript𝑓superscriptquantum-operator-product𝑓1𝑁superscriptsubscript𝑖1𝑁superscript𝑒𝜄𝑞subscript𝑟𝑖superscriptsubscript𝑂𝑖𝑙𝑁𝑆𝐶𝑔2𝛿subscript𝐸𝑓subscript𝐸𝑔𝜔\begin{split}\chi_{l}^{NSC}(q,\omega)=\frac{1}{\Gamma^{2l+2}}&\sum_{f}\left|%\langle f|\frac{1}{\sqrt{N}}\sum_{i=1}^{N}e^{\iota qr_{i}}O_{i,l}^{NSC}|g%\rangle\right|^{2}\\&\times\delta\left(E_{f}-E_{g}-\omega\right)\end{split}start_ROW start_CELL italic_χ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT ( italic_q , italic_ω ) = divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 2 italic_l + 2 end_POSTSUPERSCRIPT end_ARG end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_ι italic_q italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_O start_POSTSUBSCRIPT italic_i , italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL × italic_δ ( italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT - italic_ω ) end_CELL end_ROW(2)

Where |gket𝑔\ket{g}| start_ARG italic_g end_ARG ⟩ (|fket𝑓\ket{f}| start_ARG italic_f end_ARG ⟩) is the ground (final) state with energy Egsubscript𝐸𝑔E_{g}italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT (Efsubscript𝐸𝑓E_{f}italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT) of H𝐻Hitalic_H, and summation i𝑖iitalic_i is over the sites.Schematics of connectivity in the various RIXS scattering operators Oi,lNSC/SCsuperscriptsubscript𝑂𝑖𝑙𝑁𝑆𝐶𝑆𝐶O_{i,l}^{NSC/SC}italic_O start_POSTSUBSCRIPT italic_i , italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N italic_S italic_C / italic_S italic_C end_POSTSUPERSCRIPT are shown in Fig.2. Panel (a) (panel (b)) depicts the connectivity for the first order (second order) in the NSC and SC channels. The connectivity of the RIXS scattering operators arising from the UCL expansion for the NSC and SC channels are identical. The difference between the processes originates from spin-flip at the core-hole creation site (marked in red in Fig.2) in the NSC channel.

Zeroth order: 0thsuperscript0𝑡0^{th}0 start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT order term in NSC channel,

Oi,0NSC=Sixsubscriptsuperscript𝑂𝑁𝑆𝐶𝑖0subscriptsuperscript𝑆𝑥𝑖O^{NSC}_{i,0}=S^{x}_{i}italic_O start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 0 end_POSTSUBSCRIPT = italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT(3)

First order: 1stsuperscript1𝑠𝑡1^{st}1 start_POSTSUPERSCRIPT italic_s italic_t end_POSTSUPERSCRIPT order correction in NSC channel,

if ia𝑖𝑎i\in aitalic_i ∈ italic_a;

Oi,1NSC=J1Six(SiSi+1)+J2Six(SiSi1)+J3Six(SiSi+2)subscriptsuperscript𝑂𝑁𝑆𝐶𝑖1subscript𝐽1subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽2subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽3subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{NSC}_{i,1}=&J_{1}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i+1})+J_%{2}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i-1})\\&+J_{3}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i+2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(4)

if ib𝑖𝑏i\in bitalic_i ∈ italic_b;

Oi,1NSC=J1Six(SiSi+1)+J1Six(SiSi1)subscriptsuperscript𝑂𝑁𝑆𝐶𝑖1subscript𝐽1subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽1subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1O^{NSC}_{i,1}=J_{1}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i+1})+J_{1}{S}^{x}_{i}%(\vec{S}_{i}\cdot\vec{S}_{i-1})italic_O start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT )(5)

if ic𝑖𝑐i\in citalic_i ∈ italic_c;

Oi,1NSC=J1Six(SiSi1)+J2Six(SiSi+1)+J3Six(SiSi2)subscriptsuperscript𝑂𝑁𝑆𝐶𝑖1subscript𝐽1subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽2subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽3subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{NSC}_{i,1}=&J_{1}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i-1})+J_%{2}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i+1})\\&+J_{3}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i-2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(6)

Second order: 2ndsuperscript2𝑛𝑑2^{nd}2 start_POSTSUPERSCRIPT italic_n italic_d end_POSTSUPERSCRIPT order correction in NSC channel,

if ia𝑖𝑎i\in aitalic_i ∈ italic_a;

Oi,2NSC=J1J2Six(SiSi1)(SiSi+1)+J1J3Six(SiSi+1)(SiSi+2)+J2J3Six(SiSi1)(SiSi+2)subscriptsuperscript𝑂𝑁𝑆𝐶𝑖2subscript𝐽1subscript𝐽2subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽1subscript𝐽3subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2subscript𝐽2subscript𝐽3subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{NSC}_{i,2}=&J_{1}J_{2}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i-1%})(\vec{S}_{i}\cdot\vec{S}_{i+1})+J_{1}J_{3}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S%}_{i+1})\\&(\vec{S}_{i}\cdot\vec{S}_{i+2})+J_{2}J_{3}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}%_{i-1})(\vec{S}_{i}\cdot\vec{S}_{i+2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(7)

if ib𝑖𝑏i\in bitalic_i ∈ italic_b;

Oi,2NSC=J1J1Six(SiSi+1)(SiSi1)subscriptsuperscript𝑂𝑁𝑆𝐶𝑖2subscript𝐽1subscript𝐽1subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖1O^{NSC}_{i,2}=J_{1}J_{1}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i+1})(\vec{S}_{i}%\cdot\vec{S}_{i-1})italic_O start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT )(8)

if ic𝑖𝑐i\in citalic_i ∈ italic_c;

Oi,2NSC=J1J2Six(SiSi+1)(SiSi1)+J1J3Six(SiSi1)(SiSi2)+J2J3Six(SiSi+1)(SiSi2)subscriptsuperscript𝑂𝑁𝑆𝐶𝑖2subscript𝐽1subscript𝐽2subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽1subscript𝐽3subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2subscript𝐽2subscript𝐽3subscriptsuperscript𝑆𝑥𝑖subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{NSC}_{i,2}=&J_{1}J_{2}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}_{i+1%})(\vec{S}_{i}\cdot\vec{S}_{i-1})+J_{1}J_{3}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S%}_{i-1})\\&(\vec{S}_{i}\cdot\vec{S}_{i-2})+J_{2}J_{3}{S}^{x}_{i}(\vec{S}_{i}\cdot\vec{S}%_{i+1})(\vec{S}_{i}\cdot\vec{S}_{i-2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 2 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(9)

II.2.2 Spin conserving (SC) channel

The lthsuperscript𝑙𝑡l^{th}italic_l start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT order terms in the UCL expansion in the spin-conserving channelis,

χlSC(q,ω)=1Γ2l+2f|f|1Ni=1NeιqriOi,lSC|g|2×δ(EfEgω)superscriptsubscript𝜒𝑙𝑆𝐶𝑞𝜔1superscriptΓ2𝑙2subscript𝑓superscriptquantum-operator-product𝑓1𝑁superscriptsubscript𝑖1𝑁superscript𝑒𝜄𝑞subscript𝑟𝑖superscriptsubscript𝑂𝑖𝑙𝑆𝐶𝑔2𝛿subscript𝐸𝑓subscript𝐸𝑔𝜔\begin{split}\chi_{l}^{SC}(q,\omega)=\frac{1}{\Gamma^{2l+2}}&\sum_{f}\left|%\langle f|\frac{1}{\sqrt{N}}\sum_{i=1}^{N}e^{\iota qr_{i}}O_{i,l}^{SC}|g%\rangle\right|^{2}\\&\times\delta\left(E_{f}-E_{g}-\omega\right)\end{split}start_ROW start_CELL italic_χ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT ( italic_q , italic_ω ) = divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 2 italic_l + 2 end_POSTSUPERSCRIPT end_ARG end_CELL start_CELL ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_ι italic_q italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_O start_POSTSUBSCRIPT italic_i , italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL × italic_δ ( italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT - italic_ω ) end_CELL end_ROW(10)

Where |gket𝑔\ket{g}| start_ARG italic_g end_ARG ⟩ (|fket𝑓\ket{f}| start_ARG italic_f end_ARG ⟩) is the ground (final) state with energy Egsubscript𝐸𝑔E_{g}italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT (Efsubscript𝐸𝑓E_{f}italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT) of H𝐻Hitalic_H and summation i𝑖iitalic_i is over the sites. Operators belonging to each order are written explicitly below.

First order:

if ia𝑖𝑎i\in aitalic_i ∈ italic_a;

Oi,1SC=J1(SiSi+1)+J2(SiSi1)+J3(SiSi+2)subscriptsuperscript𝑂𝑆𝐶𝑖1subscript𝐽1subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽2subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽3subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{SC}_{i,1}=&J_{1}(\vec{S}_{i}\cdot\vec{S}_{i+1})+J_{2}(\vec{S}_%{i}\cdot\vec{S}_{i-1})\\&+J_{3}(\vec{S}_{i}\cdot\vec{S}_{i+2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(11)

if ib𝑖𝑏i\in bitalic_i ∈ italic_b;

Oi,1SC=J1(SiSi+1)+J1(SiSi1)subscriptsuperscript𝑂𝑆𝐶𝑖1subscript𝐽1subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽1subscript𝑆𝑖subscript𝑆𝑖1O^{SC}_{i,1}=J_{1}(\vec{S}_{i}\cdot\vec{S}_{i+1})+J_{1}(\vec{S}_{i}\cdot\vec{S%}_{i-1})italic_O start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT )(12)

if ic𝑖𝑐i\in citalic_i ∈ italic_c;

Oi,1SC=J2(SiSi+1)+J1(SiSi1)+J3(SiSi2)subscriptsuperscript𝑂𝑆𝐶𝑖1subscript𝐽2subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽1subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽3subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{SC}_{i,1}=&J_{2}(\vec{S}_{i}\cdot\vec{S}_{i+1})+J_{1}(\vec{S}_%{i}\cdot\vec{S}_{i-1})\\&+J_{3}(\vec{S}_{i}\cdot\vec{S}_{i-2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(13)

Second order:

if ia𝑖𝑎i\in aitalic_i ∈ italic_a;

Oi,2SC=J1J2(SiSi+1)(SiSi1)+J1J3(SiSi+1)(SiSi+2)+J2J3(SiSi1)(SiSi+2)subscriptsuperscript𝑂𝑆𝐶𝑖2subscript𝐽1subscript𝐽2subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽1subscript𝐽3subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2subscript𝐽2subscript𝐽3subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{SC}_{i,2}=&J_{1}J_{2}(\vec{S}_{i}\cdot\vec{S}_{i+1})(\vec{S}_{%i}\cdot\vec{S}_{i-1})+J_{1}J_{3}(\vec{S}_{i}\cdot\vec{S}_{i+1})\\&(\vec{S}_{i}\cdot\vec{S}_{i+2})+J_{2}J_{3}(\vec{S}_{i}\cdot\vec{S}_{i-1})(%\vec{S}_{i}\cdot\vec{S}_{i+2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(14)

if ib𝑖𝑏i\in bitalic_i ∈ italic_b;

Oi,2SC=J1J1(SiSi+1)(SiSi1)subscriptsuperscript𝑂𝑆𝐶𝑖2subscript𝐽1subscript𝐽1subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖1O^{SC}_{i,2}=J_{1}J_{1}(\vec{S}_{i}\cdot\vec{S}_{i+1})(\vec{S}_{i}\cdot\vec{S}%_{i-1})italic_O start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT )(15)

if ic𝑖𝑐i\in citalic_i ∈ italic_c;

Oi,2SC=J1J2(SiSi+1)(SiSi1)+J1J3(SiSi1)(SiSi2)+J2J3(SiSi+1)(SiSi2)subscriptsuperscript𝑂𝑆𝐶𝑖2subscript𝐽1subscript𝐽2subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖1subscript𝐽1subscript𝐽3subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2subscript𝐽2subscript𝐽3subscript𝑆𝑖subscript𝑆𝑖1subscript𝑆𝑖subscript𝑆𝑖2\begin{split}O^{SC}_{i,2}=&J_{1}J_{2}(\vec{S}_{i}\cdot\vec{S}_{i+1})(\vec{S}_{%i}\cdot\vec{S}_{i-1})+J_{1}J_{3}(\vec{S}_{i}\cdot\vec{S}_{i-1})\\&(\vec{S}_{i}\cdot\vec{S}_{i-2})+J_{2}J_{3}(\vec{S}_{i}\cdot\vec{S}_{i+1})(%\vec{S}_{i}\cdot\vec{S}_{i-2})\end{split}start_ROW start_CELL italic_O start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , 2 end_POSTSUBSCRIPT = end_CELL start_CELL italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 2 end_POSTSUBSCRIPT ) + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) ( over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i - 2 end_POSTSUBSCRIPT ) end_CELL end_ROW(16)

III Results

Within the UCL approximation in the KH formalism, the RIXS spectra are mapped to a set of correlation functions in the SC and NSC channels, as previously described. We present the responses up to the second-order corrections of the UCL approximation. RIXS intensity is calculated by computing individual multi-spin correlation functions in the SC and NSC channels using the F-RGF techniquePrabhakarandMukherjee (2023) on 18 sites containing six trimer unit cells and on 24 sites using Lanczos-based diagonalization Dagotto (1994b) with eight trimer unit cells for J1=1.0subscript𝐽11.0J_{1}=1.0italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1.0, α=0.18𝛼0.18\alpha=0.18italic_α = 0.18, and β=0.18𝛽0.18\beta=0.18italic_β = 0.18. We employ periodic boundary conditions in both cases. The real-space correlation functions are Fourier transformed to compute the momentum-dependent contributions to the RIXS intensity for all orders discussed in the previous section. The results of both methods are identical within numerical resolution for the same system size (N=18𝑁18N=18italic_N = 18). We present the Lanczos results on N=24𝑁24N=24italic_N = 24.

Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (3)

III.1 Non-spin conserving channel

The zeroth-order term within the NSC channel corresponds to the conventional dynamical susceptibility typically probed in inelastic neutron scattering experiments. In Fig.3 (a), the RIXS intensity is calculated using equations (2) and (3). This plot reveals five distinct features. The lowest feature, ‘A’, is a gapless excitation extending up to ω/J10.3similar-to𝜔subscript𝐽10.3\omega/J_{1}\sim 0.3italic_ω / italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∼ 0.3. This is followed by four distinct gapped excitations centered around ω/J1=0.9𝜔subscript𝐽10.9\omega/J_{1}=0.9italic_ω / italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.9, ω/J1=1.5𝜔subscript𝐽11.5\omega/J_{1}=1.5italic_ω / italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1.5, ω/J1=2.4𝜔subscript𝐽12.4\omega/J_{1}=2.4italic_ω / italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2.4, and ω/J1=3.0𝜔subscript𝐽13.0\omega/J_{1}=3.0italic_ω / italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 3.0 labeled by B, C, D, and E respectively. To understand these features, we begin by analyzing the spectrum of the isolated trimer Hamiltonian. The trimer Hamiltonian is as follows:

Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (4)
HTrimer=J1SaSb+J1SbSc+J3SaScsubscript𝐻𝑇𝑟𝑖𝑚𝑒𝑟subscript𝐽1subscript𝑆𝑎subscript𝑆𝑏subscript𝐽1subscript𝑆𝑏subscript𝑆𝑐subscript𝐽3subscript𝑆𝑎subscript𝑆𝑐H_{Trimer}=J_{1}\vec{S}_{a}\cdot\vec{S}_{b}+J_{1}\vec{S}_{b}\cdot\vec{S}_{c}+J%_{3}\vec{S}_{a}\cdot\vec{S}_{c}italic_H start_POSTSUBSCRIPT italic_T italic_r italic_i italic_m italic_e italic_r end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⋅ over→ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT

For the unfrustrated trimer we set J3=0subscript𝐽30J_{3}=0italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 in Htrimersubscript𝐻𝑡𝑟𝑖𝑚𝑒𝑟H_{trimer}italic_H start_POSTSUBSCRIPT italic_t italic_r italic_i italic_m italic_e italic_r end_POSTSUBSCRIPT. The three exact eigenvalues are ES=J1subscript𝐸𝑆subscript𝐽1E_{S}=-J_{1}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = - italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, ED=0subscript𝐸𝐷0E_{D}=0italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = 0 and EQ=0.5J1subscript𝐸𝑄0.5subscript𝐽1E_{Q}=0.5J_{1}italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 0.5 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The eigenstates in the |ST,SzTketsuperscript𝑆𝑇subscriptsuperscript𝑆𝑇𝑧|S^{T},S^{T}_{z}\rangle| italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⟩ representation are shown in Fig.4 below the schematic of the unfrustrated trimer, where STsuperscript𝑆𝑇S^{T}italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT and SzTsubscriptsuperscript𝑆𝑇𝑧S^{T}_{z}italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT are respectively, the total spin quantum number and total magnetic quantum number. The corresponding eigenvalues are shown in the middle column. The nomenclature of the states was introduced in literatureBeraetal. (2022). The lowest energy state is doubly degenerate, with the energy of ES=J1subscript𝐸𝑆subscript𝐽1E_{S}=-J_{1}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = - italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and labeled by |Sket𝑆\ket{S}| start_ARG italic_S end_ARG ⟩. The next excited state, also doubly degenerate, has an energy of ED=0subscript𝐸𝐷0E_{D}=0italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = 0 with label |Dket𝐷\ket{D}| start_ARG italic_D end_ARG ⟩. While the |Sket𝑆\ket{S}| start_ARG italic_S end_ARG ⟩ and |Dket𝐷\ket{D}| start_ARG italic_D end_ARG ⟩ have the same STsuperscript𝑆𝑇S^{T}italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT and SzTsubscriptsuperscript𝑆𝑇𝑧S^{T}_{z}italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, they differ in spatial character of the wavefunctions. From analyzing the eigenstates, we find that the states belonging to |Sket𝑆\ket{S}| start_ARG italic_S end_ARG ⟩ with SzT=1/2subscriptsuperscript𝑆𝑇𝑧12S^{T}_{z}=1/2italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 1 / 2 and SzT=1/2subscriptsuperscript𝑆𝑇𝑧12S^{T}_{z}=-1/2italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = - 1 / 2 correspond to 13(bc¯ab¯)\frac{1}{\sqrt{3}}(\uparrow\bar{bc}-\bar{ab}\uparrow)divide start_ARG 1 end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG ( ↑ over¯ start_ARG italic_b italic_c end_ARG - over¯ start_ARG italic_a italic_b end_ARG ↑ ) and 13(ab¯bc¯)\frac{1}{\sqrt{3}}(\bar{ab}\downarrow-\downarrow\bar{bc})divide start_ARG 1 end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG ( over¯ start_ARG italic_a italic_b end_ARG ↓ - ↓ over¯ start_ARG italic_b italic_c end_ARG ) respectively, where ij¯12(|ijij)\bar{ij}\equiv\frac{1}{\sqrt{2}}(|\uparrow_{i}\downarrow_{j}\rangle-\downarrow%_{i}\uparrow_{j}\rangle)over¯ start_ARG italic_i italic_j end_ARG ≡ divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | ↑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ↓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ - ↓ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ↑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ ), a singlet between spins at sites i𝑖iitalic_i and j𝑗jitalic_j. Similarly the two states belonging to |Dket𝐷\ket{D}| start_ARG italic_D end_ARG ⟩ with SzT=1/2subscriptsuperscript𝑆𝑇𝑧12S^{T}_{z}=1/2italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 1 / 2 and SzT=1/2subscriptsuperscript𝑆𝑇𝑧12S^{T}_{z}=-1/2italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = - 1 / 2 respectively correspond to a\uparrowc and a\downarrowc, where a singlet is formed between the ‘a’ and ‘c’ sites of the isolated trimer.The subsequent excited state exhibits four-fold degeneracy with the energy of EQ=0.5J1subscript𝐸𝑄0.5subscript𝐽1E_{Q}=0.5J_{1}italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 0.5 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and is denoted by |Qket𝑄\ket{Q}| start_ARG italic_Q end_ARG ⟩. The SzT=±3/2superscriptsubscript𝑆𝑧𝑇plus-or-minus32S_{z}^{T}=\pm 3/2italic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = ± 3 / 2 states belonging to |Qket𝑄\ket{Q}| start_ARG italic_Q end_ARG ⟩ correspond to fully polarized states (up/down) for all spins of the trimer. The SzT=+1/2superscriptsubscript𝑆𝑧𝑇12S_{z}^{T}=+1/2italic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = + 1 / 2 and SzT=1/2superscriptsubscript𝑆𝑧𝑇12S_{z}^{T}=-1/2italic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = - 1 / 2states correspond to σ=𝜎\sigma=\uparrowitalic_σ = ↑ and σ=𝜎\sigma=\downarrowitalic_σ = ↓ respectively in the three spin state: 16(σ+σ+σ)16𝜎𝜎𝜎\frac{1}{\sqrt{6}}(\leavevmode\hbox to5.29pt{\vbox to4.31pt{\pgfpicture%\makeatletter\hbox{\hskip 2.64294pt\lower 0.0pt\hbox to0.0pt{%\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hboxto%0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}}\hbox{\hbox{{\pgfsys@beginscope%\pgfsys@invoke{ 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}{{{}{}{}{}{}}{{{}}{{}}{{}}}{}{{}}{}{{}{}{}{}{}}{{{}}{{}}{{}}}{}{{}{}}{{}}{}{}{}{{{}}{{}}{{}}}{{{}}{{}}{{}}}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{-2.84528pt}{5.92818pt}%\pgfsys@curveto{-0.88383pt}{7.88962pt}{2.30647pt}{7.88962pt}{1.72237pt}{8.4737%2pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{%\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071%}{0.7071}{1.72237pt}{8.47372pt}\pgfsys@invoke{ }\pgfsys@invoke{ %\lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}%\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%\lxSVG@closescope\endpgfpicture}}+\sigma\leavevmode\hbox to4.29pt{\vbox to6.94%pt{\pgfpicture\makeatletter\hbox{\hskip 2.14583pt\lower 0.0pt\hbox to0.0pt{%\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hboxto%0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}}\hbox{\hbox{{\pgfsys@beginscope%\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}{{}{{}}}{{}{}}{}{{}{}}{}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%.0}{-2.14583pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb%}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b$}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}%\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%\lxSVG@closescope\endpgfpicture}}\leavevmode\hbox to4.33pt{\vbox to4.31pt{%\pgfpicture\makeatletter\hbox{\hskip 2.16377pt\lower 0.0pt\hbox to0.0pt{%\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hboxto%0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}}\hbox{\hbox{{\pgfsys@beginscope%\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}{{}{{}}}{{}{}}{}{{}{}}{}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%.0}{-2.16377pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb%}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$c$}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}%\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%\lxSVG@closescope\endpgfpicture}}\leavevmode\hbox to7.51pt{\vbox to4.28pt{%\pgfpicture\makeatletter\hbox{\hskip 3.04527pt\lower 5.72818pt\hbox to0.0pt{%\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hboxto%0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}{}{}{}}{{{}}{{}}{{}}}{}{{}}{}{{}{}{}{}{}}{{{}}{{}}{{}}}{}{{}{}}{{}}{}{}{}{{{}}{{}}{{}}}{{{}}{{}}{{}}}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{-2.84528pt}{8.56706pt}%\pgfsys@curveto{-0.14548pt}{9.80574pt}{3.02924pt}{8.62798pt}{2.76672pt}{9.2001%8pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{%\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.417}{-0.9089}{0.9089}%{0.417}{2.76671pt}{9.20018pt}\pgfsys@invoke{ }\pgfsys@invoke{ %\lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}%\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%\lxSVG@closescope\endpgfpicture}}~{})divide start_ARG 1 end_ARG start_ARG square-root start_ARG 6 end_ARG end_ARG ( italic_a italic_b italic_σ + italic_a italic_σ italic_c + italic_σ italic_b italic_c ). Here 12(|ij+|ij)\leavevmode\hbox to3.45pt{\vbox to6.6pt{\pgfpicture\makeatletter\hbox{\hskip 1%.72256pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }%\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}%\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }%\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{%\pgfsys@beginscope\pgfsys@invoke{ }{{{}}\hbox{\hbox{{\pgfsys@beginscope%\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}{{}{{}}}{{}{}}{}{{}{}}{}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%.0}{-1.72256pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb%}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}%\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%\lxSVG@closescope\endpgfpicture}}\leavevmode\hbox to4.69pt{\vbox to8.54pt{%\pgfpicture\makeatletter\hbox{\hskip 2.34525pt\lower-1.94443pt\hbox to0.0pt{%\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hboxto%0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}}\hbox{\hbox{{\pgfsys@beginscope%\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}{{}{{}}}{{}{}}{}{{}{}}{}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1%.0}{-2.34525pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb%}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }%\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$j$}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}%\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%\lxSVG@closescope\endpgfpicture}}\leavevmode\hbox to7.51pt{\vbox to2.36pt{%\pgfpicture\makeatletter\hbox{\hskip 3.04527pt\lower 8.01788pt\hbox to0.0pt{%\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}%\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}%{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hboxto%0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}{}{}{}}{{{}}{{}}{{}}}{}{{}}{}{{}{}{}{}{}}{{{}}{{}}{{}}}{}{{}{}}{{}}{}{}{}{{{}}{{}}{{}}}{{{}}{{}}{{}}}{{}}{}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{-2.84528pt}{8.21788pt}%\pgfsys@curveto{-0.88383pt}{10.17932pt}{2.30647pt}{10.17932pt}{1.72237pt}{10.7%6343pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{%\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.7071}{-0.7071}{0.7071%}{0.7071}{1.72237pt}{10.76343pt}\pgfsys@invoke{ }\pgfsys@invoke{ %\lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}%\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}%\lxSVG@closescope\endpgfpicture}}\equiv\frac{1}{\sqrt{2}}(|\uparrow_{i}%\downarrow_{j}\rangle+|\downarrow_{i}\uparrow_{j}\rangle)italic_i italic_j ≡ divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | ↑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ↓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ + | ↓ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ↑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ ) is a triplet between spins at sites i𝑖iitalic_i and j𝑗jitalic_j.We now consider the frustrated trimer. Up on switching on the J3=0.18J1subscript𝐽30.18subscript𝐽1J_{3}=0.18J_{1}italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.18 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in Htrimersubscript𝐻𝑡𝑟𝑖𝑚𝑒𝑟H_{trimer}italic_H start_POSTSUBSCRIPT italic_t italic_r italic_i italic_m italic_e italic_r end_POSTSUBSCRIPT, we find the exact eigenvalues are altered and now occur at ES=J1+J3subscript𝐸𝑆subscript𝐽1subscript𝐽3E_{S}=-J_{1}+J_{3}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = - italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, ED=3J34subscript𝐸𝐷3subscript𝐽34E_{D}=-\frac{3J_{3}}{4}italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = - divide start_ARG 3 italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG and EQ=0.5J1+J34subscript𝐸𝑄0.5subscript𝐽1subscript𝐽34E_{Q}=0.5J_{1}+\frac{J_{3}}{4}italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 0.5 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG. These altered eigenvalues are shown in the right column in Fig.4. It is clear the frustrating interaction J3subscript𝐽3J_{3}italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT raises the energy of the degenerate |Sket𝑆\ket{S}| start_ARG italic_S end_ARG ⟩ and |Qket𝑄\ket{Q}| start_ARG italic_Q end_ARG ⟩ states while lowering the energy of the degenerate |Dket𝐷\ket{D}| start_ARG italic_D end_ARG ⟩ states. Importantly, there are no level crossings or degeneracy lifting due to frustration, and hence, the unfrustrated state labels are carried over to frustrated cases.

Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (5)

In previous literature Beraetal. (2022), these three energy levels have been argued to be the dominant contribution excitations seen in INS-measured dynamical spin susceptibility for single spin-flip excitation. In this albeit simple but illuminating argument, the single spin-flip in a trimer ‘a’ or ‘c’ sites upon coupling to other trimers was expected to broaden into a gapless fractionalized spinon continuum as would happen in an antiferromagnetic ground state. Thus setting ES=0subscript𝐸𝑆0E_{S}=0italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = 0 the energy levels EDsubscript𝐸𝐷E_{D}italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT, and EQsubscript𝐸𝑄E_{Q}italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT are determined to be ω/J1=0.82𝜔subscript𝐽10.82\omega/J_{1}=0.82italic_ω / italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.82 and ω/J1=1.5𝜔subscript𝐽11.5\omega/J_{1}=1.5italic_ω / italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1.5. The locations of these features agree with the centroids of the ‘B’ and the ‘C’ features in the single spin-flip INS excitation Beraetal. (2022) and to the zeroth order of the NSC RIXS spectra shown in Fig.3 (a). These excitations were denoted as fractionalized spinon (|Sket𝑆|S\rangle| italic_S ⟩), doublon (|Dket𝐷|D\rangle| italic_D ⟩), and quarton (|Qket𝑄|Q\rangle| italic_Q ⟩) modes. While the frustrating exchange J3subscript𝐽3J_{3}italic_J start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT is small, we find that it is important to include it in the analysis (unlike the J1J2subscript𝐽1subscript𝐽2J_{1}-J_{2}italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT trimer spin chainChengetal. (2022b)) for correctly predicting the energy location of both the INS and the RIXS excitation spectra.

However, the high energy excitations in Fig.3 in the lowest order in the RIXS NSC channel, ‘D’ and ‘E’, cannot be captured with a single spin trimer. Moreover, as seen from the connectivity in the RIXS operators, the first and second order in the UCL expansion (Fig.2) couple two spin-trimers. We, therefore, provide a more accurate analysis by considering two spin trimers and show that the excitations of SC and NSC channels can be understood in a comprehensive manner.

We consider two isolated frustrated trimers with three energy levels: ESsubscript𝐸𝑆E_{S}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT, EDsubscript𝐸𝐷E_{D}italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT, and EQsubscript𝐸𝑄E_{Q}italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT individually. The Hilbert space is spanned by nine states constructed from the single trimer states |Sket𝑆|S\rangle| italic_S ⟩, |Dket𝐷|D\rangle| italic_D ⟩ and |Qket𝑄|Q\rangle| italic_Q ⟩. It easy to see that there are nine eigenvalues 2ES2subscript𝐸𝑆2E_{S}2 italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT, 2ED2subscript𝐸𝐷2E_{D}2 italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT, 2EQ2subscript𝐸𝑄2E_{Q}2 italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT, with doubly degenerate ES+EDsubscript𝐸𝑆subscript𝐸𝐷E_{S}+E_{D}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT, 2ED+EQ2subscript𝐸𝐷subscript𝐸𝑄2E_{D}+E_{Q}2 italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT and ES+EQsubscript𝐸𝑆subscript𝐸𝑄E_{S}+E_{Q}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT, yielding six distinct energy levels. In Fig.5, we show these six eigenvalues of two with circles. These eigenvalues in increasing order of magnitude are 2ES=1.91J12subscript𝐸𝑆1.91subscript𝐽12E_{S}=-1.91J_{1}2 italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = - 1.91 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, ES+ED=1.09J1subscript𝐸𝑆subscript𝐸𝐷1.09subscript𝐽1E_{S}+E_{D}=-1.09J_{1}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = - 1.09 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT,ES+EQ=0.41J1subscript𝐸𝑆subscript𝐸𝑄0.41subscript𝐽1E_{S}+E_{Q}=-0.41J_{1}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = - 0.41 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, 2ED=0.27J12subscript𝐸𝐷0.27subscript𝐽12E_{D}=-0.27J_{1}2 italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = - 0.27 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, ED+EQ=0.41J1subscript𝐸𝐷subscript𝐸𝑄0.41subscript𝐽1E_{D}+E_{Q}=0.41J_{1}italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 0.41 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 2EQ=1.09J12subscript𝐸𝑄1.09subscript𝐽12E_{Q}=1.09J_{1}2 italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 1.09 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.

Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (6)

The stars denote the eigenvalues for the coupled timers (α=0.18𝛼0.18\alpha=0.18italic_α = 0.18, β=0.18𝛽0.18\beta=0.18italic_β = 0.18). When the two frustrated trimers are coupled, the eigenvalue spectrum is altered non-trivially. Firstly, we identify a SzT=0subscriptsuperscript𝑆𝑇𝑧0S^{T}_{z}=0italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 0 eigenvalue with the lowest energy as the ground state labeled as ‘G’ with EG=2.03539J1subscript𝐸𝐺2.03539subscript𝐽1E_{G}=-2.03539J_{1}italic_E start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT = - 2.03539 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. This is followed by a state with energy close to 2ES2subscript𝐸𝑆2E_{S}2 italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT labeled as ‘A.’ Since in the spinon state of an isolated frustrated trimer, there was one spin 1/2 and a singlet, a state dominantly made out of two spinon states on individual trimers can have SZT=1,0,1superscriptsubscript𝑆𝑍𝑇101S_{Z}^{T}=-1,0,1italic_S start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = - 1 , 0 , 1 as indicated on the right of the feature ‘A’ in Fig.5.Similarly, the ‘B’ state is centered around ES+ED=1.09J1subscript𝐸𝑆subscript𝐸𝐷1.09subscript𝐽1E_{S}+E_{D}=-1.09J_{1}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = - 1.09 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, primarily composed of a doublon and a spinon. The result of coupling the trimers further results in the overlap of the energy levels 2ED=0.27J12subscript𝐸𝐷0.27subscript𝐽12E_{D}=-0.27J_{1}2 italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = - 0.27 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ES+EQ=0.41J1subscript𝐸𝑆subscript𝐸𝑄0.41subscript𝐽1E_{S}+E_{Q}=-0.41J_{1}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = - 0.41 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT resulting in a single feature ’C’. The next two energy states are broadened around ED+EQ=0.41J1subscript𝐸𝐷subscript𝐸𝑄0.41subscript𝐽1E_{D}+E_{Q}=0.41J_{1}italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 0.41 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT marked as ‘D’ and around 2EQ=1.09J12subscript𝐸𝑄1.09subscript𝐽12E_{Q}=1.09J_{1}2 italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = 1.09 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT marked as ‘E’.We note that at least two timers are needed to capture the SzT=0subscriptsuperscript𝑆𝑇𝑧0S^{T}_{z}=0italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 0 ground state. Also, we find that only five well-separated features remain apart from the ground state. Measuring the separation of the these from the ground state, we find the energy locations of the five features to be ΔωA=0.125J1Δsubscript𝜔𝐴0.125subscript𝐽1\Delta\omega_{A}=0.125J_{1}roman_Δ italic_ω start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = 0.125 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, ΔωB=0.945J1Δsubscript𝜔𝐵0.945subscript𝐽1\Delta\omega_{B}=0.945J_{1}roman_Δ italic_ω start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 0.945 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, ΔωC=1.625J1Δsubscript𝜔𝐶1.625subscript𝐽1\Delta\omega_{C}=1.625J_{1}roman_Δ italic_ω start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = 1.625 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, ΔωD=2.445J1Δsubscript𝜔𝐷2.445subscript𝐽1\Delta\omega_{D}=2.445J_{1}roman_Δ italic_ω start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT = 2.445 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ΔωE=3.125J1Δsubscript𝜔𝐸3.125subscript𝐽1\Delta\omega_{E}=3.125J_{1}roman_Δ italic_ω start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT = 3.125 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, in close agreement with the feature locations in Fig.3(a). In addition, we identify that the ‘D’ and the ‘E’ features arise dominantly from a ‘doublon and a quarton’, and two quarton excitations, respectively. We note that ΔωA=0.125J1Δsubscript𝜔𝐴0.125subscript𝐽1\Delta\omega_{A}=0.125J_{1}roman_Δ italic_ω start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = 0.125 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is expected to go to zero with increasing system size. We also see that the energy centroids of the ‘B’ and ‘C’ features are approximately equal to EDsubscript𝐸𝐷E_{D}italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT and EQsubscript𝐸𝑄E_{Q}italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT, respectively, since ES<<ED<<EQmuch-less-thansubscript𝐸𝑆subscript𝐸𝐷much-less-thansubscript𝐸𝑄E_{S}<<E_{D}<<E_{Q}italic_E start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT < < italic_E start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT < < italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT. While crude, this explains why the single frustrated trimer eigenvalues could label the INS data as discussed earlier. However, our analysis shows that there is a contribution of spinons to the B feature and that the C feature is an admixture of the spinon, doublon and quarton states.

The first order correction in the NSC channel are computed using Eq.(2) and Eq.(4),(5),(6) and shown in Fig 3 (b). Feature locations in energy in this order are the same as for the zeroth order, except that the spectral weight of the quarton modes is increased while spinon and doublon modes are softened. We have shown the second-order correction in the NSC channel in Fig.3 (c) and calculated by using Eq.(2) and Eq.(7),(8),(9). This demonstrates a similarity to the zeroth-order correction, highlighting five different features. At this order, spinon stiffens at the edge of the Brillouin zone, and quarton modes and feature ‘D’ appear throughout the whole Brillouin zone. The higher-order operators include the possibility of double spin-flip excitations at and around the core-hole creation site. This results in renormalization of the single-spin flip excitation at the core-hole site. We note that the NSC channel operators contain a single spin-flip operator multiplying scalar two spin-flip operators and hence only connect states differing by one unit of spin angular momentum. In Fig.5 on the right side, we provide the SzTsubscriptsuperscript𝑆𝑇𝑧S^{T}_{z}italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT of the ground state and the five excitations for the coupled frustrated trimers. Thus, the excitations in the NSC channel occur between states separated by ΔSzT=±1Δsubscriptsuperscript𝑆𝑇𝑧plus-or-minus1\Delta S^{T}_{z}=\pm 1roman_Δ italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = ± 1. This also implies that while the features renormalize due to two spin-flip excitations, they create only those excitation energies as in the zeroth order. Thus, the energy locations of the features remain the same in higher orders.

III.2 Spin conserving channel

We now consider the SC channel RIXS spectra. As seen from Appendix A Eq.(21), the zeroth order in the SC channel contains contributions of charge correlations. Thus, the lowest-order spin correlation term in the SC channel comes from the first-order contribution of the UCL expansion. The first-order term in the SC channel involves a double spin flip that conserveds SzTsubscriptsuperscript𝑆𝑇𝑧S^{T}_{z}italic_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT. Fig.6 (a) shows the first order correction in the SC channel computed by using Eq.(10) and (11),(12),(13). A key characteristic of this order is that the spectral weight vanishes at q=0𝑞0q=0italic_q = 0 and π𝜋\piitalic_π. At q=0𝑞0q=0italic_q = 0, matrix element ff|1Ni=1NOi,1SC|gsubscript𝑓quantum-operator-product𝑓1𝑁superscriptsubscript𝑖1𝑁superscriptsubscript𝑂𝑖1𝑆𝐶𝑔\sum_{f}\langle f|\frac{1}{\sqrt{N}}\sum_{i=1}^{N}O_{i,1}^{SC}|g\rangle∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_O start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT | italic_g ⟩ vanishes because operator Oi,1SCsuperscriptsubscript𝑂𝑖1𝑆𝐶O_{i,1}^{SC}italic_O start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT at q=0 commutes with the Hamiltonian resulting in vanishing RIXS intensity. At q=π𝑞𝜋q=\piitalic_q = italic_π, the operator can be written as i=1NeιqriOi,1SC=ievenOi,1SCioddOi,1SCsuperscriptsubscript𝑖1𝑁superscript𝑒𝜄𝑞subscript𝑟𝑖superscriptsubscript𝑂𝑖1𝑆𝐶subscript𝑖𝑒𝑣𝑒𝑛superscriptsubscript𝑂𝑖1𝑆𝐶subscript𝑖𝑜𝑑𝑑superscriptsubscript𝑂𝑖1𝑆𝐶\sum_{i=1}^{N}e^{\iota qr_{i}}O_{i,1}^{SC}=\sum_{i\in even}O_{i,1}^{SC}-\sum_{%i\in odd}O_{i,1}^{SC}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_ι italic_q italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_O start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i ∈ italic_e italic_v italic_e italic_n end_POSTSUBSCRIPT italic_O start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_i ∈ italic_o italic_d italic_d end_POSTSUBSCRIPT italic_O start_POSTSUBSCRIPT italic_i , 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT and this leads to the vanishing of matrix element which leads to the vanishing intensity.The excitation created by two spin flips is fractionalized and gives five distinct features labeled A, B, C, D, and E, as in the NSC channel. Since the operators evaluated in the SC channel are scalars, the only allowed transitions conserve SzT=0superscriptsubscript𝑆𝑧𝑇0S_{z}^{T}=0italic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = 0 between the ground and excited states. This selection rule constrains the number of states in the excitation manifold to have non-zero matrix elements with the SzT=0superscriptsubscript𝑆𝑧𝑇0S_{z}^{T}=0italic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = 0 ground state, as is apparent from the reduced number of possible states excitation states connecting to the ground state as seen on the right side of Fig. 5. We note for a single trimer, due to the selection rule, the contribution to the SC channel is identically zero for the ’A’ feature. We find a highly suppressed but finite (102)similar-toabsentsuperscript102(\sim 10^{-2})( ∼ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ) contribution for coupled two trimers at ‘A.’ It is, however, not visible in the color plot in Fig.6 (a).

Most of the spectral weight in this order appears in ‘B’ and ‘C’ features. Additionally, high-energy features ‘D’ and ‘E’, with highly suppressed contributions, also appear in the SC channel. The second-order correction in the SC channel is four spin flip terms. It is calculated by using Eq.(14),(15),(16) in Eq.(10) and depicted in Fig.6 (b). In contrast to the first-order correction in the SC channel, the’ B’ feature hardens in second-order correction and appears through the whole Brillouin zone, while ‘C’, D, and E modes exhibit the three sub-structures.

IV Conclusion

In this study, we have investigated the Resonant Inelastic X-ray Scattering (RIXS) cross-section for a quasi-1D frustrated antiferromagnetic Heisenberg model. The model parameters are chosen to mimic the quasi-1D frustrated spin timers that are weakly coupled in Na2Cu3Ge4O12. We have computed the RIXS spectra for the NSC and SC channels within the framework of the UCL expansion of the KH formalism, extending up to the second order. In this paper, we have focused only on the RIXS-induced many-body excitations.

The intra-trimer interaction in the spin-1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG trimer chain in Na2Cu3Ge4O12 is reported to be J1=20.21subscript𝐽120.21J_{1}=20.21italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 20.21 meV in recent experimental study. With this interaction strength, the lowest three features are dominantly made of fractionalized spinon, doublon, and quarton modes at approximately ω=0.3J1=6.063𝜔0.3subscript𝐽16.063\omega=0.3J_{1}=6.063italic_ω = 0.3 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 6.063 meV, ω=0.9J1=18.199𝜔0.9subscript𝐽118.199\omega=0.9J_{1}=18.199italic_ω = 0.9 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 18.199 meV, and ω=1.5J1=30𝜔1.5subscript𝐽130\omega=1.5J_{1}=30italic_ω = 1.5 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 30 meV, respectively, which is consistent with the recent work.

Our analysis further shows that the lowest feature eventually forms a gapless spinon continuum. The next two features, ‘B’ and ‘C’, have an admixture of ‘spinon and doublon’, and ‘spinon, doublon and quarton’ modes, respectively, improving on the previous reports. Moreover, the study predicts two new and significant features, D and E, which are expected to emerge at approximately ω=2.4J1=48𝜔2.4subscript𝐽148\omega=2.4J_{1}=48italic_ω = 2.4 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 48 meV and ω=3.0J1=60𝜔3.0subscript𝐽160\omega=3.0J_{1}=60italic_ω = 3.0 italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 60 meV, respectively. These are made up of a combination of ‘doublon and quarton’, and ‘two quarton’ modes respectively.We have further shown that the SC channels RIXS show spectral features at the same energy as in the NSC channel but are significantly suppressed in comparison. We also predict the existence of band-like low-energy excitations, spreading over the entire Brillouin zone in the second order of the NSC and SC channels. The energy range of these features is easily accessible to K𝐾Kitalic_K and L𝐿Litalic_L edge RIXS resolution. This, therefore, makes RIXS an ideal probe to explore these higher energy excitations.

Understanding the fractionalization of multi-spin correlations is an actively pursued area and holds the key to understanding the nature of collective excitations in correlated systems. Our results highlight the potential of RIXS to understand fractionalized excitations in quantum materials.

V Acknowledgements:

All computations were performed in the NOETHER, VIRGO high-performance clusters at NISER. AM acknowledges helpful discussions with V. Ravi Chandra at NISER.

APPENDIX

Appendix A Ultra-short core-hole lifetime expansion of Kramers-Heisenberg formalism

Here, we provide ultra-short core-hole lifetime expansion of Kramers-Heisenberg formalism. The RIXS intensity depends on the Kramers-Heisenberg scattering amplitude (Afgsubscript𝐴𝑓𝑔A_{fg}italic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT) between the initial state of the system |gket𝑔|g\rangle| italic_g ⟩, which is the ground state, and a final state |fket𝑓|f\rangle| italic_f ⟩, following the decay of the core-hole. This evolution from the |gket𝑔|g\rangle| italic_g ⟩ to the final state |fket𝑓|f\rangle| italic_f ⟩occurs through the states of the intermediate state Hamiltonian Hintsubscript𝐻𝑖𝑛𝑡H_{int}italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT which comprises of the unperturbed Hamiltonian H0subscript𝐻0H_{0}italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT before photo-excitation and perturbative corrections (Hsuperscript𝐻H^{\prime}italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT), from the core hole induced by X-ray photons. Similar to the initials state, the possible final states |fket𝑓|f\rangle| italic_f ⟩ also do not contain the core-hole. Unlike these states, however, the core hole is present during the evolution between |iket𝑖|i\rangle| italic_i ⟩ and |fket𝑓|f\rangle| italic_f ⟩ by intermediate Hintsubscript𝐻𝑖𝑛𝑡H_{int}italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT, whose eigenproblem is defined by Hint|n=En|nsubscript𝐻𝑖𝑛𝑡ket𝑛subscript𝐸𝑛ket𝑛H_{int}|n\rangle=E_{n}|n\rangleitalic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT | italic_n ⟩ = italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n ⟩. These definitions will be useful below. The final cross-section is summed over all possible final states as described below.

d2σdΩdωf|Afg|2δ(EfEgω)proportional-tosuperscript𝑑2𝜎𝑑Ω𝑑𝜔subscript𝑓superscriptsubscript𝐴𝑓𝑔2𝛿subscript𝐸𝑓subscript𝐸𝑔𝜔\frac{d^{2}\sigma}{d\Omega d\omega}\propto\sum_{f}|A_{fg}|^{2}\delta(E_{f}-E_{%g}-\omega)divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ end_ARG start_ARG italic_d roman_Ω italic_d italic_ω end_ARG ∝ ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_δ ( italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT - italic_ω )(17)

Here

Afg=ωresnf|D^|nn|D^|gωinEn+iΓsubscript𝐴𝑓𝑔subscript𝜔𝑟𝑒𝑠subscript𝑛quantum-operator-product𝑓^𝐷𝑛quantum-operator-product𝑛^𝐷𝑔subscript𝜔𝑖𝑛subscript𝐸𝑛𝑖ΓA_{fg}=\omega_{res}\sum_{n}\frac{\langle f|\hat{D}|n\rangle\langle n|\hat{D}|g%\rangle}{\omega_{in}-E_{n}+i\Gamma}italic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT italic_r italic_e italic_s end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT divide start_ARG ⟨ italic_f | over^ start_ARG italic_D end_ARG | italic_n ⟩ ⟨ italic_n | over^ start_ARG italic_D end_ARG | italic_g ⟩ end_ARG start_ARG italic_ω start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_i roman_Γ end_ARG(18)

The Kramers-Heisenberg amplitude Afgsubscript𝐴𝑓𝑔A_{fg}italic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT is defined in Eq.(18), representing the transition amplitude between the initial state |gket𝑔|g\rangle| italic_g ⟩ with energy Egsubscript𝐸𝑔E_{g}italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT (used as reference energy: Eg=0subscript𝐸𝑔0E_{g}=0italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT = 0) and the final state |fket𝑓|f\rangle| italic_f ⟩. The initial state |gket𝑔|g\rangle| italic_g ⟩ is photo-excited to an intermediate state described by the dipole operator D^^𝐷\hat{D}over^ start_ARG italic_D end_ARG, and ΓΓ\Gammaroman_Γ is a broadening factor stemming from the lifetime of the core hole. The dipole operator for the K𝐾Kitalic_K-edge and L𝐿Litalic_L-edge RIXS cases is well documented. The main approach to simplifying Afgsubscript𝐴𝑓𝑔A_{fg}italic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT is ultra-short core-hole lifetime, implying ΓΓ\Gammaroman_Γ as the largest scale in the denominator of Afgsubscript𝐴𝑓𝑔A_{fg}italic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT. The detuning of the incoming photon energy is given by ωin=ωin0ωressubscript𝜔𝑖𝑛superscriptsubscript𝜔𝑖𝑛0subscript𝜔𝑟𝑒𝑠\omega_{in}=\omega_{in}^{0}-\omega_{res}italic_ω start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - italic_ω start_POSTSUBSCRIPT italic_r italic_e italic_s end_POSTSUBSCRIPT, where ωressubscript𝜔𝑟𝑒𝑠\omega_{res}italic_ω start_POSTSUBSCRIPT italic_r italic_e italic_s end_POSTSUBSCRIPT is the energy of the target K𝐾Kitalic_K or L𝐿Litalic_L edge core-level. A part of the incoming photon energy is absorbed during the relaxation process of the core hole so that the energy of the outgoing photon is smaller than ωin0superscriptsubscript𝜔𝑖𝑛0\omega_{in}^{0}italic_ω start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT. The δ𝛿\deltaitalic_δ-function in Eq (17), ensures that the energy of the outgoing photon and core-hole relaxation respect energy conservation.

Now, by identifying a detuning parameter Δ=ωiniΓΔsubscript𝜔𝑖𝑛𝑖Γ\Delta=\omega_{in}-i\Gammaroman_Δ = italic_ω start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT - italic_i roman_Γ in Eq.(17), we expand Afgsubscript𝐴𝑓𝑔A_{fg}italic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT as, ωresΔl=0f|DHintlD|g/Δ(l)subscript𝜔𝑟𝑒𝑠Δsuperscriptsubscript𝑙0quantum-operator-product𝑓𝐷superscriptsubscript𝐻𝑖𝑛𝑡𝑙𝐷𝑔superscriptΔ𝑙\frac{\omega_{res}}{\Delta}\sum_{l=0}^{\infty}\langle f|DH_{int}^{l}D|g\rangle%/\Delta^{(l)}divide start_ARG italic_ω start_POSTSUBSCRIPT italic_r italic_e italic_s end_POSTSUBSCRIPT end_ARG start_ARG roman_Δ end_ARG ∑ start_POSTSUBSCRIPT italic_l = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ⟨ italic_f | italic_D italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT italic_D | italic_g ⟩ / roman_Δ start_POSTSUPERSCRIPT ( italic_l ) end_POSTSUPERSCRIPT.Also, for H0subscript𝐻0H_{0}italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, the Heisenberg Hamiltonian, and with the usual assumption of local core-holeForteetal. (2008), Hintsubscript𝐻𝑖𝑛𝑡H_{int}italic_H start_POSTSUBSCRIPT italic_i italic_n italic_t end_POSTSUBSCRIPT becomes H0+Hsubscript𝐻0superscript𝐻H_{0}+H^{\prime}italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, where H=ηijhihiJijSi.Sjformulae-sequencesuperscript𝐻𝜂subscript𝑖𝑗subscript𝑖superscriptsubscript𝑖subscript𝐽𝑖𝑗subscript𝑆𝑖subscript𝑆𝑗H^{\prime}=\eta\sum_{ij}h_{i}h_{i}^{\dagger}J_{ij}S_{i}.S_{j}italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_η ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT . italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. Here hisubscript𝑖h_{i}italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (hisuperscriptsubscript𝑖h_{i}^{\dagger}italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT) creates (destroys) a core hole at site i𝑖iitalic_i. In the cases of exact resonance ωin=0subscript𝜔𝑖𝑛0\omega_{in}=0italic_ω start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT = 0, we have:

Afg=ωres(iΓ)l=0f|D(H0+H)liΓlD|gsubscript𝐴𝑓𝑔subscript𝜔𝑟𝑒𝑠𝑖Γsuperscriptsubscript𝑙0quantum-operator-product𝑓𝐷superscriptsubscript𝐻0superscript𝐻𝑙𝑖superscriptΓ𝑙𝐷𝑔A_{fg}=\frac{\omega_{res}}{(i\Gamma)}\sum_{l=0}^{\infty}\langle f|D\frac{(H_{0%}+H^{\prime})^{l}}{i\Gamma^{l}}D|g\rangleitalic_A start_POSTSUBSCRIPT italic_f italic_g end_POSTSUBSCRIPT = divide start_ARG italic_ω start_POSTSUBSCRIPT italic_r italic_e italic_s end_POSTSUBSCRIPT end_ARG start_ARG ( italic_i roman_Γ ) end_ARG ∑ start_POSTSUBSCRIPT italic_l = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ⟨ italic_f | italic_D divide start_ARG ( italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT end_ARG start_ARG italic_i roman_Γ start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT end_ARG italic_D | italic_g ⟩(19)

The dipole operator D=D0+D0𝐷subscript𝐷0superscriptsubscript𝐷0D=D_{0}+D_{0}^{\dagger}italic_D = italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT , whereD^0(ϵ,ϵ)=iσ(ni,σrϵeiqoutRidipi+rϵeiqinRipidi)subscript^𝐷0italic-ϵsuperscriptitalic-ϵsubscript𝑖𝜎subscript𝑛𝑖𝜎subscript𝑟superscriptitalic-ϵsuperscript𝑒𝑖subscript𝑞𝑜𝑢𝑡subscript𝑅𝑖subscriptsuperscript𝑑𝑖subscript𝑝𝑖subscript𝑟italic-ϵsuperscript𝑒𝑖subscript𝑞𝑖𝑛subscript𝑅𝑖subscriptsuperscript𝑝𝑖subscript𝑑𝑖\hat{D}_{0}(\epsilon,\epsilon^{\prime})=\sum_{i\sigma}(n_{i,\sigma}r_{\epsilon%^{\prime}}e^{iq_{out}R_{i}}d^{\dagger}_{i}p_{i}+r_{\epsilon}e^{-iq_{in}R_{i}}p%^{\dagger}_{i}d_{i})over^ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_ϵ , italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i italic_σ end_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT italic_i , italic_σ end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q start_POSTSUBSCRIPT italic_o italic_u italic_t end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_d start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_i italic_q start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) for spin-conserving L𝐿Litalic_L-edge scattering andD^0(ϵ,ϵ)=i(SixrϵeiqoutRidipi+rϵeiqinRipidi)subscript^𝐷0italic-ϵsuperscriptitalic-ϵsubscript𝑖subscriptsuperscript𝑆𝑥𝑖subscript𝑟superscriptitalic-ϵsuperscript𝑒𝑖subscript𝑞𝑜𝑢𝑡subscript𝑅𝑖subscriptsuperscript𝑑𝑖subscript𝑝𝑖subscript𝑟italic-ϵsuperscript𝑒𝑖subscript𝑞𝑖𝑛subscript𝑅𝑖subscriptsuperscript𝑝𝑖subscript𝑑𝑖\hat{D}_{0}(\epsilon,\epsilon^{\prime})=\sum_{i}(S^{x}_{i}r_{\epsilon^{\prime}%}e^{iq_{out}R_{i}}d^{\dagger}_{i}p_{i}+r_{\epsilon}e^{-iq_{in}R_{i}}p^{\dagger%}_{i}d_{i})over^ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_ϵ , italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q start_POSTSUBSCRIPT italic_o italic_u italic_t end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_d start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - italic_i italic_q start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) for the spin non-conserving L𝐿Litalic_L-edge scattering. In the later expression, Sixsuperscriptsubscript𝑆𝑖𝑥S_{i}^{x}italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT can induce a single spin flip (leading to the spin-conserving scattering), rϵsubscript𝑟italic-ϵr_{\epsilon}italic_r start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT and rϵsubscript𝑟superscriptitalic-ϵr_{\epsilon^{\prime}}italic_r start_POSTSUBSCRIPT italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT denote the amplitude of the incoming and outgoing photons with polarization ϵitalic-ϵ\epsilonitalic_ϵ and ϵsuperscriptitalic-ϵ\epsilon^{\prime}italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, respectively. They give rise to the polarization-dependent matrix elements for SC and NSC channels. This polarization dependence is different for the SC and NSC channels. Using the fact that [D,H0]=0𝐷subscript𝐻00[D,H_{0}]=0[ italic_D , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] = 0 and the ground state energy Egsubscript𝐸𝑔E_{g}italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT is defined to be zero, RIXS intensity for the SC and NSC channels can be expressed as,

INSC(q,ω)(1Γ2f|f|1NieiqRiSix|g|2+1Γ4f|f|1Ni,jeiiqRJi,jSix(S^iS^j)|g|2\displaystyle I^{NSC}(q,\omega)\propto\big{(}\frac{1}{\Gamma^{2}}\sum_{f}\Big{%|}\langle f|\frac{1}{\sqrt{N}}\sum_{i}e^{iqR_{i}}S_{i}^{x}|g\rangle\Big{|}^{2}%+\frac{1}{\Gamma^{4}}\sum_{f}\Big{|}\langle f|\frac{1}{\sqrt{N}}\sum_{i,j}e^{%iqR}_{i}J_{i,j}S_{i}^{x}(\hat{S}_{i}\cdot\hat{S}_{j})|g\rangle\Big{|}^{2}italic_I start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT ( italic_q , italic_ω ) ∝ ( divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q italic_R end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ( over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
+1Γ6f|f|1Ni,j,keiqRiJi,jJi,kSix(S^iS^j)(S^iS^k)|g|2+)δ(EfEgω)\displaystyle+\frac{1}{\Gamma^{6}}\sum_{f}\Big{|}\langle f|\frac{1}{\sqrt{N}}%\sum_{i,j,k}e^{iqR_{i}}J_{i,j}J_{i,k}S_{i}^{x}(\hat{S}_{i}\cdot\hat{S}_{j})(%\hat{S}_{i}\cdot\hat{S}_{k})|g\rangle\Big{|}^{2}+\cdot\cdot\cdot\Big{)}\delta%\left(E_{f}-E_{g}-\omega\right)+ divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i , italic_j , italic_k end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_i , italic_k end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ( over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ( over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ⋯ ) italic_δ ( italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT - italic_ω )
=lχlNSC(q,ω)absentsubscript𝑙superscriptsubscript𝜒𝑙𝑁𝑆𝐶𝑞𝜔\displaystyle=\sum_{l}\chi_{l}^{NSC}(q,\omega)= ∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N italic_S italic_C end_POSTSUPERSCRIPT ( italic_q , italic_ω )(20)
ISC(q,ω)(1Γ2f|f|1NieiqRini,σ|g|2+1Γ4f|f|1Ni,jeiqRiJi,jS^iS^j|g|2\displaystyle I^{SC}(q,\omega)\propto\big{(}\frac{1}{\Gamma^{2}}\sum_{f}\Big{|%}\langle f|\frac{1}{\sqrt{N}}\sum_{i}e^{iqR_{i}}n_{i,\sigma}|g\rangle\Big{|}^{%2}+\frac{1}{\Gamma^{4}}\sum_{f}\Big{|}\langle f|\frac{1}{\sqrt{N}}\sum_{i,j}e^%{iqR_{i}}J_{i,j}\hat{S}_{i}\cdot\hat{S}_{j}|g\rangle\Big{|}^{2}italic_I start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT ( italic_q , italic_ω ) ∝ ( divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i , italic_σ end_POSTSUBSCRIPT | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
+1Γ6f|f|1Ni,j,keiqRiJi,jJi,k(S^iS^j)(S^iS^k)|g|2+)δ(EfEgω)\displaystyle+\frac{1}{\Gamma^{6}}\sum_{f}\Big{|}\langle f|\frac{1}{\sqrt{N}}%\sum_{i,j,k}e^{iqR_{i}}J_{i,j}J_{i,k}(\hat{S}_{i}\cdot\hat{S}_{j})(\hat{S}_{i}%\cdot\hat{S}_{k})|g\rangle\Big{|}^{2}+\cdot\cdot\cdot\textbf{}\Big{)}\delta%\left(E_{f}-E_{g}-\omega\right)+ divide start_ARG 1 end_ARG start_ARG roman_Γ start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT | ⟨ italic_f | divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_N end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_i , italic_j , italic_k end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_q italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_i , italic_k end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ( over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ over^ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | italic_g ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ⋯ ) italic_δ ( italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT - italic_ω )
=lχlSC(q,ω)absentsubscript𝑙superscriptsubscript𝜒𝑙𝑆𝐶𝑞𝜔\displaystyle=\sum_{l}\chi_{l}^{SC}(q,\omega)= ∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S italic_C end_POSTSUPERSCRIPT ( italic_q , italic_ω )(21)

In the NSC channel, O((1/Γ)2)𝑂superscript1Γ2O((1/\Gamma)^{2})italic_O ( ( 1 / roman_Γ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) term is the single spin-flip spin excitation scattering, and the O((1/Γ)4)𝑂superscript1Γ4O((1/\Gamma)^{4})italic_O ( ( 1 / roman_Γ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) term is a combination single spin-flip at the site where the core-hole is created and a double flip at i𝑖iitalic_i and j𝑗jitalic_j site. O((1/Γ)6)𝑂superscript1Γ6O((1/\Gamma)^{6})italic_O ( ( 1 / roman_Γ ) start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ) combination single spin-flip at the site where the core-hole is created and a double flip at j𝑗jitalic_j and k𝑘kitalic_k.For the spin-conserving channel, the zeroth order term contributes only to elastic scattering.O((1/Γ)4)𝑂superscript1Γ4O((1/\Gamma)^{4})italic_O ( ( 1 / roman_Γ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) term involves double spin flip at i𝑖iitalic_i and j𝑗jitalic_j. In contrast, O((1/Γ)6)𝑂superscript1Γ6O((1/\Gamma)^{6})italic_O ( ( 1 / roman_Γ ) start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ) is the combination of a double spin flip at i𝑖iitalic_i and k𝑘kitalic_k followed by another double spin flip at i𝑖iitalic_i and j𝑗jitalic_j.

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Predicting Fractionalized Multi-Spin Excitations in Resonant Inelastic X-ray Spectra of Frustrated Spin-1/2 Trimer Chains (2024)
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