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# Ultrafast coupled charge and spin dynamics in strongly correlated NiO

## Abstract

Charge excitations across an electronic band gap play an important role in opto-electronics and light harvesting. In contrast to conventional semiconductors, studies of above-band-gap photoexcitations in strongly correlated materials are still in their infancy. Here we reveal the ultrafast dynamics controlled by Hund’s physics in strongly correlated photoexcited NiO. By combining time-resolved two-photon photoemission experiments with state-of-the-art numerical calculations, an ultrafast (10 fs) relaxation due to Hund excitations and related photo-induced in-gap states are identified. Remarkably, the weight of these in-gap states displays long-lived coherent THz oscillations up to 2 ps at low temperature. The frequency of these oscillations corresponds to the strength of the antiferromagnetic superexchange interaction in NiO and their lifetime vanishes slightly above the Néel temperature. Numerical simulations of a two-band t-J model reveal that the THz oscillations originate from the interplay between local many-body excitations and antiferromagnetic spin correlations.

## Introduction

Employing light to understand and control ordered states in solid-state systems is a key element of modern science and technology. The optical control of spins in ferromagnets is an especially important research area, related to the ultrafast writing of magnetic information, which has recently been extended to antiferromagnets (AFM)1. An interesting class of AFM materials is the family of strongly correlated electron systems, where the local electron correlations lead to insulating behavior and simultaneously stabilize the long-range magnetic ordering via Anderson’s superexchange mechanism2. The complex electronic structure of correlated materials reflects the coupling mechanisms between the magnetic order and the orbital, spin, and lattice degrees of freedom. These mechanisms lead to numerous competing metastable phases accessible via ultrafast photoexcitation. In recent years, several exciting scenarios for the ultrafast manipulation of ordered states have been reported, including possible high-temperature superconducting states in light-driven cuprates3,4 and fullerides5, hidden states in 1T-TaS26, as well as photo-induced non-thermal magnetic and orbital-ordered states7,8. Less attention has been paid to the feedback mechanisms between the long-range electronic or magnetic ordering and the dynamics of the photoexcited charge carriers.

In correlated transition-metal oxides (TMO), the strong electron-electron repulsion splits the d bands into an occupied lower Hubbard band (LHB) and an empty upper Hubbard band (UHB). The presence of additional oxygen (ligand L) p-bands led to the Zaanen, Sawatzky, and Allen (ZSA) classification scheme9, which suggests that the electronic properties of TMOs are essentially determined by two parameters: the Hubbard U and the charge transfer Δ. Mott-Hubbard insulators are characterized by U < Δ. In these systems, the ligand band is located well below the TM d band and plays a minor role in the low-energy dynamics. In charge-transfer insulators, Δ is smaller than U, and the p-band is located between the LHB and the UHB, as schematically depicted in Fig. 1a.

While NiO is a charge-transfer insulator according to this classification scheme, the actual band structure is more complex. This is due to the d8 ground state electronic configuration with formally fully occupied t2g and half-filled eg orbitals, as indicated in the dark and light blue insets of Fig. 1b, respectively. The established energy scales for NiO are10,11:

$$\begin{array}{ccc}\Delta &=&E({d}^{9}\underline{L})-E({d}^{8}) \sim 4\,\,{\text{eV}}\,,\\ U&=&E({d}^{7})+E({d}^{9})-2E({d}^{8}) \sim 7.5\,\,\text{eV}\,,\end{array}$$

where $$\underline{L}$$ denotes a hole at the ligand site. Thus, NiO should be considered an intermediate charge-transfer insulator with the p-bands located in the energy range of the LHB. The latter situation leads to a strong hybridization between the 2p and 3d bands and their bonding combination corresponds to the Zhang–Rice doublet. A consistent theoretical description requires a treatment of the correlated 2p and 3d bands on equal footing as has been demonstrated by Kuneš et al.12 using the local density approximation (LDA) plus dynamical mean-field theory (DMFT) scheme.

In this work, we explore photoexcited transient states in NiO by exploiting recent advances in photoemission spectroscopy and numerical simulations. More specifically, using pump-probe time-resolved two-photon photoemission (2PPE) with two independently tunable ultraviolet (UV) sources, we study the fate of the excited electrons in the UHB of the transition-metal oxide NiO triggered by photoexcitation across the band gap. Upon above-band-gap optical excitation, we find: (a) an ultrafast decay of the excited electrons within less than 10 fs, and (b) a photo-induced in-gap state well below the bottom of the UHB of the originally undoped system. These two effects represent a non-equilibrium manifestation of the Hund’s coupling in NiO. We demonstrate that the many-body in-gap state is coupled to the antiferromagnetic spin background providing conditions for coherent THz oscillations of the 2PPE signal. The frequency of the THz oscillations corresponds to the superexchange interaction in NiO, and their amplitude vanishes slightly above the Néel temperature (TN). These experimental findings are complemented by theoretical modeling based on a two-band t-J model, which reveals that the coherent oscillations originate from the strong coupling between the excited Hund states and the antiferromagnetically correlated spin background.

## Results

### Time-resolved two-photon photoemission

The excitation across the optical band gap as studied in the present work is sketched in Fig. 1a, b. The photoexcitation transfers an electron into the UHB, resulting in a hole in the ligand p orbital and a triply occupied site in the Ni eg manifold (a triplon state). We explore these states and their ultrafast dynamics by a subsequent UV probe pulse, which photoemits the excited electron and projects the whole system to a final state with a ligand hole. Pump-probe 2PPE spectra are shown in Fig. 1c, d for a NiO(001)-(1 × 1) ultrathin film grown on Ag(001) at zero delay between the UV pulses. The 2PPE spectra reveal one occupied state at  −1.3 eV with respect to the Fermi level EF, marked as V in Fig. 1b, d, and two unoccupied states at 2.0 and 2.5 eV above EF marked as S and C, respectively, in Fig. 1b, c. The state V is at the top of the oxygen 2p band and is assigned to the Zhang–Rice doublet ($$3{d}^{8}\underline{Z}$$) similarly to the one-hole final state in conventional photoemission experiments12,13,14. The unoccupied states C and S are visible for all film thicknesses and their binding energies depend only weakly on the NiO film thickness for epitaxial films between 4 and 20 ML as summarized in Fig. 1b. The state C is assigned to photoemission from the lower edge of the UHB based on its energy-difference of Δ = (3.8 ± 0.2) eV with respect to photoemission from occupied states and the known value of the NiO charge-transfer gap15. The state S corresponds to photoemission from an unoccupied 3$${d}_{{z}^{2}}$$ surface state that has been predicted by theory16,17. This state is also visible in surface-sensitive scanning tunneling spectroscopy (STS) as shown in the Supplementary Note 1. A pronounced STS signal between 1.8 and 1.9 eV above EF demonstrates its surface state character.

The time-resolved 2PPE spectra are depicted in Fig. 2a for pump-probe delays up to 3 ps. The pump-photon hνpump = 4.2 eV promotes the electron into the UHB with a maximum excess energy of about 0.4 eV. The relaxation within the UHB and the population of the in-gap state takes place on ultrafast time scales well below the temporal width of our pump-probe cross correlation, which has been measured to be about 80 fs (Fig. 2b). A detailed analysis of the time traces around zero delay results in an upper limit of 10 fs for the electron lifetime in the UHB (Supplementary Note 1). We emphasize that the observed electron lifetime at the bottom of the (undoped) UHB is three orders-of-magnitude shorter than the value expected at the conduction band minimum of conventional semiconductors with a similar band gap18,19,20.

In order to understand the efficient energy dissipation of the UHB electrons, we take a closer look at the 2PPE spectra in the low-energy region. In Fig. 2, we observe a long-lived oscillatory contribution at an intermediate state energy of 1.2 eV, slightly above the low-energy photoemission cut-off. An intensity profile across this energy region reveals an oscillatory component on top of a slowly decaying background (see Fig. 2 as well as Fig. 3a). It is well described by a damped oscillation $${I}_{\text{osc}}(t)\ =\ A\cdot {e}^{-t/\tau }\cdot \cos (2\pi {\nu }_{\mathrm{b}}\cdot t\ +\ \phi )$$ with a frequency of νb = (4.20 ± 0.26) THz, a phase shift ϕ ≈ 0, and a dephasing time of τ = (558 ± 35) fs. The oscillation frequency corresponds to hνb =17 meV.

Figure 3a shows the oscillatory signal for the 9 ML ultrathin film in a temperature range from 150 to 455 K. Whereas the oscillation amplitude at t = 0 and the oscillation frequency do not vary significantly, the damping of the oscillations increases strongly at higher temperatures. This temperature dependence is summarized in Fig. 3b, d together with the data for a thicker film of 20 ML. In both cases, we find long oscillation lifetimes at low temperatures that can be extrapolated to about 683 and 587 fs at 0 K for 9 and 20 ML, respectively. The temperature dependence of the oscillation lifetime follows the functional form $${(1-T/{T}^{\prime})}^{\nu }$$, where ν is the critical exponent for the spin correlation length ν = 0.64 ± 0.03 extracted from neutron scattering21. This observation provides a first hint that the oscillations are connected with spin correlations. The temperature $${T}^{\prime}$$, where the oscillation lifetime drops to zero, is close to the Néel temperature TN: $${T}^{\prime}\simeq 1.15\,{T}_{\mathrm{N}}$$, where TN has been determined experimentally by the disappearance of the antiferromagnetic (2 × 1) superstructure seen in low-energy electron diffraction (LEED), see Fig. 3c and Supplementary Note 1. We argue that the slightly larger value of $$T^{\prime}$$ is due to short-range spin correlations, which persist above TN and influence the time-resolved photoemission experiments. A similar persistence of short-range magnetic order above the Curie temperature has been observed in ferromagnetic iron22. The temperature dependence of the oscillation lifetime shown in Fig. 3b is valid for the 20 ML and the 9 ML film, despite their different TN and $${T}^{\prime}$$ (cf. Fig. 3d). Note that the reduction of TN for the thin NiO films with respect to the NiO bulk value is in agreement with ref. 23. The observed temperature dependence of the oscillation lifetime with a critical transition close to the Néel temperature points clearly to a dominant role of the local AFM spin ordering and constitutes our main finding.

Whereas the observed amplitude at t = 0 is only weakly temperature dependent, it shows a strong threshold behavior with respect to the pump-photon energy hνpump. The hνpump dependence is exemplified in Fig. 3e for the 9 ML film, with a clear onset at hνpump = 3.8 eV that agrees with the magnitude of the charge-transfer energy Δ. Note that the silver substrate has a plasmon resonance at a similar energy of 3.8 eV, which in principle could lead to plasmonic coupling across the interface24. However, we consider an excitation across the interface less likely for two reasons: the substrate plasmon should lead to a peaked response at 3.8 eV24, whereas the THz oscillations show an onset beginning at 3.8 eV with an increasing excitation cross section with higher energy. Secondly, the oscillations are observed over a wide range of NiO thicknesses with no indication for an interface-related transport mechanism nor a loss of coherence with increasing film thickness. Therefore, we consider the THz oscillations as triggered by the excitation of an O 2p electron across the charge-transfer gap into the UHB. In the following, we will focus on the electron dynamics activated by the photoexcitation into the UHB, while the lower t2g states will not be considered. On the intermediate state energy scale, the photoelectrons that exhibit THz oscillations originate from the photoemission onset up to 1.5 eV above EF (Supplementary Note 1), which implies an in-gap state located at around 1 eV below the UHB as marked in Fig. 1b in light red.

### Two-band t-J model

To describe the observed electron dynamics in NiO, we developed a two-band extension of the t-J model in the AFM phase, as detailed in “Methods” section. The relevant parameters of this model are the Hund coupling JH and the exchange coupling Jex. In equilibrium, the ground state is dominated by local high-spin states S = 1 (S1 in Fig. 5) forming an AFM spin configuration. The remaining local many-body states are illustrated in Fig. 5 and include the triply occupied state (triplon, T), the minority high-spin states (S1) and two types of low-spin S = 0 Hund excitations. In the first type, the two electrons occupy two different orbitals with an energy cost of JH (S0) relative to the ground state, whereas in the second type the same orbital is occupied by two electrons with an energy cost of 3JH (S0). These localized excitations have been the subject of intense experimental25,26,27 and theoretical28,29,30 investigations. In addition, a spin-flip excitation S1 at energy zJex is taken into account, where z denotes the number of exchange-coupled neighbors.

In the calculated equilibrium spectral function (black line in Fig. 4a), subbands of the UHB at energies 4.5 and 6 eV can be identified. These dominant peaks are related to Hund excitations31. In addition, we can identify small sidebands which originate from the coupling of charge carriers to the AFM background and the presence of string states32,33,34,35,36. To model the pumping by UV excitations across the charge-transfer gap Δ in the experiments, we excite the system in the theoretical description by photo-doping of electrons near the lower edge of the UHB. During the photo-doping process, triplons T are produced. In the experiments, the photoexcited system is subsequently probed by photoemission, which corresponds to the removal of an electron from the system. The removal of one electron from the triplon state results in a low-spin S0 or S0 state, which produces the in-gap features evident in the light blue spectrum in Fig. 4a. The energy splitting of the Hund excitation S0 (S0) from the lowest subband of the UHB is given by the Hund coupling JH (3JH), see white arrows. There is an additional set of in-gap states shifted down in energy by JH, see Supplementary Note 2 for details. As shown in Fig. 4b, which plots the calculated time-dependent photoemission spectrum37 (see Eq. (7) in “Methods” section), these in-gap states are substantially populated. This analysis allows us to identify the dominant feature in the experiments, shown in Fig. 2a, as the Hund excitation S0. In agreement with the experimental findings, all these in-gap features exhibit transient oscillations, which will be analyzed in the following.

The time-resolved photoemission intensity shows dominant fast oscillations associated with Hund excitations. In Fig. 4c, we plot the energy-integrated intensity in the black energy window marked in Fig. 4b by a thin green line. The corresponding Fourier transform, shown in Fig. 4d exhibits a low-energy peak at ω of the order of Jex and pronounced high-energy peaks near ω ≈ 0.7JH and ω ≈ JH. As the fast Hund oscillations cannot be resolved in the experiment, we apply a low-pass filter that extracts the slow oscillations associated with the low-energy peak (see “Methods” section for details). The slow oscillations are shown by the black line and are found to scale with Jex. We thus attribute their origin to the AFM spin correlations in NiO. To further support this interpretation, we compare the time evolution of the kinetic energy below and at TN in Fig. 4e. Indeed, oscillations with long coherence time only appear at temperatures below TN. As shown in Fig. 4f, the frequency of these oscillations scales approximately linearly with the superexchange Jex. We find ω/Jex ≈ 3, but cannot exclude lower frequency contributions due to the limited propagation time. The dominant contribution roughly matches the distance between the string states in the two-band t-J model. We determined the exchange interaction Jex=100 meV by matching the equilibrium TN with the experiment. This choice leads to exchange interactions which are larger than the realistic values, see “Methods”, and thus to faster oscillations in the numerical data.

The initial fast relaxation of the photoexcited triplons happens on a time scale of 10 fs and results from the scattering of triplons with the high-spin states, producing Hund excitations S = 038. Since the Hund excitations are mainly determined by the local JH, they are little influenced by the AFM background. As a result, the initial relaxation process in Fig. 4e is similar below and above TN. The initial relaxation is strongly enhanced for high-frequency excitations, where photoexcited electrons are injected up to the upper edge of the UHB. In these processes, the local spin configurations can absorb energy quanta of JH or 3JH on the time scale of the inverse hopping amplitude 1/t0, which is a very efficient dissipation mechanism. This relaxation channel closes once the excess kinetic energy drops below JH, since this is the minimum energy that can be dissipated by local spin excitations. This is the Hund’s equivalent of the phonon-window effect39,40.

## Discussion

Coherent electron dynamics has been observed in isolated quantum systems with few relevant energy levels41 or for image potential states42,43. While these phenomena are related to well isolated quasiparticle physics44, coherent many-body phenomena are more exotic. Recently, such an observation has been reported in a large array of trapped cold atoms45. Various solid-state systems exhibit coherent oscillations, which can emerge for impurity46 and surfaces states47,48 or due to the strong coupling to collective49,50 and phononic modes51. However, our work provides the first observation of coherent long-lived many-body dynamics resulting from a coupling to the AFM background in a solid. Similar phenomena have been detected in cuprates exhibiting oscillations with a very short lifetime52, but their origin remains controversial53,54.

The experimental observations raise the central question how a strongly correlated solid can support coherent oscillations on such a long-time scale (up to 2 ps). In order to avoid dephasing due to the coupling with a continuum of states, there must exist a small subset of many-body states that are well isolated. The theoretical analysis in Fig. 5b allows us to follow the occupation of the local many-body states in the time evolution. After the photoexcitation, the majority high-spin states (dark blue) are depopulated at the expense of low-spin Hund excitations. The subsequent dynamics is governed by coherent oscillations between the high- and the low-spin states, which can be more clearly seen in Fig. 5c, where the slow background has been subtracted. Since the kinetic energy of the triplons is too small to produce Hund excitations (Ekin < JH), the triplons are trapped between nearest-neighbor sites as exemplarily shown in Fig. 5d(iii). This also explains why the in-gap states in the PES are isolated. An analogous process is possible for the high-energy Hund excitations S0 as well. Furthermore, each creation of a low-spin excitation is accompanied by a ferromagnetic (FM) disturbance in the spin background, which produces a feedback on the spin system in the form of an effective magnetic field B(t) and induces the coherent oscillations at a frequency controlled by the superexchange interaction Jex. A string of FM distortions is rapidly created by the pump pulse, which acts as a displacive excitation within the string potential. A natural consequence is a cosine-like response, as evident in the experimental data (Fig. 3), and the simulation results (Fig. 4).

The lifetime of the coherent oscillations is determined by their decay channels. A higher-order process leads to the formation of a FM domain wall and results in a steady increase in the population of the minority high-spin configuration S1 (Supplementary Note 2). This formation of FM strings is one possible relaxation process. However, being a higher-order process with rather low probability, it cannot be responsible alone for the decay of the THz oscillations with a time constant of about τ = 500 fs as observed in the experiments. A spreading of localized FM excitations via magnons can provide an alternative decay channel, which is not captured by the approximate theoretical description considered here and would be an interesting subject for future investigations. The binding of singlons and triplons and its impact on the recombination rate is not properly captured in DMFT, but should not play an important role in the dilute photoexcitation limit explored in the experiments55. The experimental observation of oscillations slightly above TN cannot be explained by our theoretical treatment, which relies on a mean-field decoupling of the spin–spin interaction. It would be interesting to extend the formalism to fluctuating spins53 in a future study.

In conclusion, we have used two-photon photoemission experiments and a two-band t-J model to investigate the evolution of ultrafast photo-induced in-gap states in strongly correlated NiO. Long-lived coherent oscillations at THz frequencies are observed in the experiments and identified in the theoretical modeling as a signature of the coupling between antiferromagnetically correlated spins and local Hund excitations. Contrary to common belief, this work shows that many-body coherent dynamics can be observed in solid-state systems despite the intrinsic coupling to numerous active degrees of freedom. Our findings indicate an important interplay between local electronic excitations and magnetic order. The combined experimental and theoretical results pave the way to systematic explorations of the many-body physics in correlated oxides driven out of equilibrium by ultrafast above-band-gap excitations. An obvious extension of our work would be the investigation of similar multi-band effects in cold-atom experiments, where two-dimensional antiferromagnetism has been realized recently56. The experimental observation of oscillations above the Néel temperature has interesting implications for cold-atom experiments on doped AFMs, where even in the absence of long-range order magnetic correlations have been detected in quantum gas spectroscopy57.

## Methods

### Theoretical methods

Crystallographically, NiO is a binary centrosymmetric cubic system of the m3m symmetry class in a paramagnetic state. Each Ni2+ ion is surrounded by six O2− ions and has an effective spin S = 1. The ligand ions mediate the superexchange interaction between the next-nearest neighbour (NNN) spins (along the 〈100〉 and equivalent directions) resulting in a J2 > 0 exchange constant of the antiferromagnetic type (Fig. 5a). The spin coupling between the nearest neighbour (NN) Ni-ions (along the 〈110〉 and equivalent directions) is much weaker and is of the ferromagnetic type (J1 < 0). Below the Néel temperature, the two exchange interactions result in type-II magnetic order in which the spins of the Ni2+ ions are aligned ferromagnetically within {111} planes that form an antiferromagnetic order26. The exchange constants J1 and J2 are well-known from experiments and according to Hutchings and Samuelsen58 are  − 1.37 and 19.0 meV, respectively. The ferromagnetic coupling J1, which is much smaller than the observed oscillation frequency hνb ≈ 17 meV, is neglected resulting in four independent interpenetrating simple cubic lattices58. Within each sublattice, there are only the antiferromagnetic interactions between the spins mediated by the ligand p-orbitals (predominately a kinetic superexchange which is second order in the hopping between the Ni2+ d-states and O2−p-states59). Since we do not treat the superexchange mechanism explicitly, the ligand O-sites are removed from the consideration. Thus, the focus of the theory is a cubic sublattice of spins with AFM nearest neighbour spin-interactions and coordination number z = 6.

In the microscopic modeling, we have to consider that each on-site spin S = 1 is formed by two elemental spins (s = 1/2) located on the respective eg orbitals. We only retain doubly occupied and triply occupied states (doublons and triplons) and project out the local many-body states which are either unoccupied, singly or fully occupied by introducing the projected operators $${\tilde{c}}_{i\alpha \sigma }={P}^{\dagger }{c}_{i\alpha \sigma }P,$$ where ciασ is the annihilation operator for site i, orbital α and spin σ and P is the local projector. The resulting Hamiltonian is composed of the local, kinetic and the superexchange parts, H = Hloc + Hkin + Hex.  The local part captures the Hubbard and the Hund physics,

$${H}_{\text{loc}}= \,U\sum\limits _{i,\alpha }{\tilde{n}}_{i,\alpha \uparrow }{\tilde{n}}_{i\alpha \downarrow }-\mu \sum\limits _{i\alpha \sigma }{\tilde{n}}_{i\alpha \sigma }+\sum\limits _{i,\alpha <\beta }\sum\limits _{\sigma ,\sigma ^{\prime} }(U^{\prime} -{J}_{\mathrm{H}}{\delta }_{\sigma \sigma ^{\prime} }){\tilde{n}}_{i\alpha \sigma }{\tilde{n}}_{i\beta \sigma ^{\prime} }\\ \,+\gamma {J}_{\mathrm{H}}\sum\limits _{i,\alpha <\beta }\left({\tilde{c}}_{i\alpha \uparrow }^{\dagger }{\tilde{c}}_{i\alpha \downarrow }^{\dagger }{\tilde{c}}_{i\beta \downarrow }{\tilde{c}}_{i\beta \uparrow }+{\tilde{c}}_{i\alpha \uparrow }^{\dagger }{\tilde{c}}_{i\beta \downarrow }^{\dagger }{\tilde{c}}_{i\alpha \downarrow }{\tilde{c}}_{i\beta \uparrow }\right),$$
(1)

where U, $$U^{\prime}$$, and JH is the intra-orbital Coulomb, inter-orbital Coulomb, and the Hund’s exchange interaction, respectively. The interaction in Eq. (1) is the Kanamori interaction, which becomes rotationally invariant when $$U^{\prime} =U-2{J}_{\mathrm{H}}$$ and γ = 1. Here we set $$U^{\prime} =U-2{J}_{\mathrm{H}}$$ but employ the Ising approximation (with γ = 0). We have assumed no crystal-field splitting between the two eg orbitals. The half-filling condition is imposed by adjusting the chemical potential to μ = (3U − 5JH)/2.

$${H}_{\text{ex}}={J}_{\text{ex}}\sum _{\alpha \left\langle ij\right\rangle }{{\bf{s}}}_{i\alpha }\cdot {{\bf{s}}}_{j\alpha },$$
(2)

where s is a spin-$$\frac{1}{2}$$ operator, 〈ij〉 denotes nearest-neighbor indices on the AFM sublattice, and the subscript α labels different eg orbitals. Due to the presence of long-range order, a mean-field approximation to Eq. (2) is justified, which allows to treat the feedback of the magnetization on the electron dynamics at the Ising level. Comparing the energy cost of a single spin-flip 2zJ2 with that given by Eq. (2) in the Ising approximation, zJex, we infer the relation 2J2 = Jex between the macroscopic and the microscopic exchange parameters.

The formation of AFM spin-order competes with the delocalization of triplon states, with the latter described by

$${H}_{\text{kin}}=-{t}_{0}\sum _{\langle i,j\rangle }\sum _{\alpha \sigma }({\tilde{c}}_{i\alpha \sigma }^{\dagger }{\tilde{c}}_{j\alpha \sigma }+{\tilde{c}}_{j\alpha \sigma }^{\dagger }{\tilde{c}}_{i\alpha \sigma }).$$
(3)

Here t0 denotes the nearest-neighbor hopping in the AFM sublattice and $${\tilde{c}}^{\dagger }$$ the (projected) electron creation operator. The competition between the kinetic and the superexchange term results in string-like states at the lower edge of the UHB32,33,34,35.

To solve the electron dynamics, we use the non-equilibrium dynamical mean-field theory60, which maps a correlated lattice problem onto a self-consistently determined impurity problem. While recent theoretical developments have enabled the description of photo-doped charge-transfer insulators, state-of-the-art simulations are still limited to the paramagnetic phase and short-time dynamics61,62. In order to simplify the description, we treat the impurity problem using the projection onto the subspace of relevant local many-body states63. In addition, we use the lowest-order strong coupling expansion, the so-called non-crossing approximation (NCA)53,64. In order to get access to the long-time behavior, we furthermore solve the problem on a Bethe lattice rather than a cubic lattice, which does not compromise the analysis of the proposed mechanism since the realistic system is not one-dimensional (spin-charge separation does not play a role).

We explicitly break the AFM symmetry and treat the feedback of the AFM order on the mean-field level, see Eq. (2), as

$${J}_{\text{ex}}\sum _{\alpha \langle ij\rangle }{{\bf{s}}}_{i\alpha }{{\bf{s}}}_{j\alpha }\to {J}_{\text{ex}}\sum _{\alpha }\sum _{i}\sum _{j{\text{nn}}\,i}{{\bf{s}}}_{i\alpha }^{z}\langle {{\bf{s}}}_{j\alpha }^{z}\rangle$$
(4)

for both eg orbitals assuming the spin polarization in the z direction. In the large dimensionality limit, the Hartree term will be dominant and the mean-field decoupling becomes exact. In DMFT, we need to rescale the superexchange as Jex → J*/z,34 where z is the connectivity. The cavity construction, in combination with the mean-field decoupling, leads to

$${H}_{\text{ex}}=\frac{{J}^{* }}{z}\sum _{\alpha \langle i,0\rangle }{\langle {{\bf{s}}}_{i\alpha }^{{\bf{z}}}\rangle }_{0}{{\bf{s}}}_{0\alpha }^{{\bf{z}}}={J}^{* }\sum _{\alpha }{\langle {{\bf{s}}}_{0\alpha ,B}^{{\bf{z}}}\rangle }_{0}{{\bf{s}}}_{0\alpha ,A}^{{\bf{z}}}=-{J}^{* }\sum _{\alpha }\langle {{\bf{s}}}_{0\alpha ,A}^{{\bf{z}}}\rangle {{\bf{s}}}_{0\alpha ,A}^{{\bf{z}}},$$
(5)

where 〈〉0 marks the expectation value in the cavity setup. We have assumed a bipartite lattice with sublattices A and B, and the AFM relation $$\langle {{\bf{s}}}_{0\alpha ,A}^{{\bf{z}}}\rangle =-\langle {{\bf{s}}}_{0\alpha ,B}^{{\bf{z}}}\rangle$$. In the last step, we have used the assumption of a high-dimensional Bethe lattice to convert the cavity expectation value into a usual lattice expectation value. To roughly match the parameter J* to the physical Jex in our lattice with z = 6, one should choose (Jex/BWNiO) = (J*/BWBethe)/6. With the bandwidths BWNiO ≈ 2–3 eV and BWBethe = 4 eV, this leads to J* ≈ 10Jex. In practice, we choose J* = 0.1 eV (unless otherwise specified), which also yields a reasonable value for the Néel temperature, TNéel ≈ 500K. To simplify the discussion of the theoretical results in the main text, we furthermore use the notation J* → Jex.

The model parameters other than Jex are taken from equilibrium LDA+U65 and LDA+DMFT12,66 studies, which used U ≈ 8 eV, JH ≈ 1.0 eV. We infer the effective hopping t0 ≈ 1 eV from the bandwidth of the UHB in the calculation of Kuneš et al.12 and from the bandwidth of the d9 state in the Bremsstrahlung isochromate spectroscopy15. The hopping t0 is of indirect nature and is mediated by the hopping through the oxygen 2p states as explained by Zhang and Rice in the case of cuprates67.

The slow oscillations in Fig. 4 have been extracted by a low-pass filter with a Fermi-function shape $$f(\omega )=1.0/(\exp (\beta (\omega -{\omega }_{0}))+1),$$ see shaded region in Fig. 4d. The presented results are for Jex = 0.2 and the parameters β = 100 and ω0 = 0.8.  As the main limitation in the comparison between the theoretical and experimental results is the maximum propagation time and the corresponding mismatch in the superexchange, we propose this problem as a perfect testbed for future advances in the theoretical modeling of strongly correlated systems out of equilibrium.

In order to model the excitation from a deep fully occupied p-band to the UHB, we approximate the excitation by a sudden electron doping in a certain energy range of the UHB. This is justified, since the time scale for the UV excitation at hν = 4.1 eV corresponds to 0.1 fs according to Heisenberg’s uncertainty principle. The UV pump pulse may thus be modeled within the rotating wave approximation exciting electrons from fully occupied orbitals. In all simulations, we have attached the full bath at the lower edge of the UHB in the energy range ω = [3, 4] eV for the time interval [0, 2]  fs. The time-dependent spectral function A(ωt) is obtained from the retarded component of the local Green’s function GR as

$$A(\omega ,t)=-\frac{1}{\pi }{\rm{Im}}\int_{t}^{t+{t}_{\text{cut}}}{\mathrm{d}}t^{\prime} {e}^{{\rm{i}}\omega (t^{\prime} -t)}{G}^{R}(t^{\prime} ,t),$$
(6)

and the photoemission spectrum is obtained from the lesser component G< as

$$\,{\text{PES}}\,(\omega ,t)=\frac{1}{\pi }{\rm{Im}}\int_{t}^{t+{t}_{\text{cut}}}{\mathrm{d}}t^{\prime} {e}^{{\rm{i}}\omega (t^{\prime} -t)}{G}^{\,{<}\,}(t^{\prime} ,t).$$
(7)

Due to the limited propagation time, we have employed a forward integration for the Fourier transform in contrast to ref. 37, where the Fourier integral has been done in relative time.

### Experimental methods

The experiments have been performed in an ultrahigh vacuum chamber equipped with a 150 mm hemispherical electron analyzer (Phoibos 150, SPECS, Berlin) with a 2D CCD detector and low-energy electron diffraction optics as described in detail elsewhere68,69. A broadly tunable femtosecond laser system operated at a repetition rate of 1.4 MHz with two noncollinear optical parametric amplifiers (NOPA), pumped by a 20 W fiber laser (IMPULSE, Clark-MXR, Dexter) is used for the 2PPE68. The two frequency-doubled beams of the NOPAs serve as pump-probe excitation source with typical pump-probe cross-correlation widths between 70 and 100 fs (full-width-at-half-maximum).

NiO thin films have been prepared by reactive Ni deposition at room temperature on a Ag(001) single crystal in 10−6 mbar O2. Subsequent annealing is applied to achieve long-range order as monitored by LEED and high-resolution electron energy loss spectroscopy70.

## Data availability

The data that support the plots and other findings presented in this paper are available from the corresponding author upon reasonable request.

## Code availability

The numerical code used to calculate the results for this work is available from the corresponding author upon reasonable request.

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## Acknowledgements

The calculations have been performed on the Beo04 cluster at the University of Fribourg and on the Rusty cluster at the Flatiron Institute using a software library developed by M. Eckstein and H.U.R. Strand. The Flatiron Institute is a division of the Simons Foundation. K.G., C.-T.C., and W.W. acknowledge financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) through SFB 762 (A3, B8) and SFB/TRR 227 (A06). D.G., N.B. and P.W. were supported by Swiss National Science Foundation grant 200021-165539 and the European Research Council through ERC consolidator grant No. 724103. Y. P. acknowledges support of the DFG via SFB/TRR 173. D.G. would like to acknowledge R. Sesny for help with graphical representations. Open access funding provided by Projekt DEAL.

## Author information

Authors

### Contributions

K.G. and W.W. designed the experiments. K.G. prepared the samples and measured the 2PPE data. The data analysis and evaluation was done by K.G., C.-T.C., Y.P., and W.W., D.G., N.B., Y.P. and P.W. constructed the model description and D.G. carried out the DMFT calculations. All authors discussed the results and co-wrote the paper.

### Corresponding authors

Correspondence to Philipp Werner or Wolf Widdra.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Mark Dean and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Gillmeister, K., Golež, D., Chiang, CT. et al. Ultrafast coupled charge and spin dynamics in strongly correlated NiO. Nat Commun 11, 4095 (2020). https://doi.org/10.1038/s41467-020-17925-8

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41467-020-17925-8


text:
# Math Help - matlab code helpppp :(

1. ## matlab code helpppp :(

helpp!! i need a MATLAB CODE!

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest number that is evenly divisible by all of the numbers from 1 to 20?

i need the matlab code for this it is so confusing i got other codes but they do not seem to be working

2. Originally Posted by savni98
helpp!! i need a MATLAB CODE!

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest number that is evenly divisible by all of the numbers from 1 to 20?

i need the matlab code for this it is so confusing i got other codes but they do not seem to be working
First decide what algorithm you are going to employ, then post it here and explain what it is with the algorithm you are having trouble turning into code.

CB

3. i got some of the code i think i dont know what to do ??

for i = 20:2:1000000000
t = 0;
for j = 2:20
t += 1
end
end
if t == 19

without the code it goes

20=2*2*5
19
18=2*3*3
17
16=2*2*2*2
15=3*5
14=2*7
13
12=2*6
11
10=2*5
9=3*3
8=2*2*2
7
6=2*3
5
4=2*2
3
2
1

2*2*2*2*3*3*5*7*11*13*17*19=232,792,56

4. Originally Posted by savni98
i got some of the code i think i dont know what to do ??

for i = 20:2:1000000000
t = 0;
for j = 2:20
t += 1
end
end
if t == 19

without the code it goes

20=2*2*5
19
18=2*3*3
17
16=2*2*2*2
15=3*5
14=2*7
13
12=2*6
11
10=2*5
9=3*3
8=2*2*2
7
6=2*3
5
4=2*2
3
2
1

2*2*2*2*3*3*5*7*11*13*17*19=232,792,56
Start with p=20, then loop from n= 19 down to 1 with update p=lcm(n,p)

so the core code would look something like:

Code:
p=20
n=19:-1:1
p=lcm(p,n);
end
CB

format: plain


text:
# Prove that 0 is in the convex hull of points chosen from each orthant

If we arbitrarily choose a point from each orthant in $$\mathbb{R}^n$$, that is we choose $$2^n$$ points in total, how do we prove that 0 is in the convex hull of these $$2^n$$ points? It seems obvious, but when I sit down and start thinking about it, I couldn't definitely find a set of $$\lambda_i, i = 1,2,\cdots,2^n$$ such that $$0 \leq \lambda_i \leq 1, \sum_i \lambda_i = 1$$ and $$0 = \sum_i \lambda_i x_i$$, where $$x_i, i = 1,2,\cdots,2^n$$ are any set of points satisfying the condition.

In the $$n=1$$ base case, you have a positive number and a negative number; $$0$$ can be represented as a convex combination of them.
For $$n>1$$: Given your $$2^n$$ points, one from each orthant, divide them into 2 sets of size $$2^{n-1}$$, where the first set has points whose last coordinates are positive and the second set where the last coordinates are negative. By the inductive hypothesis there is a convex combo of the first set of points such that the first $$n-1$$ coordinates vanish, and similarly for the second set. The last coordinates of these two convex combos are positive and negative, so there is a convex combo of them that is zero.
Take in pairs the points that differ in sign for the coordinate $$n$$ (and only that one) and form the convex combinations such that that coordinate is cancelled.
Now you have $$2^{n-1}$$ points in each orthants of the diminished space.