Accessing Multi-triplons in Spin Ladders using Resonant Inelastic X-ray Scattering
Gary Ferkinghoff, Leanna Müller, Götz S. Uhrig, Umesh Kumar, Benedikt Fauseweh
AAccessing Multi-triplons in Spin Ladders using Resonant Inelastic X-ray Scattering
Gary Ferkinghoff, Leanna M¨uller, G¨otz S. Uhrig, Umesh Kumar, and Benedikt Fauseweh Lehrstuhl f¨ur Theoretische Physik I, Technische Universit¨at Dortmund,Otto-Hahn-Straße 4, 44221 Dortmund, Germany Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: December 4, 2020)Resonant inelastic x-ray scattering (RIXS) has proven a powerful tool to investigate dynamicmagnetic correlations, yielding complementary insights compared to neutron scattering. A keyissue is the interpretation of spectroscopic measurements in terms of quasi-particle excitations. Inthis paper we study the spin-conserving channel of RIXS spectra for two-leg quantum spin laddersusing continuous unitary transformations. We find that multi-triplon continua and bound statesare the leading contribution, opening new venues for observing and characterizing multi-triplons inquantum spin systems.
Exploring and identifying novel phases of matter is a cen-tral aim in modern solid-state physics. By identifyingtypical excitation spectra which can be related uniquelyto the underlying phases by means of their elementaryexcitations, scattering experiments provide a unique toolfor this purpose. Many exotic phases, such as disorderedspin liquids, can be identified by their excitation spec-trum, making theoretical corroboration important in theinterpretation of such experiments.A unique class of materials in this context are spin lad-der compounds. During the last three decades, they havebeen the subject of great interest in the strongly corre-lated community. Ladder compounds are intermediatebetween one-dimensional and two-dimensional systems,offering a platform to understand many novel phenom-ena common to these systems, such as fractionalization,quantum criticality, and even superconductivity.Traditionally, inelastic neutron scattering (INS) has al-lowed one to study the spin dynamics in strongly cor-related materials. For instance, one-triplon dynam-ics was observed in the ladder compound, Sr Cu O cuprates [1] following earlier theoretical predictions [2, 3].More recently, confined spinons were observed in weaklycoupled spin ladder compounds, CaCu O [4] followingthe predictions in Ref. [5]. The field induced quantumcritical point in (C H N) CuBr was investigated inRef. [6]. Another major breakthrough in a ladder com-pound has been the observation of superconductivity inSr . Ca . Cu O . [7], again preceded by a theoreti-cal study [8].Several studies in the past have tried to understandthe multi-triplon dynamics in ladder compounds. Us-ing INS, S = 1 two-triplon states were observed inLa Sr Cu O [9] and analyzed using the continuousunitary transformation (CUT) method.In the recent years, resonant inelastic x-ray scatter-ing (RIXS) has become an important complementarytool to study spin dynamics [10]. It has allowedobservation of paramagnons in doped two-dimensional(2D) cuprates [11], spin-orbital separation [12] multi-spinon outside of two-spinon continua [13, 14] in a one-dimensional (1D) cuprate, inaccessible to INS probe. To the best of our knowledge, only one RIXS experimenthas been reported on a ladder cuprate, Sr Cu O [15].The RIXS signal was interpreted as S = 0 two-triplonstates, before realizing that RIXS can indeed measuresingle spin-flip excitations [16, 17]. Subsequent numeri-cal studies reported that these spectra should indeed bedominated by S = 1 two-triplon states [18, 19]. It isalso known that RIXS spectra can be resolved into spin-conserving (∆ S = 0) and spin non-conserving (∆ S (cid:54) = 0)at the Cu L edge [20, 21], allowing for new challengesand opportunities. The spin-conserving channel is uniqueto RIXS and can be well captured by the dynamical ex-change structure factor. Apart from a channel at the Cu L -edge, Cu K -edge and oxygen K -edge in cuprates con-sist of only spin conserving channels [13, 22].One key issue in these studies is the interpretation ofRIXS spectra in terms of single quasi-particle excita-tions, multi-particle continua, and (anti)-bound states.The Kramers-Heisenberg (KH) formalism captures theRIXS mechanism, which is usually evaluated using exactdiagonalization on finite systems [14, 15, 23–25]. Morerecently density matrix renormalization group has alsobeen developed to evaluate RIXS spectra using KH for-malism [26]. Also, the KH formalism can be cast into sim-pler dynamical correlation functions under the ultra-fastcore-hole lifetime (UCL) [27, 28] approximation. Thesedynamical correlation functions were also studied usingnumerical technqiues [18, 19, 29]. While these numeri-cal approaches can provide high-resolution spectral func-tions [30, 31], they do not yield an intrinsic physical un-derstanding of the underlying processes and structure ofthe spectra. In contrast, the study of dynamical corre-lation functions using exact methods in the thermody-namic limit allow for a detailed characterization in termsof quasi-particle excitations. For example, analysis ofthe RIXS spectra of the 1D antiferromagnetic chain us-ing Bethe Ansatz showed that its spin-conserving channelentirely fractionalizes into two-spinon states [32].In this paper we study the spin-conserving channel of spinladders as probed by RIXS. We use the CUT techniqueto obtain an effective model of the spin ladder in termsof triplons. CUT was used in a variety of problems, such a r X i v : . [ c ond - m a t . s t r- e l ] D ec as the renormalization of the electron-phonon coupling[33, 34], dissipative systems [35], impurity problems [36],the 2D Hubbard model [37] as well as 1D [38, 39] and 2D[40] spin systems. It gives us direct access to the renor-malized multiparticle interactions and spectral weightsin the thermodynamic limit. Using this approach, we re-veal how the RIXS response is sensitive to multiparticleexcitations and bound states, which can not be capturedby conventional INS experiments.We study a spin model in a two-leg ladder geometry. TheHamiltonian reads H = J rung N (cid:88) i =1 S i, · S i, + J leg (cid:88) i,τ =1 , S i,τ · S i +1 ,τ (1)Here, S i,τ is a spin operator at site i along the legand on rung τ = 0 , J leg ( J rung ) are the superexchangealong the leg (rung) direction of the ladder, respectively.This model describes the spin dynamics of a number ofstrongly correlated ladder compounds at half-filling. Tocompute the RIXS response at the Cu L -edge of cuprates,we use the dynamical correlation functions given by theUCL approximation [28]. At this edge, the spectra con-sist of the non-spin-conserving channel (NSC) and thespin-conserving channel (SC). The NSC is dominated bythe dynamical spin structure factor (DSF) which is alsoaccessible to INS. We provide data for the DSF in theSupplemental Material [41]. The SC channel can be wellcaptured by the dynamical spin-exchange structure fac-tor (DESF) given by S exch ( q , ω ) = 1 N (cid:88) f |(cid:104) f | (cid:88) i,τ e i qR i,τ O exch i,τ | g (cid:105)| × δ ( E f − E g + ω ) (2)where O exch i,τ = S i,τ · [ J leg ( S i +1 ,τ + S i − ,τ ) + J rung S i, ¯ τ ] isthe spin exchange observable. | g (cid:105) and | f (cid:105) are the ground,and final states with energies E g and E f respectively, and ω is the energy loss to the system. Fig. 1 illustrates thespin excitation via double spin-flip in the SC channel.In order to evaluate the SC contribution to the RIXSspectra we decouple the interacting multi-triplon sectorsfrom each other by applying a CUT to the Hamilto-nian and the relevant observables which is systematicallycontrolled up to high order in the expansion parameter x = J leg /J rung . This mapping renders the subsequentcomputations for few quasi-particles possible, as the ef-fective Hamiltonian is decomposed into n -particle irre-ducible parts. The starting point for this approach is thestrong-coupling regime, in which spin singlets are formedon the rungs. We then reformulate (1) in terms of triploncreation and annihilation operators t αi | s (cid:105) = | t αi (cid:105) , where | s (cid:105) is the singlet ground state for x = 0 and | t αi (cid:105) is an S = 1 triplet state of flavor α ∈ { x, y, z } on rung i . Ex-plicit expressions for the Hamiltonian and the observablesin the triplon language are given in the Supplemental Ma-terial [41]. We compute H eff and the corresponding effec-tive observables up to order 10 in the directly evaluated FIG. 1: Schematics for double spin-flips in the spinladder. The photon interacts with a local site (a), thelocal core-hole potential in the spin-conserving channelperturbs the system locally (b) and allows forexcitations via double spin-flips along the chain (c) andthe rung (d).enhanced perturbative CUT (deepCUT). The deepCUTapproach allows for a non-perturbative evaluation of H eff and was proven to yield reliable results for the spin ladderup to x = 3 [42], while being stable even in the presenceof frustration [43, 44]. We choose a particle conservinggenerator m : n , which decouples only the first m quasi-particle sectors from all other sectors n [45, 46]. In thispaper we decouple the zero-, one-, two- and three-triplonsector [41]. Based on the effective observables obtainedfrom the CUT we use a Lanczos algorithm to compute acontinued fraction expansion of the DSF and the DESF[41]. This allows for negligible finite size effects and atthe same time identifies the important physical processesthat contribute to the spectral functions. The accuracyof the CUT approach is examined by comparing it tolower order CUT calculations as well as to results fromexact diagonlization on finite systems in the Supplemen-tal Material [41].We compute the RIXS response for the values x = { . , . , . , . } . This choice covers the strong- ,intermediate- and weak-coupling regime. The value x =1 . Cu O . In the strong-coupling regime, the systemis dimerized on the rungs. The elementary excitations arerung triplons which couple due to the leg coupling, let-ting the triplons hop and interact. In the weak-couplingregime x (cid:29) J leg > J rung in terms of multi-triplon contributions.In this paper we focus on the SC channel of RIXS, whichis inaccessible to INS. In Figure 2 and 3 we show theresults for the SC channel as function of momentum q = ( q x , q y ) and energy ω . Note that q x = [0 , π ) de-notes the continuous direction along the ladder, while q y is oriented along the rungs. q y takes the discrete values { , π } . q x [ a ] [ J r un g ] x = 0.25 o100
14 12 34 x = 0.5 o100
14 12 34 x = 1.2 o100
14 12 34 x = 2.0 o10 O exch+ ( q y = 0)2QP2QP-int ( q )bound state( S = 0)0
14 12 34
FIG. 2: RIXS response in the SC channel for q y = 0. Results from the deepCUT in order 10 for the parameter x using the 2: n generator. The solid white line shows the boundary of the two-triplon continuum and the dashedwhite line shows the S = 0 bound state, broadened with a Lorentzian of width 0 . J rung . q x [ a ] [ J r un g ] x = 0.25 o10 01e-3 2345 (b) x = 0.5 o10 01e-2 68 (c) x = 1.2 o101234 03e-1 1015 (d) x = 2.0 o10 O exch ( q y = )3QP3QP-int246 bare3 ( q ) int3 ( q )012.53.03.54.0 (e) x = 0.25 o100
14 12 34 x = 0.5 o100
14 12 34 x = 1.2 o101234 0
14 12 34 x = 2.0 o10 O exch ( q y = )3QP2QP-int246 bare3 ( q ) int3 ( q )0
14 12 34 FIG. 3: RIXS response in the SC channel for q y = π . (a)-(d) results from the mixed deepCUT calculations, 2: n generator in order 10 for the single- and two-triplon matrix elements, 3: n generator in order ≥ n generator.(e)-(h) results from the 2: n deepCUT calculations in order 10 excluding irreducible three-triplon interactions. q x [ a ] [ J r un g ]
310 410 510 610 710 x =1.2 o10 03e-1
310 410 510 610 710 x =2.0 o10 O exch ( q y = )3QP3QP-int bare3 ( q ) int3 ( q )bound05e-1 FIG. 4: RIXS response in the SC channel for q y = π .Detailed view of the lower three-triplon boundary andthe formation of a three-triplon bound state outside ofthe continuum, broadened with a Lorentzian of width0 . J rung .The system has a reflection symmetry P around the cen-ter line of the ladder along the x direction. The corre-sponding parity is a conserved quantity before and afterthe CUT. This implies that the channels with even andodd numbers of triplons are decoupled and that observ-ables that have a even (odd) parity for x = 0 will injectan even (odd) number of triplons for any x [47].Similar to the DSF, the q y = 0 results in Fig. 2 for theDESF is dominated by the two-triplon contributions, due to the even parity of the observable. For fixed total mo-mentum the two triplons form a spectral continuum asfunction of energy. The boundary of the continuum aredetermined by the single-triplon dispersion and are indi-cated in the plots as solid white lines. Additionally two-triplon bound states are formed, due to triplon-triploninteraction. The dispersion of these bound states de-pends on the total spin of the two triplons. Due to theselection rule ∆ S = 0 the S = 1 bound state is for-bidden in the spectra. Instead the S = 0 bound stateemerges below the two-triplon continuum. Importantlythis bound states is lower in energy than the S = 1 boundstate [41, 47, 48]. This bound state was first reported inthe ladder material (Ca,La) Cu O using optical two-triplon-plus-phonon absorption spectroscopy [49]. In thestrong-coupling regime x < .
2, the dominating spectralweight is in the continuum at q x ≈ π , with a smallerfraction of the spectral weight in the bound state. Thecontinuum itself is rather featureless. This changes uponapproaching the weak-coupling limit. Here the boundstate holds the leading contribution in the two-triplonchannel around q x = π/ q y = π .Here the effective observable has odd parity, but does notcouple to the single-triplon channel, because the singletriplon is a ∆ S = 1 excitation. The leading contribu-tions comes instead from the three-triplon sector.For the results shown in Fig. 3(a)-(d) we use a mixedHamiltonian calculation, i.e., the irreducible single- andtwo-triplon terms in the Hamiltonian are computed upto order 10, while the three-triplon interactions are com-puted up to at least order 5, see also Supplemental Mate-rial [41, 47]. Higher order calculations in the three-triplonsector are not accessible within the deepCUT approachdue to a divergence in the flow equations resulting fromthe strong quasi-particle overlap.The CUT approach allows us to disentangle the effect ofdifferent quasi-particle spaces. Thus, we computed a sec-ond data set, where we performed a 2: n calculation andneglect the three-triplon interactions for the computationof the RIXS response. Comparing the results from Fig.3 (a)-(d) with the 2: n calculations in Fig. 3 (e)-(h) wecan clearly distinguish the spectral features induced bythree-triplon interactions from pure two-triplon effects.In the strong-coupling regime, e.g., for x = 0 .
25 and x = 0 .
5, the response is mostly centered around q x = π/ n and the 2: n calculations indicatingthat the irreducible three-triplon interactions are negligi-ble for small values of x . However this changes drasticallyin the weak-coupling regime.For x > . n calculations predict that the dominatingspectral weight is within the bare three-triplon contin-uum. This suggests that a three-triplon bound state witha very small energy separation from the three-triplon con-tinuum can exist similar to multi-magnon bound statesdiscovered only very recently [50]. Indeed a closer in-spection reveals, that a single eigenvalue of the effective Hamiltonian lies outside of the three triplon continuumfor x = 1 . x = 2 . q x ≥ q x ≈ . π . Depending on q x , the bound states carriesa significant portion of the total weight, i.e., up to 33 %for x = 1 . q x ≈ . π . Since the energy difference tothe boundary is very small, i.e., ≈ . J rung , this boundstate cannot be distinguished from the three-triplon con-tinuum in exact diagonalization, see Supplemental Ma-terial [41]. Previously multi-triplon bound states havebeen reported in strongly frustrated spin ladders [51],but so far not in the unfrustrated case for more than twotriplons. Thus our calculations prove a significant three-triplon interaction effect onto the spectra, that gets evenstronger as we enter into the weak-coupling regime. HereRIXS provides a unique opportunity to study these multi-triplon interaction effects in real materials.In this paper we investigated the RIXS response of spinladder systems. With the capability of RIXS to distin-guish spin conserving and spin non-conserving channelsnew insights into these systems are possible. Our re-sults show, that the spin-conserving channel provides apromising approach to measure multi-triplon excitations,including S = 0 bound states, which are inaccessibleto INS experiments. 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