Anyons are quasiparticles with fractional charge and statistics that arise in strongly correlated two-dimensional systems such as the fractional quantum Hall (FQH) effect and fractional Chern insulators (FCI). Interactions between anyons can lead to emergent phenomena, such as anyon superconductivity as well as anyon condensation which allows for a hierarchical construction of quantum Hall states. In this work, we study how quasihole anyons in a $ν=1/3$ Laughlin fractional quantum Hall state can be bound together by a sufficiently strong attractive impurity potential. The competition between the repulsive interaction between the quasiholes themselves and the attractive interaction between the quasiholes and the impurity leads to states with different numbers of quasiholes bound to the impurity. Tuning the chemical potential via gating while remaining within a quantum Hall plateau changes the number of quasiholes bound to the impurity. We propose methods for studying these states experimentally, for example using scanning tunneling microscopy and exciton spectroscopy. While the impurities in traditional platforms such as GaAs heterostructures are typically too weak to observe the binding of anyons, the recently discovered zero-field fractional Chern insulators in twisted MoTe$_2$ offer a platform which may realize the strong-impurity regime.

We demonstrate significant tunability of the plasma frequency in a wire medium by mechanically deforming a lattice of parallel metallic wires arranged at the nodes of a honeycomb structure. Numerical simulations predict up to 78% tunability and a proof-of-concept experiment confirms 64%, surpassing previously reported values for tunable wire media.

Forward regression is a classical and effective tool for variable screening in ultra-high dimensional linear models, but its standard projection-based implementation can be computationally costly and numerically unstable when predictors are strongly collinear. Motivated by this limitation, we propose an orthogonalized forward regression procedure, implemented recursively through Gram-Schmidt updates, that ranks predictors according to their unique contributions after removing the effects of variables already selected. This approach preserves the interpretability of forward regression while substantially reducing the cost of repeated projections. We further develop a path-based model size selection rule using statistics computed directly from the forward sequence, thereby avoiding cross-validation and extensive tuning. The resulting method is particularly well suited to settings in which the number of predictors far exceeds the sample size and strong collinearity renders the conventional forward fitting ineffective. Theoretically, we derive the optimal convergence rate for the proposed Gram-Schmidt forward regression, thereby extending existing results for projection-based forward regression, and further show that it enjoys sure screening property and variable selection consistency under suitable conditions. Simulation studies and empirical examples demonstrate that it provides a favorable balance among computational efficiency, numerical stability, screening accuracy, and predictive performance, especially in highly correlated ultra-high dimensional settings.

Cherenkov detectors have been used in space missions for many decades, and for a variety of purposes, including for example, for Galactic Cosmic Ray (GCR) and Solar Energetic Particle (SEP) measurements. Cherenkov detectors are sensitive to many types of particles that are present in the environment of space, including gamma rays, trapped particles and cosmic particles, and each particle component acts as essentially a background when trying to view another specific particle component. In this research, GRAS/Geant4 simulations were performed to characterise the count rates that a simple Cherenkov detector design would experience in a low Earth orbit, and we find that Cherenkov count rates due to most particle components vary significantly depending on many different factors, including the location in the orbit, the date of the orbit, whether or not the detector is within the van Allen belts, and whether or not a solar particle event is occurring. We find that a small Cherenkov detector is readily able to gather detailed data on both trapped particles and spectral information during Ground-Level Enhancements. We also investigate the use of coincidence as a method to remove count rates due to trapped particles and delta electrons, finding that this method is generally very effective for resolving count rates due to GLEs amongst intense trapped particle environments, but that some Cherenkov count rates due to trapped particles are still observed in the simulated south Atlantic anomaly region.

Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any particular subsequence of length n occurs at most once in a period in either direction. They were introduced in the early 1990s in the context of possible applications in position sensing. Bounds on the period of such sequences and a range of methods of construction have been devised, although apart from very small cases a significant gap remains between the largest known period for such a sequence and the best known upper bound. In this paper we first give improved upper bounds on the period of such sequences. We then give a new general method of construction for orientable sequences involving subgraphs of the de Bruijn graph with special properties, and describe two different approaches for generating such subgraphs. This enables us to construct orientable sequences with periods meeting the improved upper bounds when n is 2 or 3, as well as n=4 and k odd. For 4\leq n\leq 8, in some cases the sequences produced by the methods described have periods larger than for any previously known sequences.

Various ``cosmic shorelines" have been proposed to delineate which planets have atmospheres. The fates of individual planet atmospheres may be set by a complex sea of growth and loss processes, driven by unmeasurable environmental factors or unknown historical events. Yet, defining population-level boundaries helps illuminate which processes matter and identify high-priority targets for future atmospheric searches. Here, we provide a statistical framework for inferring the position, shape, and fuzziness of an instellation-based cosmic shoreline, defined in the three-dimensional space of planet escape velocity, planet bolometric flux received, and host star luminosity. We circumvent the need to estimate individual host stars' historical X-ray and extreme ultraviolet fluences by including luminosity in the definition of the shoreline, explicitly modeling how sharply such drivers of atmospheric escape intensify toward lower-luminosity M dwarf stars and marginalizing over the associated uncertainties. Using Solar System and exoplanet atmospheric constraints, under the assumption that one planar boundary applies across a wide parameter space, we find the critical flux threshold for atmospheres scales with escape velocity with a power-law index of $p=5.9_{-0.43}^{+0.61}$, steeper than the canonical literature slope of $p=4$, and scales with stellar luminosity with a power-law index of $q=1.17_{-0.20}^{+0.28}$, steep enough to disfavor atmospheres on Earth-sized planets out to the habitable zone for stars less luminous than $\log_{10} (L_\star/L_\odot) = -2.23_{-0.21}^{+0.18}$ (roughly spectral type M4V). This model provides quantitative predictions for the probability any planet may have an atmosphere, which can be rigorously tested by upcoming JWST Rocky Worlds observations.

One of the outstanding challenges in contemporary science and technology is building a quantum computer that is useful in applications. By starting from an estimate of the algorithm success rate, we can explicitly connect gate fidelity to quantum system size targets and define a quantitative roadmap that maximizes performance while avoiding distractions. We identify four distinct phases for quantum hardware and enabling technology development. The aim is to improve performance as we scale and increase the algorithmic complexity the quantum hardware is capable of running, the algorithmic radius, towards a point that sets us up for quantum advantage with deep noisy intermediate-scale quantum computing (NISQ) as well as building a large-scale error-corrected quantum computer (QEC). Our hope is that this document contributes to shaping the discussion about the future of the field.

We study a spinful, time-reversal symmetric lowest Landau level model for a flatband quantum spin Hall system at total filling fraction $ν_\mathrm{T}=2/3$. Such models are relevant, e.g. for spin-valley locked moiré transition metal dichalcogenides. The opposite Chern number of the two spins hinders the formation of a quantum Hall ferromagnet, instead favouring other phases. We study the phase diagram in dependence on different short-range Haldane pseudopotentials $V_m$ and uncover several phases: A fractional topological insulator, a phase separated state, a spin-polarized fractional quantum Hall state, and the partially particle-hole transformed Halperin (111) state. The effect of the pseudopotentials $V_m$ depends on the parity of $m$, the relative angular momentum.

We provide an iterative solution approach for the indefinite Helmholtz equation discretised using finite elements, based upon a Hermitian Skew-Hermitian Splitting (HSS) iteration applied to the shifted operator, and prove that the iteration is k- and mesh-robust when O(k) HSS iterations are performed. The HSS iterations involve solving a shifted operator that is suitable for approximation by multigrid using standard smoothers and transfer operators, and hence we can use O(N) parallel processors in a high performance computing implementation, where N is the total number of degrees of freedom. We argue that the algorithm converges in O(k) wallclock time when within the range of scalability of the multigrid. We provide numerical results in both 2D and 3D verifying our proofs and demonstrating this claim, establishing a method that can make use of large scale high performance computing systems.

We revisit the study of $G_2$-structures with special torsion, and isolated singularities. Many of the known examples with conical singularities admit additional symmetries, and we describe circle-invariant $G_2$-structures in this context. Finally, we show that collapsing the circle fibres of a contact Calabi-Yau manifold at isolated points cannot produce a $G_2$-structure with bounded torsion.

We deal with weighted Hardy-Sobolev type inequalities for functions on $\mathbb{R}^d$, $d\geq 2$. The weights involved are anisotropic, given by products of powers of the distance to the origin and to a nontrivial subspace. We establish necessary and sufficient conditions for validity of these inequalities, and investigate the existence/nonexistence of extremal functions.

The limiting probability distribution is one of the key characteristics of a Markov chain since it shows its long-term behavior. In this paper, for a higher order Markov chain, we establish some properties related to its exact limiting probability distribution, including a sufficient condition for the existence of such a distribution. Our results extend the corresponding conclusions on first order chains. Besides, they complement the existing results concerning higher order chains which rely on approximation schemes or two-phase power iterations. Several illustrative example are also given.

A single-mode squeezed vacuum is a foundational quantum state that, despite its nonclassical nature, exhibits classical-like, super-Poissonian photon statistics. This feature motivates a ``quantum-of-quantum'' inquiry: can the superposition of two such states generate the opposite behavior -- strongly sub-Poissonian light? We demonstrate that the ``Janus state,'' a coherent superposition of two squeezed vacua with opposing orientations, achieves precisely this. Our exact analytic solution reveals a universal lower bound on second-order coherence, showing that $g^{(2)}$ cannot be driven below 1/2. The mechanism is tuned interference that suppresses two-photon events. Beyond this asymptotic bound, we identify a practical minimum of $g^{(2)} \approx 0.567$ at moderate squeezing, defining an accessible ``sweet spot.'' While requiring a minimal non-Gaussian element for its creation, the Janus state establishes a definitive performance limit for engineering sub-Poissonian photon statistics from Gaussian resources, with a clear path toward quantum applications.

Linearly polarized elastic waves propagating in a linear path have been extensively explored in numerous applications from biomedical imaging to structural health monitoring. However, elastic waves propagating in a circularly polarized helical path are less explored because of the challenges in their generation and control. In this paper, we harness conventional actuation methods combined with sheets decorated by Archimedean spirals to generate elastic helical waves. We show that our metamaterials can support the propagation of such waves along a curved path without backscattering in a topologically protected manner. Moreover, we also show the creation and propagation of helical waves in our metamaterials without the need to wave guiding or domain interfaces, all with a single mode excitation source. We establish our methodology for flat plates and show the wave evolution as the metamaterials transition from slightly curved plates to fully curved cylinders. We observe the preservation of topologically protected edge states and helical wave beaming at different frequencies without the need to domain interferences. Furthermore, we leverage the mode to tune the helicity of the propagating waves. Our methodology can open new avenues for the generation and control of elastic helical waves using single source actuation that can be used in numerous applications.

Record-breaking temperature events are now frequently in the news, proffered as evidence of climate change, and often bring significant economic and human impacts. Our previous work undertook the first substantial spatial modelling investigation of temperature record-breaking across years for any given day within the year, employing a dataset consisting of over sixty years of daily maximum temperatures across peninsular Spain. That dataset also supplies daily minimum temperatures (which, in fact, are now available through 2023). Here, the dataset is converted into a daily pair of binary events, indicators, for that day, of whether a yearly record was broken for the daily maximum temperature and/or for the daily minimum temperature. Joint modelling addresses several inference issues: (i) defining/modelling record-breaking with bivariate time series of yearly indicators, (ii) strength of relationship between record-breaking events, (iii) prediction of joint, conditional and marginal record-breaking, (iv) persistence in record-breaking across days, (v) spatial interpolation across peninsular Spain. We substantially expand our previous work to enable investigation of these issues. We observe strong correlation between both processes but a growing trend of climate change that is well differentiated between them both spatially and temporally as well as different strengths of persistence and spatial dependence.

It is well-known that the unitary nature of black hole evaporation enforces its entropy evolution to follow the Page curve. In this Letter, we find that the quantum speed limit on how fast a quantum system can evolve unitarily, when the maximal information transmission bound is saturated, will slow down the would-be divergent decreasing rate of dynamical black-hole entropy at the very end of the Hawking evaporation, during which the Penrose inequality from cosmic censorship conjecture is exactly saturated. Therefore, an evaporating Schwarzschild black hole is the fastest transmitter of information in nature. Further applying the maximal information transmission bound to an effective-field-theory description of a de Sitter space would roughly reproduce the trans-Planckian censorship conjecture, indicating the de Sitter space as the fastest receiver of information in nature.

Interacting many-particle systems can self-organize into a rich variety of crystalline structures. While symmetry provides a powerful framework for predicting whether transitions between crystal states are continuous or discontinuous, collective lattice dynamics offer complementary insights into the microscopic mechanisms that drive these transitions. Trapped laser-cooled ions present a pristine and highly controllable few-body system for studying this interplay of symmetry and dynamics. Here, we use real-time fluorescence imaging while deforming the trap potential to observe a variety of structural transitions in three-dimensional (3D), unit-cell-like ion clusters. We identify a set of transitions signaled by parity-odd octupole order parameters, and probe their distinct dynamical signatures. Our observations reveal the softening of a collective Higgs-like mode indicating spontaneous symmetry-breaking, hysteresis resulting from a catastrophe where a metastable state vanishes abruptly, and stochastic switching between metastable states of differing symmetries. We also uncover a remarkable coincidence of symmetry-breaking and discontinuous transitions, analogous to a thermodynamic triple point. Our results establish 3D trapped-ion clusters as a versatile platform to engineer complex potential energy landscapes, opening new avenues for studies of reaction kinetics, geometric frustration, and related phenomena in mesoscopic platforms.

BPS states in holographic CFTs naturally split into those describing black holes in the bulk and those that do not, with black hole states only existing above a certain energy threshold. In the context of the AdS$_3$/CFT$_2$ duality this can be seen from the agreement of the CFT and supergraviton supersymmetric indices up to a certain central charge-scaling conformal dimension. However, there also exist additional smooth horizonless bulk configurations called singletons that have not been previously accounted for when distinguishing black hole states from non-black hole states. These singletons describe bulk degrees of freedom that are non-trivial on the AdS$_3$ boundary. From a detailed analysis of BPS states in the D1-D5 system we identify singleton states in $\mathrm{Sym}^N(T^4)$ and $\mathrm{Sym}^N(K3)$ and explicitly incorporate them into a generalised supergraviton index for low levels for the case of $N=2$, leading to an enhanced matching with the CFT index in the case of $T^4$. Singleton states are monotone and, under the assumption that together with supergraviton states they span the monotone Hilbert space, the generalised supergraviton index represents the full monotone index. This allows us to define fortuitous indices for these theories and for $\mathrm{Sym}^2(K3)$ we construct the first explicit fortuitous states.

Application of a displacement field opens a gap and enhances the Van-Hove singularities in the band structure of Bernal-stacked bilayer graphene. By adjusting the carrier density so that the Fermi energy lies in the vicinity of these singularities, recent experiments observe a plethora of highly correlated electronic phases including isospin polarized phases and high-resistance states with non-linear electric transport indicative of a possible Wigner crystal. We perform Hartree-Fock calculations incorporating long-range Coulomb interactions and allowing for translational and rotational symmetry breaking. We obtain the displacement field vs. carrier density phase diagram which shows isospin polarized metallic phases tracking the Van-Hove singularity in the valence band. Between these metallic phases we observe regions where the ground state is a Wigner crystal. The isospin polarization of the Wigner crystals tracks the isospin polarization of the nearby metallic phases. Depending on whether we have four, three, two or one isospin flavours, we obtain a full, three-quarter, half or quarter Wigner crystal.

The latest results from the Atacama Cosmology Telescope Collaboration have moved the preferred perturbation spectral index $n_{\rm s}$ closer towards one. We reanalyse constraints on the simplest version of the curvaton model, using $n_{\rm s}$ and the tensor-to-scalar ratio $r$. We show that in the massless curvaton case the model gives the same locus of $n_{\rm s}$-$r$ predictions as the power-law model $V(ϕ) \propto ϕ^p$, but with a different and more elegant physical interpretation. The model gives an excellent account of current observational data, both in the regime where the curvaton dominates the observed perturbations and with an admixture of inflaton-originated perturbations of up to about one-third of the total. Addition of the newest South Pole Telescope SPT-3G D1 dataset maintains this conclusion in favor of the curvaton model. Forthcoming large-scale structure surveys, such as that of the recently-launched SPHEREx probe, could discriminate between curvaton and single-field inflation models using the bispectrum.

We establish the connection between the Steinitz problem for ordering vector families in arbitrary norms and its variant for not necessarily zero-sum families consisting of `nearly unit' vectors.

Access/egress travel to train stations continues to pose a significant barrier to increasing the number of train travellers. Shared micromobility (SMM), including bicycles, e-bikes, steps and mopeds, is often cited as a prominent solution, especially for the activity-end of the trip chain. Using a stated preference survey, we analyse activity-end mode-choice preferences for SMM, walking and public transport (PT) among the Dutch population. By means of a latent class choice model, we uncover three user groups with respect to activity-end mode choice behaviour. The largest (58%) are Multimodal sharing enthusiast, who choose based on the trade-offs between various travel characteristics, while not having strong modal preferences. They are the most open, ready and able to use SMM. Sharing hesitant cyclists (16%) have a strong preference for cycling and while they are open to using SMM , they may not feel themselves ready, stating that use of SMM can be difficult and dangerous. Sharing-averse PT users (27%) are most likely to use PT and avoid SMM as they find it too difficult and dangerous to use. The high preference to walk for short egress distances reaffirms the need for transit-oriented development policies. For longer egress distances, PT should be the primary focus at stations in high-density areas with high demand, where high frequencies and dense networks are justified, while stations in lower demand areas are better served by SMM. Providing multiple SMM options would result mainly in competition for the same travellers.

We consider the formation of metastable cosmic strings in a left-right unified theory. The produced monopoles are diluted by a stage of F-term hybrid inflation (FHI) which is realized consistently with the SUSY breaking and a global $U(1)$ $R$ symmetry in the context of a $U(1)_{\rm R}\times U(1)_{B-L}$ extension of MSSM. The hidden-sector Kaehler manifold enjoys an enhanced $SU(1,1)/U(1)$ symmetry with the scalar curvature determined by the achievement of a SUSY-breaking de Sitter vacuum without ugly tuning. FHI turns out to be compatible with data -- including the recent ACT results --, provided that the magnitude of the emergent soft tadpole term is confined in the range $(0.1-70)$ TeV, and it is accompanied with the production of cosmic strings. Their dimensionless tension $Gμ_{\rm cs}\simeq(1-11)\cdot10^{-8}$ interprets the present observations from PTA experiments on the stochastic background of gravitational waves. The $μ$ parameter of MSSM arises by appropriately adapting the Giudice-Masiero mechanism and facilitates the out-of-equilibrium decay of the $R$ saxion at a reheat temperature lower than about $34$ GeV. The SUSY mass scale turns out to lie in the PeV region.

We present a new algorithm for solving the reduction problem in the context of holonomic integrals, which in turn provides an approach to integration with parameters. Our method extends the Griffiths--Dwork reduction technique to holonomic systems and is implemented in Julia. While not yet outperforming creative telescoping in D-finite cases, it enhances computational capabilities within the holonomic framework. As an application, we derive a previously unattainable differential equation for the generating series of 8-regular graphs.

Consider the special linear group of degree $2$ over an arbitrary finite field, acting on the full space of $2 \times 2$-matrices by transpose. We explicitly construct a generating set for the corresponding modular matrix invariant ring, demonstrating that this ring is a hypersurface. Using a recent result on $a$-invariants of Cohen-Macaulay algebras, we determine the Hilbert series of this invariant ring, and our method avoids seeking the generating relation. Additionally, we prove that the modular matrix invariant ring of the group of upper triangular $2 \times 2$-matrices is also a hypersurface.

The set of real matrices of upper-bounded rank is a real algebraic variety called the real generic determinantal variety. An explicit description of the tangent cone to that variety is given in Theorem 3.2 of Schneider and Uschmajew [SIAM J. Optim., 25 (2015), pp. 622-646]. The present paper shows that the proof therein is incomplete and provides a proof. It also reviews equivalent descriptions of the tangent cone to that variety. Moreover, it shows that the tangent cone and the algebraic tangent cone to that variety coincide, which is not true for all real algebraic varieties.

We report measurements of the ratios of branching fractions ${\cal R}(D^{(*)+}) = \frac{{\cal B}(\overline{B}{}^0 \to D^{(*)+} \,τ^- \, \overlineν_τ)}{{\cal B}(\overline{B}{}^0 \to D^{(*)+} \, \ell^- \, \overlineν_\ell)}$, where $\ell$ denotes either an electron or a muon. These ratios test the universality of the charged-current weak interaction. The results are based on a $365\, \mathrm{fb}^{-1}$ data sample collected with the Belle II detector at the SuperKEKB $e^+e^-$ collider, which operates at a center-of-mass energy corresponding to the $Υ(4S)$ resonance, just above the threshold for $B\overline{B}{}$ production. Signal candidates are reconstructed by selecting events in which the companion $B$ meson from the $Υ(4S) \to B\overline{B}{}$ decay is identified in semileptonic modes. The $τ$ lepton is reconstructed via its leptonic decays. We obtain ${\cal R}(D^+) = 0.418 \pm 0.074 ~({\mathrm{stat}}) \pm 0.051 ~({\mathrm{syst}})$ and ${\cal R}(D^{*+}) = 0.306 \pm 0.034 ~({\mathrm{stat}}) \pm 0.018 ~({\mathrm{syst}})$, which are consistent with world average values. Accounting for the correlation between them, these values differ from the Standard Model expectation by a collective significance of $1.7$ standard deviations.

Shared micromobility (SMM), including bicycles, e-bikes, scooters, etc., is often cited as a solution to the first and especially the last mile problem of public transport (PT), yet when implemented, they often do not get adopted by a broader travelling public. As behavioural adaption is largely related to peoples' attitudes and perceptions, we develop a behavioural framework based on the UTAUT2 framework to gain better understanding why individuals may (not) be willing to use SMM. Through an exploratory factor analysis (EFA) and a latent class cluster analysis (LCCA), we study the adoption potential of SMM and assess drivers and barriers as perceived by different user groups. Our findings uncover six user groups; Shared mobility positives, Car-oriented sharing neutrals, Older apprehensive sharers, Young eager adopters, (Shared) Mobility avoiders and Skilled sharing sceptics. The Young eager adopters and Shared mobility positives tend to be the most open to adopting SMM and able to do so. Older apprehensive sharers would like to, but find it difficult or dangerous to use, while Skilled sharing sceptics are capable and confident, but have limited intention of using it. Car-oriented sharing neutrals and (Shared) Mobility avoiders are most negative about SMM, finding it difficult to use and dangerous. Factors relating to technological savviness, ease-of-use, physical safety and societal perception seem to be the strongest adoption predictors. Younger, high-educated males are the group most likely and open to using SMM, while older individuals with lower incomes and a lower level of education tend to be the least likely.

We propose a grid-based methodology for online changepoint detection that allows offline changepoint tests to be applied to sequentially observed data. The methodology achieves low update and storage costs by testing for changepoints over a dynamically updating grid of candidate changepoint locations. For a broad class of test statistics, including those based on empirical averages and certain likelihood ratios, we show that the resulting online procedure has update and storage costs that grow at most logarithmically with the sample size. We further show that finite-sample power guarantees for the offline test translate directly into non-asymptotic upper bounds on the detection delay, under a mild robustness assumption. Building upon the methodology, we construct methods for detecting changes in the mean and in the covariance matrix of multivariate data, and prove near-optimal non-asymptotic upper bounds on their detection delays. The effectiveness of the methodology is supported by a simulation study, where we compare its performance for detecting mean changes with that of state-of-the-art online methods. To illustrate its practical applicability, we use the methodology to detect structural changes in currency exchange rates in real time.

Topologically ordered quantum liquids are highly sought-after quantum phases of matter, and recently, fractional Chern insulators (FCIs) joined the few experimental realizations of such phases. Here, we ask whether a gapped classical, highly degenerate liquid can be the birthplace of FCIs upon the addition of suitable quantum fluctuations. Two competing tendencies can be anticipated: (i) following the quantum order-by-disorder paradigm, quantum fluctuations could induce symmetry-breaking (charge) order, or (ii) the classical liquid builds up long-range entanglement and turns into a quantum liquid. We study spinless fermions on a honeycomb lattice subject to cluster-charging interactions and introduce quantumness through a Haldane kinetic term, featuring complex second-nearest-neighbor hopping. Based on extensive exact diagonalization calculations and high-order perturbation theory, we find that neither scenario (i) nor (ii) prevails, but (i) and (ii) manifest sequentially as the kinetic energy is increased. We demonstrate how the gradual lifting of kinematic constraints gives rise to this sequence of phases. Our results relate to the regime of intermediate-scale interactions present in moiré systems, where band projections are not suitable to model FCIs and competing charge-ordered phases have been identified.