We introduce a compact and stable moment representation for real-frequency Green's functions, hybridization functions, and self-energies for machine-learning applications, avoiding the inefficiency of dense frequency grids as well as the ill-posed analytic continuation of Matsubara approaches. The representation is constructed from Cayley-mapped trigonometric moments with the Jacobian included, which preserve spectral-weight normalization, tie the moment sequence to a positive matrix-valued spectral measure, and admit a systematic route to a pole representation via ESPRIT. This provides a fixed-dimensional learning target in which physical constraints such as normalization and positivity can be imposed directly. Using a graph-attention neural network with FiLM conditioning, we benchmark the representation on single-orbital DMFT, antiferromagnetic DMFT, and a two-orbital impurity model. The results demonstrate accuracy matching or exceeding that of direct frequency-domain learning, reliable reproduction of the density and staggered magnetization, stable self-energy reconstruction through Dyson equation inversion, and accurate recovery of matrix-valued spectra with orbital mixing.

Low-order hybridization expansion methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA) are widely used impurity solvers in the study of strongly correlated systems, yet their accuracy in genuine multi-orbital settings remains poorly understood. Using the decoupled orbital limit as a controlled reference point, we derive analytic results connecting multi-orbital restricted propagators and Green's functions to their single-orbital counterparts, identify the diagrammatic mechanisms responsible for the breakdown of low-order methods in multi-orbital settings, and determine their regimes of applicability. Our central finding is that the accuracy of these methods is governed by the least correlated orbital: i.e., the orbital with the most rapidly decaying retarded Green's function. That orbital's properties are transferred to all other orbitals through a spurious coupling generated by the truncated expansion, thereby suppressing correlation-induced features such as the Kondo resonance. This occurs even in orbitals that are themselves strongly correlated within single-orbital calculations using the same approximation scheme. We confirm this numerically across representative two-orbital model systems in the steady-state, systematically identifying the parameter regimes in which low-order methods succeed or fail. Our results provide a practical guide for assessing when insights from single-orbital calculations carry over to multi-orbital settings, and serve as a benchmark for the development and validation of higher-order multi-orbital impurity solvers.

Representing real-time data as a sum of complex exponentials provides a compact form that enables both denoising and extrapolation. As a fully data-driven method, the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm is agnostic to the underlying physical equations, making it broadly applicable to various observables and experimental or numerical setups. In this work, we consider applications of the ESPRIT algorithm primarily to extend real-time dynamical data from simulations of quantum systems. We evaluate ESPRIT's performance in the presence of noise and compare it to other extrapolation methods. We demonstrate its ability to extract information from short-time dynamics to reliably predict long-time behavior and determine the minimum time interval required for accurate results. We discuss how this insight can be leveraged in numerical methods that propagate quantum systems in time, and show how ESPRIT can predict infinite-time values of dynamical observables, offering a purely data-driven approach to characterizing quantum phases.

We present numerically exact inchworm quantum Monte Carlo results for the real-time dynamics of the spin polarization in the sub-Ohmic spin-boson model at finite temperature. We focus in particular on the localization and coherence behavior of the model, extending our previous study at low temperature [Goulko et al., Phys. Rev. Lett. 134, 056502 (2025)]. As the temperature increases, the system becomes less localized and less coherent. The loss of coherence, which is controlled by two independent mechanisms-a smooth damping-driven crossover and a sharp frequency-driven transition-exhibits a nontrivial temperature dependence. While both types of coherence loss occur at lower coupling in the high temperature regime, the frequency exhibits a sharper drop at high temperatures and this drop is observed for all values of the sub-Ohmic exponent, in contrast to the zero-temperature case. We discuss the full temperature-dependent dynamical phase diagram of the system and the interplay between coherence and localization across a wide range of physical parameters.

We investigate the real-time dynamics of the sub-Ohmic spin-boson model across a broad range of coupling strengths, using the numerically exact inchworm quantum Monte Carlo algorithm. From short- and intermediate-time dynamics starting from an initially decoupled state, we extract signatures of the zero-temperature quantum phase transition between localized and delocalized states. We show that the dynamical phase diagram thus obtained differs from the equilibrium phase diagram in both the values of critical couplings and the associated values of the critical exponent. We also identify and quantitatively analyze two competing mechanisms for the crossover between coherent oscillations and incoherent decay. Deep in the sub-Ohmic regime, the crossover is driven by the damping of the oscillation amplitude, while closer to the Ohmic regime the oscillation frequency itself drops sharply to zero at large coupling.

Combinatorial mechanical metamaterials are made of anisotropic, flexible blocks, such that multiple metamaterials may be constructed using a single block type, and the system's response depends on the frustration (or its absence) due to the mutual orientations of the blocks within the lattice. Specifically, any minimal loop of blocks that may not simultaneously deform in their softest mode defines a mechanical defect at the vertex (in two dimensions) or edge (in three dimensions) that the loop encircles. Defects stiffen the metamaterial, and allow one to design the spatial patterns of stress and deformation as the system is externally loaded. We study the ability to place defects at arbitrary positions in metamaterials made of a family of block types that we recently introduced for the square, honeycomb, and cubic lattices. Alongside blocks for which we show that any defect configuration is possible, we identify situations in which not all sets are realizable as defects. One of the restrictions is that in three dimensions, defected edges form closed curves. Even in cases when not all geometries of defect lines are possible, we show how to produce defect lines of arbitrary knottedness.

The investigation of quantum impurity models plays a crucial role in condensed matter physics because of their wide-ranging applications, such as embedding theories and transport problems. Traditional methods often fall short of producing accurate results for multi-orbital systems with complex interactions and off-diagonal hybridizations. Recently, tensor-train-based integration and summation techniques have shown promise as effective alternatives. In this study, we use tensor train methods to tackle quantum impurity problems formulated within the imaginary-time inchworm hybridization expansion framework. We identify key challenges in the inchworm expansion itself and its interplay with tensor-train-based methods. We demonstrate the accuracy and versatility of our approach by solving general quantum impurity problems. Our results suggest that tensor-train decomposition schemes offer a viable path toward accurate and efficient multi-orbital impurity solvers.

Teachers play a crucial role in promoting self-regulated learning (SRL) and problem solving (PS) in classrooms, yet their knowledge in these domains remains understudied and challenging to evaluate. While traditional assessment tools like surveys exist, there is a need for diversification, particularly through situational judgment tests (SJT). This study leverages human-AI collaboration to develop an SJT tool for evaluating teachers’ SRL-PS knowledge. Human interaction with generative artificial intelligence (Gen-AI) enables overcoming difficulties and mutually complementing each other. The purpose of this paper is to present an initial attempt to develop an SJT tool for evaluating teachers’ SRL-PS knowledge using the assistance of ChatGPT and to examine how the researcher-ChatGPT interaction contributed to the development of this tool. Through systematic interaction between researchers and ChatGPT, we developed a comprehensive tool comprising 15 difficulty categories and 20 scenarios. The research demonstrates how scenarios created by researchers can be effectively complemented by ChatGPT-generated content. A synthesized map of difficulty categories and scenarios from both researchers and ChatGPT revealed patterns of over- and under-representation across categories, and identified opportunities for synthesized new difficulty categories that could facilitate new scenarios. The study advances SRL knowledge evaluation tools through SJT methodology while demonstrating the benefits of human-Gen-AI collaborative interactions.

As education shifts toward student-centered learning, evaluating pre-service teachers' ability to foster self-regulated learning (SRL) is essential. This study examines lesson plans as performance-based assessment tools for evaluating pre-service teachers' competence in designing SRL-supportive instruction in online courses. Through qualitative analysis of 37 lesson plans created by 73 pre-service teachers, the research applies a three-dimensional analytical framework focusing on SRL strategy implementation, technology integration, and pedagogical competence. Findings reveal significant variability in how teachers structure SRL, scaffold autonomy development, and incorporate instructional technologies. High-quality lesson plans explicitly embedded metacognitive scaffolds and included structured support for student autonomy, while weaker plans provided inadequate SRL elements and lacked clear instructional rationales. Technology integration varied, with some lesson plans leveraging digital tools to enhance reflection and feedback, while others used technology in a passive, content-delivery manner. These findings reinforce the validity of lesson plans as assessment tools, offering teacher educators a structured alternative to self-reports in evaluating pre-service teachers' instructional reasoning and readiness to promote SRL in diverse learning environments.

A description of long-lived photodoped states in Mott insulators is challenging, as it needs to address exponentially separated timescales. We demonstrate how properties of such states can be computed using numerically exact steady state techniques, in particular, the quantum Monte Carlo algorithm, by using a time-local ansatz for the distribution function with separate Fermi functions for the electron and hole quasiparticles. The simulations show that the Mott gap remains robust to large photodoping, and the photodoped state has hole and electron quasiparticles with strongly renormalized properties.

Developing self-regulated learning (SRL) skills among young adolescents is crucial for lifelong learning, and teachers play a vital role in fostering these skills. Problem-solving is a key SRL skill; however, both learners and teachers encounter challenges during the problem-solving process and in evaluating its incorporated strategies. To tackle these challenges, this study examined students’ problem-solving difficulties using a generic video-assisted SRL training programme, based on teachers’ reports, and analysed these challenges across STEM and non-STEM domains. This study focusses on identifying and classifying the difficulties of 241 students within the problem-solving framework, as reported by teachers across various disciplines and grades (6–11). The analysis revealed that the most significant difficulties arose during the problem identification, followed by time planning and management, problem exploration, solution development and solution evaluation. Specifically, STEM teachers reported relatively fewer difficulties in problem exploration than in other steps, while non-STEM teachers reported fewer challenges in presenting and communicating solutions to peers. A dedicated map named the problem-solving sort out setbacks (PS-SOS) map was created to pinpoint challenges within the problem-solving process. The application of this mapping technique can further support the development of technology-based feedback systems, including digital assistants, which offer valuable assistance to students.

A precise dynamical characterization of quantum impurity models with multiple interacting orbitals is challenging. In quantum Monte Carlo methods, this is embodied by sign problems. A dynamical sign problem makes it exponentially difficult to simulate long times. A multi-orbital sign problem generally results in a prohibitive computational cost for systems with multiple impurity degrees of freedom even in static equilibrium calculations. Here, we present a numerically exact inchworm method that simultaneously alleviates both sign problems, enabling simulation of multi-orbital systems directly in the equilibrium or nonequilibrium steady-state. The method combines ideas from the recently developed steady-state inchworm Monte Carlo framework [Erpenbeck et al., Phys. Rev. Lett. 130, 186301 (2023)] with other ideas from the equilibrium multi-orbital inchworm algorithm [Eidelstein et al., Phys. Rev. Lett. 124, 206405 (2020)]. We verify our method by comparison with analytical limits and numerical results from previous methods.

Describing the ground states of continuous, real-space quantum many-body systems, like atoms and molecules, is a significant computational challenge with applications throughout the physical sciences. Recent progress was made by variational methods based on machine learning (ML) ansatzes. However, since these approaches are based on energy minimization, ansatzes must be twice differentiable. This (a) precludes the use of many powerful classes of ML models; and (b) makes the enforcement of bosonic, fermionic, and other symmetries costly. Furthermore, (c) the optimization procedure is often unstable unless it is done by imaginary time propagation, which is often impractically expensive in modern ML models with many parameters. The stochastic representation of wavefunctions (SRW), introduced in Nat Commun 14, 3601 (2023), is a recent approach to overcoming (c). SRW enables imaginary time propagation at scale, and makes some headway towards the solution of problem (b), but remains limited by problem (a). Here, we argue that combining SRW with path integral techniques leads to a new formulation that overcomes all three problems simultaneously. As a demonstration, we apply the approach to generalized ``Hooke's atoms'': interacting particles in harmonic wells. We benchmark our results against state-of-the-art data where possible, and use it to investigate the crossover between the Fermi liquid and the Wigner molecule within closed-shell systems. Our results shed new light on the competition between interaction-driven symmetry breaking and kinetic-energy-driven delocalization.

Dynamical mean-field theory allows access to the physics of strongly correlated materials with nontrivial orbital structure, but relies on the ability to solve auxiliary multi-orbital impurity problems. The most successful approaches to date for solving these impurity problems are the various continuous time quantum Monte Carlo algorithms. Here, we consider perhaps the simplest realization of multi-orbital physics: the bilayer Hubbard model on an infinite-coordination Bethe lattice. Despite its simplicity, the majority of this model's phase diagram cannot be predicted by using traditional Monte Carlo methods. We show that these limitations can be largely circumvented by recently introduced Inchworm Monte Carlo techniques. We then explore the model's phase diagram at a variety of interaction strengths, temperatures and filling ratios.

We study bulk particle transport in a Fermi-Hubbard model on an infinite-dimensional Bethe lattice, driven by a constant electric field. Previous numerical studies showed that one dimensional analogs of this system exhibit a breakdown of diffusion due to Stark many-body localization (Stark-MBL) at least up to time which scales exponentially with the system size. Here, we consider systems initially in a spin density wave state using a combination of numerically exact and approximate techniques. We show that for sufficiently weak electric fields, the wave's momentum component decays exponentially with time in a way consistent with normal diffusion. By studying different wavelengths, we extract the dynamical exponent and the generalized diffusion coefficient at each field strength. Interestingly, we find a non-monotonic dependence of the dynamical exponent on the electric field. As the field increases towards a critical value proportional to the Hubbard interaction strength, transport slows down, becoming sub-diffusive. At large interaction strengths, however, transport speeds up again with increasing field, exhibiting super-diffusive characteristics when the electric field is comparable to the interaction strength. Eventually, at the large field limit, localization occurs and the current through the system is suppressed.

Self-regulated learning (SRL) is crucial in today‟s educational landscape, championing students' autonomy over their learning journey. Yet, SRL, especially in problem-solving, poses significant difficulties for students. This study innovatively bridges this gap with a set of design principles for a generative AI chatbot specifically shaped to promote students' SRL skills during problem-solving tasks. Leveraging insights from feedback collected from both educators and students involved in a blended SRL program, this research shines a light on the range of difficulties involves in SRL within problem solving context. It pinpoints three main difficulties: Inconsistencies in assessment and perception that create a gap between student self-evaluations and educator assessments; Cognitive strategies difficulties that hinder skill application; and beyond cognitive difficulties such as motivation and task self-management obstacles toward independent learning. Responding to these difficulties, the design principles of the proposed AI chatbot are focused on providing structured guidance, which includes explicit instruction in cognitive and motivational skills, offering personalized feedback based on the specific difficulties, and facilitating the assessment of these skills during task performance. This approach not only advances the theoretical framework of SRL but also translates these concepts into tangible support mechanisms, equipping educators with an innovative tool to enhance SRL. The chatbot stands as a testament to the power of generative AI in transforming educational practices, paving the way for learners to become more self-reliant and adept in navigating the complexities of the digital learning era.

Quantum transport is often characterized not just by mean observables like the particle or energy current but by their fluctuations and higher moments, which can act as detailed probes of the physical mechanisms at play. However, relatively few theoretical methods are able to access the full counting statistics (FCS) of transport processes through electronic junctions in strongly correlated regimes. While most experiments are concerned with steady state properties, most accurate theoretical methods rely on computationally expensive propagation from a tractable initial state. Here, we propose a simple approach for computing the FCS through a junction directly at the steady state, utilizing the propagator noncrossing approximation. Compared to time propagation, our method offers reduced computational cost at the same level of approximation, but the idea can also be used within other approximations or as a basis for numerically exact techniques. We demonstrate the method's capabilities by investigating the impact of lead dimensionality on electronic transport in the nonequilibrium Anderson impurity model at the onset of Kondo physics. Our results reveal a distinct signature of one dimensional leads in the noise and Fano factor not present for other dimensionalities, showing the potential of FCS measurements as a probe of the environment surrounding a quantum dot.

Entropy is a central concept in physics, but can be challenging to calculate even for systems that are easily simulated. This is exacerbated out of equilibrium, where generally little is known about the distribution characterizing simulated configurations. However, modern machine learning algorithms can estimate the probability density characterizing an ensemble of images, given nothing more than sample images assumed to be drawn from this distribution. We show that by mapping system configurations to images, such approaches can be adapted to the efficient estimation of the density, and therefore the entropy, from simulated or experimental data. We then use this idea to obtain entropic limit cycles in a kinetic Ising model driven by an oscillating magnetic field. Despite being a global probe, we demonstrate that this allows us to identify and characterize stochastic dynamics at parameters near the dynamical phase transition.

Institutions, organisations, and policymakers use open educational resources (OERs) to promote student equity and social inclusion. The global COVID-19 crisis highlighted the need for lifelong learning and underscored the importance of the higher education system in this endeavour. This study describes informal learning among adults through OERs, during the COVID-19 crisis, distinguishing between employed and unemployed individuals and between professional and personal development. A questionnaire distributed during the COVID-19 lockdown focused on three themes: (1) types of OERs used for learning during this period; (2) perceived OERs’ usefulness; and (3) changes in OER use due to the crisis. Our findings revealed group differences in types of OERs used and in changes brought about by COVID-19, as well as within-group differences based on personal characteristics. Only a few participants reported using massive open online courses (MOOCs). Moreover, videoconferencing usage increased despite low perceived usefulness ratings, pointing to a change in informal learning modes. This exploratory research provides insights into the preferences of individual groups. These insights may be used to reduce socioeconomic disparities, especially among those who have lost their jobs, and to develop effective models for open education. Implications for practice or policy: Enhancing the discussions about the future of open education by reflecting a wide picture of OERs use. Redesigning OERs for the labour market by distinguishing between employed and unemployed, and professional and personal development. OER preferences according to personal characteristics can be used to achieve better engagement with learning.

The simulation of strongly correlated quantum impurity models is a significant challenge in modern condensed matter physics that has multiple important applications. Thus far, the most successful methods for approaching this challenge involve Monte Carlo techniques that accurately and reliably sample perturbative expansions to any order. However, the cost of obtaining high precision through these methods is high. Recently, tensor train decomposition techniques have been developed as an alternative to Monte Carlo integration. In this study, we apply these techniques to the single-impurity Anderson model at equilibrium by calculating the systematic expansion in power of the hybridization of the impurity with the bath. We demonstrate the performance of the method in a paradigmatic application, examining the first-order phase transition on the infinite-dimensional Bethe lattice, which can be mapped to an impurity model through dynamical mean field theory. Our results indicate that using tensor train decomposition schemes allows the calculation of finite-temperature Green's functions and thermodynamic observables with unprecedented accuracy. The methodology holds promise for future applications to frustrated multiorbital systems, using a combination of partially summed series with other techniques pioneered in diagrammatic and continuous time quantum Monte Carlo.