Abstract
We present a method for computing the resonant inelastic xray scattering (RIXS) spectra in onedimensional systems using the density matrix renormalization group (DMRG) method. By using DMRG to address this problem, we shift the computational bottleneck from the memory requirements associated with exact diagonalization (ED) calculations to the computational time associated with the DMRG algorithm. This approach is then used to obtain RIXS spectra on cluster sizes well beyond stateoftheart ED techniques. Using this new procedure, we compute the lowenergy magnetic excitations observed in Cu Ledge RIXS for the challenging corner shared CuO_{4} chains, both for large multiorbital clusters and downfolded tJ chains. We are able to directly compare results obtained from both models defined in clusters with identical momentum resolution. In the strong coupling limit, we find that the downfolded tJ model captures the main features of the magnetic excitations probed by RIXS only after a uniform scaling of the spectra is made.
Introduction
Resonant inelastic xray scattering (RIXS) has emerged as a powerful and versatile probe of elementary excitations in quantum materials^{1,2}. One of the most commonly used approaches for computing RIXS spectra is small cluster exact diagonalization (ED)^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}. This approach is limited by the exponential growth of the Hilbert space, however, which restricts clusters to a relatively small size, thus limiting momentum resolution. For example, ED treatments of multiorbital spinchain systems such as the edgeshared CuGeO_{3} or corner shared Sr_{2}CuO_{3} have been limited to no more than six CuO_{4} plaquettes^{3,6,8,12,22}, while studies carried out using downfolded singleband Hubbard (or tJ) chains have been limited to ~16–22 sites^{4,10,19,20}.
The density matrix renormalization group (DMRG) is the most powerful method for computing the ground state properties of strongly correlated materials in one dimension (1D)^{23,24,25}. Within the DMRG framework, several efficient methods are available for computing dynamical correlation functions, including: timedependent DMRG^{26,27}, which computes dynamical correlation functions in the time domain with a subsequent Fourier transform into frequency space^{28}; correctionvector methods, which compute the dynamical correlator directly in frequency space^{29,30,31,32}; continued fraction methods^{33,34,35}; and Chebyshev polynomial expansion methods^{36,37}. In this work, we present an efficient algorithm to compute the dynamical correlation function representing the RIXS scattering cross section with DMRG directly in frequency space. We then apply this approach to computing the Cu Ledge RIXS spectra of a quasi1D cornershared cuprate (e.g., Sr_{2}CuO_{3}, see Fig. 1b), a geometry that is challenging for ED calculations due to significant finite size effects^{3,6,8}. We consider a multiorbital Hubbard model that retains the Cu and O orbital degrees of freedom, as well as a downfolded tJ model. Using our DMRGbased approach, we access systems sizes beyond those accessible to ED, thus enabling us to directly compare the results obtained from the two models on large clusters with comparable momentum resolution.
The KramersHeisenberg Formalism
In a RIXS experiment, photons with energy ω_{in} and momentum k_{in} (ℏ = 1) scatter inelastically off of a sample, transferring momentum q = k_{out} − k_{in} and energy Ω = ω_{out} − ω_{in} to its elementary excitations. The resonant nature of the probe arises because ω_{in} is tuned to match one of the elemental absorption edges, such that it promotes a core electron to an unoccupied level of the crystal.
The intensity of the RIXS process I(q, Ω) is given by the KramersHeisenberg formalism^{1,2}, with
Here, E_{g} and E_{f} are the energies of the ground g〉 and final states f〉 of the system, respectively. The scattering amplitude F_{f,g} is defined as
where \(\hat{{\mathscr{D}}}({\bf{k}})\) is the dipole transition operator describing the corehole excitation. In what follows, we consider the Cu Ledge (a Cu 2p → 3d transition). In this case, the dipole operator is defined as \(\hat{{\mathscr{D}}}({\bf{k}})={\sum }_{j,\sigma ,\alpha }\,{e}^{i{\bf{k}}\cdot {{\bf{R}}}_{j}}\times \) \([{A}_{\alpha }^{\hat{\varepsilon }}{\hat{d}}_{j,\sigma }^{\dagger }{\hat{p}}_{j,\alpha ,\sigma }+{\rm{h}}.\,{\rm{c}}.\,]\), where \({d}_{j,\sigma }^{\dagger }\) adds an electron to the valence band orbital \((3{d}_{{x}^{2}{y}^{2}})\), and \({\hat{p}}_{j,\alpha ,\sigma }\) destroys a spin σ electron (creates a hole) in a core 2p_{α} orbital on site j located at R_{j}. The prefactor \({A}_{\alpha }^{\hat{\varepsilon }}\) is the matrix element of the dipole transition between the core 2p_{α} orbital and the valence \(3{d}_{{x}^{2}{y}^{2}}\) orbital, \(\langle 3{d}_{{x}^{2}{y}^{2},\sigma }\hat{\varepsilon }\cdot \hat{r}2{p}_{\alpha ,\sigma }\rangle \), which we set to 1 for simplicity. Γ is the inverse corehole lifetime, and \({\hat{H}}_{{\rm{ch}}}=\hat{H}+{\hat{H}}^{C}\), where \({\hat{H}}^{C}={V}_{C}\,\sum _{j,\sigma ,\sigma ^{\prime} }\,{\hat{n}}_{j,\sigma }^{d}(1{\hat{n}}_{j,\sigma ^{\prime} }^{{p}_{\alpha }})\) describes the Coulomb interaction between the core hole and the valence electrons and \(\hat{H}\) is the manybody Hamiltonian of the system.
Under the assumption that the corehole is completely localized, and only one Cu 2p_{α} orbitals is involved in the RIXS process, Eq. (2) simplifies to
where we have defined the local dipoletransition operator \({\hat{D}}_{j,\sigma }\equiv {\hat{d}}_{j,\sigma }^{\dagger }{\hat{p}}_{j,\sigma }\) and \({\hat{H}}_{{\rm{ch}},j}=\hat{H}+{\hat{H}}_{j}^{C}\), with \({\hat{H}}_{j}^{C}=\) \({V}_{C}\,\sum _{\sigma ,\sigma ^{\prime} }\,{\hat{n}}_{j,\sigma }^{d}(1{\hat{n}}_{j,\sigma ^{\prime} }^{p})\).
Reformulation of the Problem for DMRG
The primary difficulty in evaluating Eq. (1) lies in computing the final states f〉. This task is often accomplished using ED on small clusters meant to approximate the infinite system. Obtaining these same final states is usually impossible with DMRG, which targets only the ground state; however, we will show that to accomplish this task one can use the Lanczos method, which projects the state onto a Krylov space^{38}. Some of the present authors introduced this alternative method to calculate the correction vectors for frequencydependent correlation functions with DMRG^{32}.
We can formulate an efficient DMRG algorithm by expanding the square in Eq. (1), yielding a real space version of the KramersHeisenberg formula. To compact the notation, we define vectors \({\alpha }_{j,\sigma }\rangle \equiv [{\omega }_{{\rm{in}}}{\hat{H}}_{{\rm{ch}},j}+\) \({E}_{g}+i{\rm{\Gamma }}{]}^{1}{\hat{D}}_{j,\sigma }g\rangle \). Using this definition, Eq. (1) can be written as
Here, η is a broadening parameter, which plays the same role as the Gaussian or Lorentzian broadening introduced in ED treatments of the energyconserving δfunction appearing in Eq. (1). Throughout this work, we set it to 75 meV, which is in the range of the typical energy resolution currently attainable in RIXS experiments^{39}. Note that the vectors α_{j,σ}〉 must be computed for each value of ω_{in} and Γ.
The Xray absorption spectrum (XAS) can be computed using a similar formalism. Its intensity is given by
Finally, we note that we have removed the elastic line from all spectra shown in this work. The precise method for doing this is discussed in Supplementary Note IV.
Computational Procedure
The algorithm to compute the RIXS spectra using Eq. (3) is as follows (see also Fig. 1a):

Step 1: Compute the ground state g〉 of \(\hat{H}\) using the standard ground state DMRG method. The vector g〉 must be stored for later use.

Step 2: Restart from the ground state calculation, reading and then targeting the ground state vector calculated earlier and using a different Hamiltonian \({\hat{H}}_{{\rm{ch}},c}=\hat{H}+{\hat{H}}_{c}^{C}\), where j = c is the center site of the chain. Construct the vector α_{c,σ}〉 at the center of the chain using the Krylovspace correction vector approach^{32} where we have performed a Lanczos tridiagonalization \({\tilde{T}}_{c}\) with starting vector \({\hat{D}}_{c,\sigma }g\rangle \), and a subsequent diagonalization \({\tilde{S}}_{c}\) of the Hamiltonian \({\hat{H}}_{{\rm{ch}},c}\), represented in its diagonal form D_{ch,c} in the Krylov basis. The vector α_{c,σ}〉 should also be stored for later use. Because the cluster is not periodic, the use of a central site here represents an approximation that will become exact in the thermodynamic limit. This central site “trick” was used for the first time in the application of timedependent DMRG^{26}.
$${\alpha }_{c,\sigma }\rangle \simeq {\tilde{T}}_{c}^{\dagger }{\tilde{S}}_{c}^{\dagger }\frac{1}{{\omega }_{{\rm{in}}}{D}_{{\rm{ch}},c}+{E}_{g}+i{\rm{\Gamma }}}{\tilde{S}}_{c}{\tilde{T}}_{c}{\hat{D}}_{c,\sigma }g\rangle ,$$(5) 
Step 3: Restart from previous run, now using a different Hamiltonian \({\hat{H}}_{{\rm{ch}},j}=\hat{H}+{\hat{H}}_{j}^{C}\). Read and then target (in the DMRG sense) the ground state vector g〉, as well as the vector α_{c,σ}〉 constructed in Step 2. For each site j, except for the center site considered in Step 2, construct the vector with a Lanczos tridiagonalization \({\tilde{T}}_{j}\) with starting vector \({\hat{D}}_{j,\gamma }g\rangle \), and a subsequent diagonalization of \({\hat{H}}_{{\rm{ch}},j}\). This step of the algorithm requires a number of runs which is equal to the number of sites minus 1, i.e., L − 1. These can be run in parallel on a standard cluster machine, restarting from Step 2. Performing Step 2 and Step 3 in this sequence is crucial for having the vectors α_{c,σ}〉 and α_{j,γ}〉 in the same DMRG basis. The vector α_{j,γ}〉 should also be stored for later use.
$${\alpha }_{j,\gamma }\rangle \simeq {\tilde{T}}_{j}^{\dagger }{\tilde{S}}_{j}^{\dagger }\frac{1}{{\omega }_{{\rm{in}}}{D}_{{\rm{ch}},j}+{E}_{g}+i{\rm{\Gamma }}}{\tilde{S}}_{j}{\tilde{T}}_{j}{\hat{D}}_{j,\gamma }g\rangle ,$$(6) 
Step 4: Restart using the original Hamiltonian \(\hat{H}\). Read and then target the ground state vector g〉, the vector α_{c,σ}〉, as well as the vector α_{j,γ}〉 constructed in Step 3. For a fixed Ω = ω_{l}, compute the correction vector of α_{c,σ}〉 using again the Krylovspace correction vector approach as with a Lanczos tridiagonalization \(\tilde{T}\) (using \({\hat{D}}_{j,\sigma ^{\prime} }^{\dagger }{\alpha }_{c,\sigma }\rangle \) as the seed) and a subsequent diagonalization \(\tilde{S}\) of the Hamiltonian \(\hat{H}\), with D being the diagonal form of \(\hat{H}\) in the Krylov basis. This is a crucial part of the algorithm, which amounts to computing the correction vector \({x}_{c,\sigma ^{\prime} ,\sigma }\rangle \) of a previously calculated correction vector α_{c,σ}〉. Execute this computation N_{Ω} times for \({\rm{\Omega }}\in [{\omega }_{0},\,{\omega }_{N1}]\).
$$\begin{array}{ccc}{x}_{c,\sigma ^{\prime} ,\sigma }\rangle & \equiv & \frac{1}{{\rm{\Omega }}\hat{H}+{E}_{g}+{\rm{i}}\eta }{\hat{D}}_{c,\sigma ^{\prime} }^{\dagger }{\alpha }_{c,\sigma }\rangle ={\mathop{T}\limits^{ \sim }}^{\dagger }{\mathop{S}\limits^{ \sim }}^{\dagger }\frac{1}{{\rm{\Omega }}D+{E}_{g}+{\rm{i}}\eta }\mathop{S}\limits^{ \sim }\mathop{T}\limits^{ \sim }\,{\hat{D}}_{c,\sigma ^{\prime} }^{\dagger }{\alpha }_{c,\sigma }\rangle ,\end{array}$$(7) 
Step 5: Finally, compute the RIXS spectrum in real space \({I}_{j,c}({\rm{\Omega }})\propto \langle {\alpha }_{j,\gamma }{\hat{D}}_{j,\gamma ^{\prime} }{x}_{c,\sigma ^{\prime} ,\sigma }\rangle \) (in I_{j,c}(Ω) we omit the spin indices γ, γ′, σ, σ′ in order to lighten the notation) and then Fourier transform the imaginary part to obtain the RIXS intensity.
$$I({\bf{q}},{\rm{\Omega }})\propto \,{\rm{Im}}\,\sum _{\begin{array}{c}j,\gamma ,\gamma ^{\prime} \\ \sigma ,\sigma ^{\prime} \end{array}}\,{e}^{i{\bf{q}}\cdot ({{\bf{R}}}_{j}{{\bf{R}}}_{c})}{I}_{j,c}({\rm{\Omega }}).$$(8)
Computational Complexity
The computational cost required for DMRG to compute the RIXS spectrum can be easily estimated, assuming that the ground state of the Hamiltonian has already been calculated. Let C_{2–3} be the computational cost (i.e., the number of hours) for a single run in Step 2 (1 run only) or Step 3 (L − 1 runs in total). Let C_{4} be the computational cost for a single run in Step 4. The total computational time needed to compute the RIXS spectrum is then CPU_{cost} = C_{2–3}L + C_{4}LN_{Ω}, where N_{Ω} is the number of frequencies needed in a given interval of energy losses. As explained in the previous section, we use a center site “trick” to reduce the computational cost by a factor of the order of L (Eqs (3–8)). For the largest system size considered in this work (20 plaquettes in the CuO_{4} multiorbital model at halffilling, using up to m = 1000 DMRG states), the typical values for CPU_{cost} on a single core of a standard computer cluster are: C_{2–3} ~ 2 hours, while C_{4} ~ 2 − 24 hours. The computational cost C_{4} for Step 4 follows the typical performance profile of the Krylovspace approach found in ref.^{32}, where less CPU time is needed to compute the spectra at lower energylosses. We also note that the calculation of each energy loss is trivially parallelizable. From these assumptions, we estimate the proposed method can compute the RIXS spectrum of a cluster as large as Cu_{20}O_{61} in less than a day if enough cores are available. In this specific case, one single core run was needed for ground state calculation, 80 single core runs for Steps 2–3, and 800 single core runs for Step 4.
Numerical Results for the tJ Model
We first apply our approach to compute the RIXS spectrum of the 1D tJ model as an effective model for the antiferromagnetic cornershared spin chain cuprate Sr_{2}CuO_{3} (see Methods). Throughout this paper, we adopt open boundary conditions, work at halffilling, and set t = 0.3 eV for the nearest neighbor hopping and J = 0.25 eV for the antiferromagnetic exchange interaction. These values are typical for Sr_{2}CuO_{3}^{10,20,21,40,41,42,43,44,45,46}.
Before scaling up our DMRG calculations to large systems, we benchmarked our method by directly comparing our DMRG results to ED. The results for a L = 16 sites tJ chain are presented in Supplementary Note I. (We provide a similar comparison for a fourplaquette multiorbital cluster in Supplementary Note II.) Our DMRG approach gives perfect agreement with the ED result for both the XAS and RIXS spectra, for the largest clusters we can access with ED. All of the DRMG simulations presented in this work used up to m = 1000 states, with a truncation error smaller than 10^{−6}.
We now turn to results obtained on a L = 64 site chain, as shown in Fig. 2. Here, we present results for the spinconserving (ΔS = 0) and nonspinconserving (ΔS = 1) contributions to the total RIXS intensity. The ΔS = 0 contribution corresponds to the σ = σ′ and γ = γ′ terms in the KramersHeisenberg formula Eq. (3). In this case, only two configurations (\(\gamma =\gamma ^{\prime} =\sigma =\sigma ^{\prime} =\uparrow \) and \(\gamma =\gamma ^{\prime} =\downarrow \), \(\sigma =\sigma ^{\prime} =\uparrow \)) have to be explicitly calculated with DMRG, as the other two possible spinconserving configurations contribute equally by symmetry. The remaining terms with σ ≠ σ′ and γ ≠ γ′ determine the nonspinconserving ΔS = 1 contributions to the spectrum. In this case, only one configuration (\(\sigma ^{\prime} =\downarrow \), \(\sigma =\uparrow \), \(\gamma ^{\prime} =\downarrow \), \(\gamma =\uparrow \)) has been simulated with DMRG, as the flipped configuration (\(\sigma ^{\prime} =\uparrow \), \(\sigma =\downarrow \), \(\gamma ^{\prime} =\uparrow \), \(\gamma =\downarrow \)) contributes equally by symmetry. The remaining two possible nonspinconserving configurations also give zero contribution to the RIXS spectrum by symmetry.
In Fig. 2, the ΔS = 1 part of the RIXS spectrum shows a continuum of excitations resembling the two spinon continuum commonly observed in the dynamical spin structure factor S(q, ω) of onedimensional spin1/2 antiferromagnets^{47,48,49,50}. The ΔS = 0 contribution in Fig. 2a shows two broad arcs with maxima at q = π/2a. Notice also the perfect cancellation of the RIXS signal at the zone boundary, which is \(q=\frac{\pi }{a}\frac{L}{L+1}\) in open boundary conditions. Our results agree with the ED results of refs^{4},^{19}, but with much better momentum resolution. We find that the finite size effects of the magnetic excitations in the tJ model are mild; we observe only small differences between results obtained on L = 16 (shown in Supplementary Note I) and L = 64 site clusters.
Magnetic Excitations in the Multiorbital pdModel
In the strong coupling limit, the lowenergy magnetic response of the spinchain cuprates are believed to be effectively described by a single orbital Hubbard or tJ model^{51,52}. According to this picture, holes predominantly occupy the Cu orbitals at halffilling, while the oxygens along the CuCu direction provide a pathway for superexchange interactions between the nearestneighbor Cu orbitals. Since our DMRG approach provides access to large cluster sizes, we now compute the RIXS spectrum of a more realistic multiorbital model. Here, we consider the challenging cornershared geometry, which suffers from slow convergence in the cluster size. To address this, we consider finite 1D Cu_{n}O_{3n+1} clusters, with open boundary conditions, as illustrated in Fig. 1b for the n = 4 case. The Hamiltonian is given in the Methods section. We evaluated the Cu Ledge RIXS intensity for this model as a function of n for up to n = 20 CuO_{4} plaquettes.
The RIXS spectra for spinconserving (ΔS = 0) and nonspinconserving contributions (ΔS = 1) calculated with our DMRG method are shown in Fig. 3. Similar to the t − J spectra, panels (a–f) in Fig. 3 show two broad arcs with maxima at ±π/2a. Here, we observe significant finite size effects in the RIXS spectra. Some of these effects are the result of our use of the “centersite approximation” in evaluating the KramersHeisenberg formula. For example, the downward dispersing lowenergy peak centered at q = 0 seen in the smaller clusters is the result of this approximation. These features in the spectra can be minimized by carrying out calculations on larger clusters. Because of this, to observe well defined spectral features, we need to consider at least fourteen plaquettes. The pd model also shows that the lowenergy ΔS = 1 part of the RIXS spectrum is characterized by a twospinonlike continuum of excitations (panels (g–l) in Fig. 3).
Comparing the MultiOrbital and Effective tJ Models
Over the past decade, there has been a considerable research effort dedicated to quantitatively understand the intensity of magnetic excitations probed by inelastic neutron scattering (INS)^{44,49,53}. This effort is motivated by the desire to understand the relationship between the spectral weight of the dynamical spin response S(q, ω) and the superconducting transition temperature T_{c} of unconventional superconductors^{54}. To this end, several studies have set out to determine whether the observed INS intensity can be accounted for by the Heisenberg model in lowdimensional strongly correlated cuprates. In this context, the highest degree of success has been achieved in quasi1D materials, where accurate theoretical predictions for S(q, ω) are available^{44,49}. Many of these studies find that the lowenergy Heisenberg model can indeed account for the INS intensity, after including corrections due to effects such as the degree of covalency, its impact on the form factor, and DebyeWaller factors.
RIXS has also been applied to study magnetic excitations in many of the same materials^{10,43,46}. It is therefore natural to ponder how covalency modifies the magnetic excitations as viewed by RIXS. In the limit of a short corehole lifetime, or under constraints in the incoming and outgoing photon polarization, the RIXS intensity for single orbital Hubbard and tJ chains is well approximated by S(q, ω)^{1,5,19,46}. However, to the best of our knowledge, no systematic comparison of the RIXS intensity, as computed by the KramersHeisenberg formalism, has been carried out for multiorbital and downfolded Hamiltonians.
Figure 3 demonstrated that DMRG grants access to large system sizes. We are, therefore, in a position to make such a comparison for the multiorbital spinchain cuprates. Figure 4 compares the spectra computed on a L = 20 site tJ chain against those computed on a Cu_{20}O_{61} cluster, such that the momentum resolution of the two clusters is the same. The parameters for the multiorbital model are identical to those used in Fig. 3. To facilitate a meaningful comparison with the tJ model, we adopted t = 0.5 eV and J = 0.325 eV. These values are obtained by diagonalizing a Cu_{2}O_{7} cluster (see methods). Note that we use the same value of the core hole potential V_{C} = 4 eV in both cases. In Supplementary Note III, we show results for a reduced value of V_{C} for the tJ model, which are very similar. To compare the two spectra, the results for the tJ model have been scaled by a factor of 0.26 such that the maximum intensity of the ΔS = 1 excitations is the same at the zone boundary. This factor presumably accounts for covalent factors and differences in how the corehole interacts with the distribution of electrons in the intermediate state.
After we have rescaled the spectra, we find excellent overall agreement between the two calculations: the amplitude of the broad arcs for the magnetic excitations, both in the ΔS = 0 and in the ΔS = 1 channels of the RIXS spectra are well captured by the tJ model. There are, however, minor quantitative differences related to the spectral weight of the excitations appearing near q = π/2a in the ΔS = 0 channel. In this channel, it is natural that the CuO_{4} spectrum is more noisy than the t − J because we are using the same number of DMRG states for both models, but the Hilbert space of the 20site CuO_{4} cluster is much larger than the 20site t − J cluster. Overall, the tJ model concentrates the magnetic excitations at slightly lower values of the energy loss in the ΔS = 0 channel. This discrepancy might be compensated for by taking a different value of J; however, this would come at the expense of the agreement in the ΔS = 1 channel. These differences should be kept in mind when one calculates the lowenergy magnetic RIXS spectra using an effective tJ or singleband Hubbard model. Nevertheless, our results suggest that in the strong coupling limit, the magnetic RIXS spectrum can be described qualitatively by the effective tJ model.
Figure 4 shows that the overall agreement between the full multiorbital model and the tJ model is much better in the ΔS = 1 channel than in the ΔS = 0 channel. We have verified that this discrepancy is not linked to loss of accuracy in our computational algorithm in the ΔS = 0 channel. Rather, we can naively understand this difference by recalling the role of charge fluctuations in the two magnetic excitation pathways. The ΔS = 1 RIXS excitations are possible in a system with strong spinorbit coupling in the Cu 2p orbitals, which allows the spin of the corehole to flip in the intermediate state of the RIXS process^{4,10,19}. The ΔS = 0 pathway, however, requires a double spinflip between neighboring Cu spins in the final state^{4,19}. At the Cu Ledge, such processes occur due to charge fluctuations between the neighboring Cu sites in the intermediate state. The multiorbital model treats such charge fluctuations differently owing to the presence of the ligand oxygen orbitals. This difference accounts for the discrepancy between the two models in the ΔS = 0 channel. At the Cu Ledge, however, the strong corehole potential suppresses this difference by repelling holes from the site where it was created resulting in only minor differences between the predictions of the two models.
Concluding Remarks
We have presented a novel DMRG approach to computing the RIXS spectra and benchmarked this method against traditional ED. Using our DMRG algorithm, we can compute the RIXS spectra on 1D clusters much larger than those accessible to stateoftheart ED methods. Using the proposed technique, we modeled the magnetic excitations probed by RIXS at the Cu Ledge in 1D antiferromagnets on the largest cluster sizes to date. We found that both the full multiorbital cluster and the effective tJ model provide comparable descriptions of the magnetic excitations in the ΔS = 1 channel, while there were slight differences in the ΔS = 0 channel. This discrepancy was explained by noting the different ways that magnetic excitations are created in these channels.
Finally, we note that the bottleneck to RIXS simulations using ED is the exponential growth of the Hilbert space. Our approach shifts the computational burden to the availability of CPUs, thus opening the door to calculations for much larger systems. For example, one can envision extending this approach to the 2D models currently under active study by the DMRG community. Indeed, our RIXSDMRG method is not restricted to 1D systems and can be applied to a 2D lattice geometry in the same sense specified in ref.^{55}. One can always map a finite 2D N × N lattice with shortrange interactions and hoppings into a 1D lattice with N^{2} sites and longrange interactions and hoppings. Once such a mapping is obtained, for instance, by drawing a 1D path that scans through the lattice following a “snake”like pattern, one can straightforwardly apply the conventional DMRG algorithm, and similarly our RIXSDMRG approach. It is well known, however, that there is a cost to this simplification. Since the interactions have longrange character, the numerical simulations become computationally more expensive as more DMRG states are needed to achieve converged results (arealaw entanglement growth). Instead, DMRG applications to 2D systems usually adopt a mapping to multileg cylinders^{55}. These lattice structures consist of coupled 1D chains or “legs” (usually up to 12 legs for spin systems^{55}, and up to 4–6 legs for fermionic systems^{56,57,58}). DMRG simulations are then performed using periodic boundary conditions along the short (“y”) direction and open boundary conditions along the long (“x”) direction. The arealaw entanglement growth is still the main limitation of the DMRG algorithm in this case as the number of legs of the cylinder increases^{56,57,58}. The computation of dynamical response functions within the DMRG framework on multileg cylinders is computationally even more challenging, but it has been shown to be feasible. Indeed, the computation of spin (S(q, ω)) and charge (N(q, ω)) dynamical response functions of 4leg tt′J cylinders has recently appeared in ref.^{59}. Our RIXSDMRG algorithm can be carried out on similar multileg cylinders.
Methods
The multiorbital pdHamiltonian describing the cornershared spinchains, given in the holepicture, is
Here, \(\langle \ldots \rangle \) denotes a sum over nearest neighbor orbitals; \({d}_{i,\sigma }^{\dagger }\) (\({p}_{j,\gamma ,\sigma }^{\dagger }\)) creates a spin σ hole on the i^{th} Cu 3\({d}_{{x}^{2}{y}^{2}}\) orbital (the j^{th} O 2p_{γ} orbital, γ = x, ±y); ε_{d} and ε_{p,γ} are the onsite energies; \({n}_{i,\sigma }^{d}\) (\({n}_{j,\gamma ,\sigma }^{p}\)) is the number operator for the Cu 3\({d}_{{x}^{2}{y}^{2}}\) orbital (the j^{th} O 2p_{γ} orbital); \({t}_{pd}^{ij}\) and \({t}_{pp}^{jj^{\prime} }\) are the CuO and OO overlap integrals, respectively (the ij and jj′ dependence only indicates the ± differences among hoppings, Fig. 1b); U_{d} and U_{p} are the onsite Hubbard repulsions of the Cu and O orbitals, respectively, and U_{pd} is the nearestneighbor CuO Hubbard repulsion. The phase convention for the overlap integrals is shown in Fig. 1b. In this work, we adopt (in units of eV) ε_{d} = 0, ε_{p,x} = 3, ε_{p,y} = 3.5, t_{(p,x)d} = 1.5 t_{(p,y)d} = 1.8, t_{pp} = 0.75, U_{d} = 8, U_{p} = 4, and U_{pd} = 1, following ref.^{21}.
In the limit of large U_{d}, one integrates out the oxygen degrees of freedom and maps Eq. (9) onto an effective spin1/2 tJ Hamiltonian^{52}
Here, \({\tilde{d}}_{i,\sigma }\) is the annihilation operator for a hole with spin σ at site i, under the constraint of no double occupancy, \({n}_{i}={\sum }_{\sigma }\,{n}_{i,\sigma }\) is the number operator, and S_{i} is the spin operator at site i.
To facilitate a direct comparison between the two models, one can extract the hopping t and exchange interaction J from an ED calculation of a twoplaquette Cu_{2}O_{7} cluster with open boundary conditions^{60}. Here, we obtain the hopping (t = 0.5 eV) by diagonalizing this cluster in the (\(2\,\uparrow \,,\,1\,\downarrow \))hole sector, and setting 2t to be equal to the energy separation between the bonding and antibonding states of the ZhangRice singlet. Similarly, we can obtain the superexchange (J = 0.325 eV) by diagonalizing the cluster in the \((1\,\uparrow \,,\,1\,\downarrow \,)\)hole sector, and setting the singlettriplet splitting of the Cu (d^{9}d^{9}) configurations equal to J.
Data and Code availability
The data that support the findings of this study are available from the corresponding author upon request. In the Supplementary Note V we provide details about the code used to obtain the DMRG results.
Additional information
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Acknowledgements
A.N., N.K., and E.D. were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. G. A. and S. J. were supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Sciences, Advanced Scientific Computing Research and Basic Energy Sciences, Division of Materials Sciences and Engineering. N.K. was also partially supported by the National Science Foundation Grant No. DMR1404375. This research used computational resources supported both by the University of Tennessee and Oak Ridge National Laboratory Joint Institute for Computational Sciences (Advanced Computing Facility). It also used computational resources at the National Energy Research Scientific Computing Center (NERSC).
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A.N., E.D. and S.J. planned the project. A.N. implemented the DMRGRIXS algorithm with the help of N.K. and G.A. A.N. performed all of the DMRG simulations. U.K. performed all of the exact diagonalization calculations. A.N., U.K. and S.J. wrote the manuscript with inputs from all other coauthors.
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The authors declare no competing interests.
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Correspondence to A. Nocera.
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