In organic light-emitting diodes (OLEDs), understanding the efficiency loss mechanism known as exciton-polaron quenching (EPQ) has been a longstanding challenge. Traditionally, EPQ was believed to occur mainly within the bulk of the OLED emission layer (EML) due to the coexistence of polarons and excitons, limiting our understanding of its full impact. Here, we report a previously overlooked phenomenon termed “interfacial EPQ (Inf. EPQ),” which occurs at the heterointerface between the EML and the adjacent charge transport layer (CTL). We observe the direct transfer of EML excitons to CTL polarons across this interface, spanning distances of up to 4 nm. Notably, Inf. EPQ exceeds EPQ within the EML bulk, even in devices with a slight interfacial energy barrier ($>0.2\mathrm{eV}$). We propose a methodology that enables independent probing of Inf. EPQ, systematic validation of its essential parameters, and the identification of Dexter energy transfer as its governing mechanism. Importantly, our findings highlight Inf. EPQ as a ubiquitous phenomenon across devices with various luminescent mechanisms, wavelengths, and injection levels. By effectively addressing Inf. EPQ, we achieve efficiency enhancements in red, green, and blue phosphorescent OLEDs, by $70\%\pm 8\%$, $47\%\pm 5\%$, and $66\%\pm 6\%$, respectively. Our work advances the physical understanding of exciton and polaron dynamics at organic heterointerfaces, providing universally applicable solutions to Inf. EPQ-related issues in practical devices.

We introduce the concept of “quantum geometric nesting” (QGN) to characterize the idealized ordering tendencies of certain flat-band systems implicit in the geometric structure of the flat-band subspace. Perfect QGN implies the existence of an infinite class of local interactions that can be explicitly constructed and give rise to solvable ground states with various forms of possible fermion bilinear order, including flavor ferromagnetism, density waves, and superconductivity. For the ideal Hamiltonians constructed in this way, we show that certain aspects of the low-energy spectrum can also be exactly computed including, in the superconducting case, the phase stiffness. Examples of perfect QGN include flat bands with certain symmetries (e.g., chiral or time reversal) and non-symmetry-related cases exemplified with an engineered model for pair-density wave. Extending this approach, we obtain exact superconducting ground states with nontrivial pairing symmetry.

We describe coupled nonequilibrium electron-phonon systems semiclassically—Ehrenfest dynamics for the phonons and quantum mechanics for the electrons—using a classical Monte Carlo approach that determines the nonequilibrium response to a large pump field. The semiclassical approach is expected to be accurate, because the phonons are excited to average energies much higher than the phonon frequency, eliminating the need for a quantum description. The numerical efficiency of this method allows us to perform a self-consistent time evolution out to very long times (tens of picoseconds), enabling us to model pump-probe experiments of a charge-density-wave (CDW) material. Our system is a half-filled, one-dimensional (1D) Holstein chain that exhibits CDW ordering due to a Peierls transition. The chain is subjected to a time-dependent electromagnetic pump field that excites it out of equilibrium, and then a second probe pulse is applied after a time delay. By evolving the system to long times, we capture the complete process of lattice excitation and subsequent relaxation to a new equilibrium, due to an exchange of energy between the electrons and the lattice, leading to lattice relaxation at finite temperatures. We employ an indirect (impulsive) driving mechanism of the lattice by the pump pulse due to the direct driving of the electrons. We identify two driving regimes, where the pump can either cause small perturbations or completely invert the initial CDW order. Our work successfully describes the ringing of the amplitude mode in CDW systems that has long been seen in experiment but never successfully explained by microscopic theory. We also describe the fluence-dependent crossover that inverts the CDW order parameter and changes the phonon dynamics. Finally, we illustrate how this method can examine a number of different types of experiments including photoemission, x-ray diffraction, and two-dimensional (2D) spectroscopy.

The interplay of dynamical correlations and electronic ordering is pivotal in shaping phase diagrams of correlated quantum materials. In magic-angle twisted bilayer graphene, transport, thermodynamic, and spectroscopic experiments pinpoint a competition between distinct low-energy states with and without electronic order, as well as between localized and delocalized charge carriers. In this study, we utilize dynamical mean-field theory on the topological heavy fermion model of twisted bilayer graphene to investigate the emergence of electronic correlations and long-range order in the absence of strain. We contrast moment formation, Kondo screening, and ordering on a temperature basis and explain the nature of emergent correlated states based on three central phenomena: (i) the formation of local spin and valley isospin moments around 100 K, (ii) the ordering of the local isospin moments around 10 K preempting Kondo screening, and (iii) a cascadic redistribution of charge between localized and delocalized electronic states upon doping. At integer fillings, we find that low-energy spectral weight is depleted in the symmetric phase, while we find insulating states with gaps enhanced by exchange coupling in the zero-strain ordered phases. Doping away from integer filling results in distinct metallic states: a “bad metal” above the ordering temperature, where scattering off the disordered local moments suppresses electronic coherence, and a “good metal” in the ordered states with coherence of quasiparticles facilitated by isospin order. This finding reveals coherence from order as the microscopic mechanism behind the Pomeranchuk effect observed experimentally by Rozen et al. [Nature (London) 592, 214 (2021)] and by Saito et al. [Nature (London) 592, 220 (2021)]. Upon doping, there is a periodic charge reshuffling between localized and delocalized electronic orbitals leading to cascades of doping-induced Lifshitz transitions, local spectral weight redistributions, and periodic variations of the electronic compressibility ranging from nearly incompressible to negative. Our findings highlight the essential role of charge transfer, hybridization, and ordering in shaping the electronic excitations and thermodynamic properties in twisted bilayer graphene and provide a unified understanding of the most puzzling aspects of scanning tunneling spectroscopy, transport, and compressibility experiments.

Open system quantum dynamics can generate a variety of long-range entangled mixed states, yet it has been unclear in what sense they constitute phases of matter. To establish that two mixed states are in the same phase, as defined by their two-way connectivity via local quantum channels, we use the renormalization group (RG) and decoders of quantum error correcting codes. We introduce a real-space RG scheme for mixed states based on local channels which ideally preserve correlations with the complementary system, and we prove this is equivalent to the reversibility of the channel’s action. As an application, we demonstrate an exact RG flow of finite temperature toric code in two dimensions to infinite temperature, thus proving it is in the trivial phase. In contrast, for toric code subject to local dephasing, we establish a mixed-state toric code phase using local channels obtained by truncating an RG-type decoder and the minimum weight perfect matching decoder. We also discover a precise relation between mixed-state phase and decodability, by proving that local noise acting on toric code cannot destroy logical information without bringing the state out of the toric code phase.

Symmetries of three-dimensional periodic scalar fields are described by 230 space groups (SGs). Symmetries of three-dimensional periodic (pseudo)vector fields, however, are described by the spin-space groups (SSGs), which were initially used to describe the symmetries of magnetic orders. In SSGs, the real-space and spin degrees of freedom are unlocked in the sense that an operation could have different spatial and spin rotations. SSGs give a complete symmetry description of magnetic structures and have natural applications in the band theory of itinerary electrons in magnetically ordered systems with weak spin-orbit coupling. Altermagnetism, a concept raised recently that belongs to the symmetry-compensated collinear magnetic orders but has nonrelativistic spin plitting, is well described by SSGs. Because of the vast number and complicated group structures, SSGs have not yet been systematically enumerated. In this work, we exhaust SSGs based on the invariant subgroups of SGs, with spin operations constructed from three-dimensional (3D) real representations of the quotient groups for the invariant subgroups. For collinear and coplanar magnetic orders, the spin operations can be reduced into lower-dimensional real representations. As the number of SSGs is infinite, we consider only SSGs that describe magnetic unit cells up to 12 times crystal unit cells. We obtain 157 289 noncoplanar, 24 788 coplanar-noncollinear, and 1421 collinear SSGs. The enumerated SSGs are stored in an online database with a user-friendly interface. We develop an algorithm to identify SSGs for realistic materials and find SSGs for 1626 magnetic materials. We also discuss several potential applications of SSGs, including the representation theory, topological states protected by SSGs, structures of spin textures, and refinement of magnetic neutron diffraction patterns using SSGs. Our results serve as a solid starting point for further studies of symmetry and topology in magnetically ordered materials.

Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and underlying properties of magnetic ordered materials. However, the basic theory of SSG has seldom been developed. In this work, we present a systematic study of the enumeration and the representation theory of the SSG. Starting from the 230 crystallographic space groups and finite translation groups with a maximum order of eight, we establish an extensive collection of over 100 000 SSGs under a four-index nomenclature as well as international notation. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. To facilitate the identification of the SSG, we develop an online program that can determine the SSG symmetries of any magnetic ordered crystal. Moreover, we derive the irreducible corepresentations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSG symmetries and physical effects beyond the framework of magnetic space groups through several representative material examples, including a candidate altermagnet ${\mathrm{RuO}}_2$, spiral spin polarization in the coplanar antiferromagnet ${\mathrm{CeAuAl}}_3$, and geometric Hall effect in the noncoplanar antiferromagnet ${\mathrm{CoNb}}_3{\mathrm{S}}_6$. Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials.

In this work, we exhaust all the spin space symmetries, which fully characterize collinear, noncollinear, and commensurate spiral as well as incommensurate spiral magnetism, etc., and investigate enriched features of electronic bands that respect these symmetries. We achieve this by systematically classifying the so-called spin space groups (SSGs)—joint symmetry groups of spatial and spin operations that leave the magnetic structure unchanged. Generally speaking, they are accurate (approximate) symmetries in systems where spin-orbit coupling (SOC) is negligible (finite but weaker than the energy scale of interest), but we also show that specific SSGs could remain valid even in the presence of strong SOC. In recent years, SSGs have played increasingly pivotal roles in various fields such as altermagnetism, topological electronic states, and topological magnon, etc. However, due to its complexity, a complete SSG classification has not been completed up to now. By representing the SSGs as $\mathrm{O}(N)$ representations, we—for the first time—obtain the complete classifications of 1421, 9542, and 56 512 distinct SSGs for collinear ($N=1$), coplanar ($N=2$), and noncoplanar ($N=3$) magnetism, respectively. SSG not only fully characterizes the symmetry of spin degrees of freedom, but also gives rise to exotic electronic states, which, in general, form projective representations of magnetic space groups (MSGs). Surprisingly, electronic bands in SSGs exhibit features never seen in MSGs, such as (i) nonsymmorphic SSG Brillouin zone, where SSG operations behave as a glide or screw when acting on momentum, (ii) effective $\pi$ flux, where translation operators anticommute with each other and yield duplicate bands, (iii) higher-dimensional representations unexplained by MSGs, and (iv) unconventional spin texture on a Fermi surface, which is completely determined by SSGs, independent of Hamiltonian details. To apply our theory, we identify the SSG for each of the 1595 published magnetic structures in the MAGNDATA database on the Bilbao Crystallographic Server. Material examples exhibiting the novel features (i)–(iv) are discussed with emphasis. We also investigate new types of SSG-protected topological electronic states that are unprecedented in MSGs. In particular, we propose a 3D ${\mathbb{Z}}_2$ topological insulator state with a fourfold degenerate Dirac point on the surface and a new scenario of anomalous ${\mathbb{Z}}_2$ helical states that appear on magnetic domain walls.

We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an $M$th-order diagram at inverse temperature $\beta$ and spectral width ${\omega }_{\max }$ from $\mathcal{O}\mathbf{(}(\beta {\omega }_{\max }{)}^{2M-1}\mathbf{)}$ for a direct quadrature to $\mathcal{O}\mathbf{(}M(\log (\beta {\omega }_{\max }){)}^{M+1}\mathbf{)}$, with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams.

We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number $n$ of eigenstates. In the case $n=2$, this ansatz reproduces ETH. For $n=4$ it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits.

Imaging of nanoscale magnetic textures within extended material systems is of critical importance to both fundamental research and technological applications. While high-resolution magnetic imaging of thin nanoscale samples is well established with electron and soft x-ray microscopy, the extension to micrometer-thick systems currently requires hard x rays, which limits high-resolution imaging to rare-earth magnets. Here, we overcome this limitation by establishing soft x-ray magnetic imaging of micrometer-thick systems using the pre-edge phase x-ray magnetic circular dichroism signal, thus making possible the study of a wide range of magnetic materials. By performing dichroic spectroptychography, we demonstrate high spatial resolution imaging of magnetic samples up to $1.7\mathrm{\mu }\mathrm{m}$ thick, an order of magnitude higher than conventionally possible with soft x-ray absorption-based techniques. We demonstrate the applicability of the technique by harnessing the pre-edge phase to image thick chiral helimagnets, and naturally occurring magnetite particles, gaining insight into their three-dimensional magnetic configuration. This new regime of magnetic imaging makes possible the study of extended non-rare-earth systems that have until now been inaccessible, including magnetic textures for future spintronic applications, non-rare-earth permanent magnets for energy harvesting, and the magnetic configuration of giant magnetofossils.

Exciton-exciton interactions are key to understanding nonlinear optical and transport phenomena in van der Waals heterobilayers, which emerged as versatile platforms to study correlated electronic states. We present a combined theory-experiment study of excitonic many-body effects based on first-principle band structures and Coulomb interaction matrix elements. Key to our approach is the explicit treatment of the fermionic substructure of excitons and dynamical screening effects for density-induced energy renormalization and dissipation. We demonstrate that dipolar blueshifts are almost perfectly compensated by many-body effects, mainly by screening-induced self-energy corrections. Moreover, we identify a crossover between attractive and repulsive behavior at elevated exciton densities. Theoretical findings are supported by experimental studies of spectrally narrow, mobile interlayer excitons in atomically reconstructed, $h$-BN-encapsulated ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ heterobilayers. Both theory and experiment show energy renormalization on a scale of a few meV even for high injection densities in the vicinity of the Mott transition. Our results revise the established picture of dipolar repulsion dominating exciton-exciton interactions in van der Waals heterostructures and open up opportunities for their external design.

Hybrid Josephson junctions (JJs) realized in superconductor-semiconductor heterostructures host fermionic modes known as Andreev bound states (ABSs). In these structures, a promising and yet unexplored avenue for harnessing spin and parity degrees of freedom is offered by JJs with three or more superconducting terminals, where phase-induced spin polarization and transitions of the ground state to an odd parity were predicted to arise. Here we spectroscopically probe the two-dimensional band structure of ABSs in a phase-controlled $\mathrm{InAs}/\mathrm{Al}$ three-terminal JJ. Andreev bands show signatures of spin-degeneracy breaking, with level splitting in excess of $\sim 9\mathrm{GHz}$, and zero-energy crossings associated to ground state fermion parity transitions. Spin splitting and parity transitions are enabled and controlled by locally applied magnetic fluxes, in the absence of Zeeman effect or Coulomb blockade. Our results underscore the potential of multiterminal hybrid devices for phase engineering ABSs, with significant implications for spin- and parity-based quantum devices.

The approach to equilibrium in interacting classical and quantum systems is a challenging problem of both theoretical and experimental interest. One useful organizing principle characterizing equilibration is the dissipative universality class, the most prevalent one being diffusion. In this paper, we use the effective field theory (EFT) of diffusion to systematically obtain universal power-law corrections to diffusion. We then employ large-scale simulations of classical and quantum systems to explore their validity. In particular, we find universal scaling functions for the corrections to the dynamical structure factor $⟨n(x,t)n⟩$, in the presence of a single $\mathrm{U}(1)$ or $\mathrm{SU}(2)$ charge in systems with and without particle-hole symmetry, and present the framework to generalize the calculation to multiple charges. Classical simulations show remarkable agreement with EFT predictions for subleading corrections, pushing precision tests of effective theories for thermalizing systems to an unprecedented level. Moving to quantum systems, we perform large-scale tensor-network simulations in unitary and noisy 1D Floquet systems with conserved magnetization. We find a qualitative agreement with EFT, which becomes quantitative in the case of noisy systems. Additionally, we show how the knowledge of EFT corrections allows for fitting methods, which can improve the estimation of transport parameters at the intermediate times accessible by simulations and experiments. Finally, we explore nonlinear response in quantum systems and find that EFT provides an accurate prediction for its behavior. Our results provide a basis for a better understanding of the nonlinear phenomena present in thermalizing systems.

First-order phase transitions produce abrupt changes to the character of both ground and excited electronic states. Here we conduct electronic compressibility measurements to map the spin phase diagram and Landau level (LL) energies of monolayer ${\mathrm{WSe}}_2$ in a magnetic field. We resolve a sequence of first-order phase transitions between completely spin-polarized LLs and states with LLs of both spins. Unexpectedly, the LL gaps are roughly constant over a wide range of magnetic fields below the transitions, which we show reflects spin-polarized ground states with opposite spin excitations. These transitions also extend into compressible regimes, with a sawtooth boundary between full and partial spin polarization. We link these observations to the important influence of LL filling on the exchange energy beyond a smooth density-dependent contribution. Our results show that ${\mathrm{WSe}}_2$ realizes a unique hierarchy of energy scales where such effects induce reentrant magnetic phase transitions tuned by density and magnetic field.

Ever since the discovery of the charge density wave (CDW) transition in the kagome metal ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$, the nature of its symmetry breaking has been under intense debate. While evidence suggests that the rotational symmetry is already broken at the CDW transition temperature ($T_{\mathrm{CDW}}$), an additional electronic nematic instability well below $T_{\mathrm{CDW}}$ has been reported based on the diverging elastoresistivity coefficient in the anisotropic channel ($m_{E_{2g}}$). Verifying the existence of a nematic transition below $T_{\mathrm{CDW}}$ is not only critical for establishing the correct description of the CDW order parameter, but also important for understanding low-temperature superconductivity. Here, we report elastoresistivity measurements of ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ using three different techniques probing both isotropic and anisotropic symmetry channels. Contrary to previous reports, we find the anisotropic elastoresistivity coefficient $m_{E_{2g}}$ is temperature independent, except for a step jump at $T_{\mathrm{CDW}}$. The absence of nematic fluctuations is further substantiated by measurements of the elastocaloric effect, which show no enhancement associated with nematic susceptibility. On the other hand, the symmetric elastoresistivity coefficient $m_{A_{1g}}$ increases below $T_{\mathrm{CDW}}$, reaching a peak value of 90 at $T^{*}=20\mathrm{K}$. Our results strongly indicate that the phase transition at $T^{*}$ is not nematic in nature and the previously reported diverging elastoresistivity is due to the contamination from the $A_{1g}$ channel.

A characteristic feature of “quantum chaotic” systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this feature, we introduce a framework that allows us to compare the ensemble properties of eigenstates in local systems with those of pure random states. In particular, our framework defines a notion of distance between quantum state ensembles that utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). This notion gives rise to a quantitative metric for quantum chaos that not only accounts for averages of the distributions but also higher moments. The differences in moments are compared on a highly resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. As a result, the metric can distinguish between chaotic and integrable behaviors and, in addition, quantify and compare the degree of chaos (in terms of proximity to RMT behavior) between two systems that are assumed to be chaotic. We implement our framework in local, minimally structured, Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed-field Ising model (MFIM). Importantly, for Hamiltonian systems, we find that the reference random distribution must be appropriately constrained to incorporate the effect of energy conservation in order to describe the ensemble properties of midspectrum eigenstates. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that “maximally chaotic” Hamiltonians may exist in fine-tuned pockets of parameter space.

Understanding and manipulating emergent phases, which are themes at the forefront of quantum-materials research, rely on identifying their underlying symmetries. This general principle has been particularly prominent in materials with coupled electronic and magnetic degrees of freedom, in which magnetic order influences the electronic band structure and can lead to exotic topological effects. However, identifying symmetry of a magnetically ordered phase can pose a challenge, particularly in the presence of small domains. Here we introduce a multimodal approach for determining magnetic structures, which combines symmetry-sensitive optical probes, scattering, and group-theoretical analysis. We apply it to ${\mathrm{EuIn}}_2{\mathrm{As}}_2$, a material that has received attention as a candidate axion insulator. While first-principles calculations predict this state on the assumption of a simple collinear antiferromagnetic structure, subsequent neutron-scattering measurements reveal a much more intricate magnetic ground state characterized by two coexisting magnetic wave vectors reached by successive thermal phase transitions. The proposed high- and low-temperature phases are a spin helix and a state with interpenetrating helical and Néel antiferromagnetic order termed a “broken helix,” respectively. Employing a multimodal approach, we identify the magnetic structure associated with these two phases of ${\mathrm{EuIn}}_2{\mathrm{As}}_2$. We find that the higher-temperature phase is characterized by a variation of the magnetic moment amplitude from layer to layer, with the moment vanishing entirely in every third Eu layer. The lower-temperature structure is similar to the broken helix, with one important difference: Because of local strain, the relative orientation of the magnetic structure and the lattice is not fixed. Consequently, the symmetry required to protect the axion phase is not generically protected in ${\mathrm{EuIn}}_2{\mathrm{As}}_2$, but we show that it can be restored if the magnetic structure is tuned with uniaxial strain. Finally, we present a spin Hamiltonian that identifies the spin interactions that account for the complex magnetic order in ${\mathrm{EuIn}}_2{\mathrm{As}}_2$. Our work highlights the importance of a multimodal approach in determining the symmetry of complex order parameters.

The fundamental optical excitations of correlated transition-metal compounds are typically identified with multielectronic transitions localized at the transition-metal site, such as $dd$ transitions. In this vein, intense interest has surrounded the appearance of sharp, below-band-gap optical transitions, i.e., excitons, within the magnetic phase of correlated ${\mathrm{Ni}}^{2+}$ van der Waals magnets. The interplay of magnetic and charge-transfer insulating ground states in ${\mathrm{Ni}}^{2+}$ systems raises intriguing questions on the roles of long-range magnetic order and of metal-ligand charge transfer in the exciton nature, which inspired microscopic descriptions beyond typical $dd$ excitations. Here we study the impact of charge transfer and magnetic order on the excitation spectrum of the nickel dihalides ($\mathrm{Ni}{X}_2$, $X=\mathrm{Cl}$, Br, and I) using Ni-$L_3$ edge resonant inelastic x-ray scattering (RIXS). In all compounds, we detect sharp excitations, analogous to the recently reported excitons, and assign them to spin-singlet multiplets of octahedrally coordinated ${\mathrm{Ni}}^{2+}$ stabilized by intra-atomic Hund’s exchange. Additionally, we demonstrate that these excitons are dispersive using momentum-resolved RIXS. Our data evidence a ligand-mediated multiplet dispersion, which is tuned by the charge-transfer gap and independent of the presence of long-range magnetic order. This reveals the mechanisms governing nonlocal interactions of on-site $dd$ excitations with the surrounding crystal or magnetic structure, in analogy to ground-state superexchange. These measurements thus establish the roles of magnetic order, self-doped ligand holes, and intersite-coupling mechanisms for the properties of $dd$ excitations in charge-transfer insulators.

A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-$1/2$ Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way.