SEARCH · Engineering Papers

Results for “Self consistent field methods”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 361 records · Page 20

All sources NASA NTRS DOE OSTI

DOE OSTI2021

On the computational solution of vector-density based continuum dislocation dynamics models: A comparison of two plastic distortion and stress update algorithms

Continuum dislocation dynamics models of mesoscale plasticity consist of dislocation transport-reaction equations coupled with crystal mechanics equations. The coupling between these two sets of equations is such that dislocation transport gives rise to the evolution of plastic distortion (strain), while the evolution of the latter fixes the stress from which the dislocation velocity field is found via a mobility law. Earlier solutions of these equations employed a staggered solution scheme for the two sets of equations in which the plastic distortion was updated via time integration of its rate, as found from Orowan's law. In this work, we show that such a direct time integration scheme can suffer from accumulation of numerical errors. We introduce an alternative scheme based on field dislocation mechanics that ensures consistency between the plastic distortion and the dislocation content in the crystal. The new scheme is based on calculating the compatible and incompatible parts of the plastic distortion separately, and the incompatible part is calculated from the current dislocation density field. Stress field and dislocation transport calculations were implemented within a finite element based discretization of the governing equations, with the crystal mechanics part solved by a conventional Galerkin method and the dislocation transport equations by the least squares method. A simple test was first performed to show the accuracy of the two schemes for updating the plastic distortion, which shows that the solution method based on field dislocation mechanics is more accurate. This method then was used to simulate an austenitic steel crystal under uniaxial loading and multiple slip conditions. By considering dislocation interactions caused by junctions, a hardening rate similar to discrete dislocation dynamics simulation results was obtained. Finally, the simulations show that dislocations exhibit some self-organized structures as the strain is increased.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Small tensor product distributed active space (STP-DAS) framework for relativistic and non-relativistic multiconfiguration calculations: Scaling from 10 9 on a laptop to 10 12 determinants on a supercomputer

Despite the power and flexibility of configuration interaction (CI) based methods in computational chemistry, their broader application is limited by an exponential increase in both computational and storage requirements, particularly due to the substantial memory needed for excitation lists that are crucial for scalable parallel computing. Here, the objective of this work is to develop a new CI framework, namely, the small tensor product distributed active space (STP-DAS) framework, aimed at drastically reducing memory demands for extensive CI calculations on individual workstations or laptops, while simultaneously enhancing scalability for extensive parallel computing. Moreover, the STP-DAS framework can support various CI-based techniques, such as complete active space (CAS), restricted active space, generalized active space, multireference CI, and multireference perturbation theory, applicable to both relativistic (two- and four-component) and non-relativistic theories, thus extending the utility of CI methods in computational research. We conducted benchmark studies on a supercomputer to evaluate the storage needs, parallel scalability, and communication downtime using a realistic exact-two-component CASCI (X2C-CASCI) approach, covering a range of determinants from 10 9 to 10 12 . Additionally, we performed large X2C-CASCI calculations on a single laptop and examined how the STP-DAS partitioning affects performance.

Complete-active space self-consistent field↗

DOE OSTI2022

Modelling of electrostatic ion-scale turbulence in divertor tokamaks with the gyrokinetic code $\mathrm{COGENT}$

Here, continuum gyrokinetic simulations of electrostatic ion scale turbulence are presented for the case of a diverted (single-null) tokamak geometry. The simulation model, implemented in the finite-volume code COGENT, solves the long-wavelength limit of the full-F gyrokinetic equation for ion species coupled to a vorticity equation for electrostatic potential variations, where a fluid model is used for an electron response. The model describes the ion scale ion temperature gradient (ITG) and resistive drift modes as well as neoclassical ion physics effects. Different turbulence regimes are observed depending on the plasma profiles, and the roles of a self-consistent background electric field and an X-point geometry are explored. In particular, increasing the pedestal density gradient and the corresponding radial electric field is demonstrated to suppress the ITG turbulence, whereas the same edge plasma background can still be destabilized by the resistive modes when the plasma resistivity is increased. The effects of X-point geometry are assessed by comparing cross-separatrix simulations with counterpart calculations performed for a toroidal annulus geometry. For the simulation parameters considered, similar global behaviour is observed in both cases, whereas strong local suppression of turbulence fluctuations is demonstrated near the X-point for the case of a single-null geometry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Self-supervised physics-informed generative networks for phase retrieval from a single X-ray hologram

X-ray phase contrast imaging significantly improves the visualization of structures with weak or uniform absorption, broadening its applications across a wide range of scientific disciplines. Propagation-based phase contrast is particularly suitable for time- or dose-critical in vivo/in situ/operando (tomography) experiments because it requires only a single intensity measurement. However, the phase information of the wave field is lost during the measurement and must be recovered. Conventional algebraic and iterative methods often rely on specific approximations or boundary conditions that may not be met by many samples or experimental setups. In addition, they require manual tuning of reconstruction parameters by experts, making them less adaptable for complex or variable conditions. Here we present a self-learning approach for solving the inverse problem of phase retrieval in the near-field regime of Fresnel theory using a single intensity measurement (hologram). A physics-informed generative adversarial network is employed to reconstruct both the phase and absorbance of the unpropagated wave field in the sample plane from a single hologram. Unlike most state-of-the-art deep learning approaches for phase retrieval, our approach does not require paired, unpaired, or simulated training data. This significantly broadens the applicability of our approach, as acquiring or generating suitable training data remains a major challenge due to the wide variability in sample types and experimental configurations. The algorithm demonstrates robust and consistent performance across diverse imaging conditions and sample types, delivering quantitative, high-quality reconstructions for both simulated data and experimental datasets acquired at beamline P05 at PETRA III (DESY, Hamburg), operated by Helmholtz-Zentrum Hereon. Furthermore, it enables the simultaneous retrieval of both phase and absorption information.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Fast and robust all-electron density functional theory calculations in solids using orthogonalized enriched finite elements

Here, we present a computationally efficient approach to perform systematically convergent real-space all-electron Kohn-Sham density functional theory calculations for solids using an enriched finite element (FE) basis. The enriched FE basis is constructed by augmenting the classical FE basis with atom-centered numerical basis functions, comprising of atomic solutions to the Kohn-Sham problem. Notably, to improve the conditioning, we orthogonalize the enrichment functions with respect to the classical FE basis, without sacrificing the locality of the resultant basis. In addition to improved conditioning, this orthogonalization procedure also renders the overlap matrix block diagonal, greatly simplifying its inversion. Subsequently, we use a Chebyshev polynomial based filtering technique to efficiently compute the occupied eigenspace in each self-consistent field iteration. We demonstrate the accuracy and efficiency of the proposed approach on periodic unit cells and supercells. The benchmark studies show a staggering 130× speedup of the orthogonalized enriched FE basis over the classical FE basis. We also present a comparison of the orthogonalized enriched FE basis with the linearized augmented plane-wave + local orbitals basis, both in terms of accuracy and efficiency. Notably, we demonstrate that the orthogonalized enriched FE basis can handle large system sizes of ~10 000 electrons. Finally, we observe good parallel scalability of our implementation with 92% efficiency at 22× speedup for a system with 620 electrons

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Physics basis for the Wisconsin HTS Axisymmetric Mirror (WHAM)

The Wisconsin high-temperature superconductor axisymmetric mirror experiment (WHAM) will be a high-field platform for prototyping technologies, validating interchange stabilization techniques and benchmarking numerical code performance, enabling the next step up to reactor parameters. A detailed overview of the experimental apparatus and its various subsystems is presented. WHAM will use electron cyclotron heating to ionize and build a dense target plasma for neutral beam injection of fast ions, stabilized by edge-biased sheared flow. At 25 keV injection energies, charge exchange dominates over impact ionization and limits the effectiveness of neutral beam injection fuelling. This paper outlines an iterative technique for self-consistently predicting the neutral beam driven anisotropic ion distribution and its role in the finite beta equilibrium. Beginning with recent work by Egedal et al. ( Nucl. Fusion , vol. 62, no. 12, 2022, p. 126053) on the WHAM geometry, we detail how the FIDASIM code is used to model the charge exchange sources and sinks in the distribution function, and both are combined with an anisotropic magnetohydrodynamic equilibrium solver method to self-consistently reach an equilibrium. We compare this with recent results using the CQL3D code adapted for the mirror geometry, which includes the high-harmonic fast wave heating of fast ions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2010

Establishing Consensus Turbulence Statistics for Hot Subsonic Jets

Many tasks in fluids engineering require knowledge of the turbulence in jets. There is a strong, although fragmented, literature base for low order statistics, such as jet spread and other meanvelocity field characteristics. Some sources, particularly for low speed cold jets, also provide turbulence intensities that are required for validating Reynolds-averaged Navier-Stokes (RANS) Computational Fluid Dynamics (CFD) codes. There are far fewer sources for jet spectra and for space-time correlations of turbulent velocity required for aeroacoustics applications, although there have been many singular publications with various unique statistics, such as Proper Orthogonal Decomposition, designed to uncover an underlying low-order dynamical description of turbulent jet flow. As the complexity of the statistic increases, the number of flows for which the data has been categorized and assembled decreases, making it difficult to systematically validate prediction codes that require high-level statistics over a broad range of jet flow conditions. For several years, researchers at NASA have worked on developing and validating jet noise prediction codes. One such class of codes, loosely called CFD-based or statistical methods, uses RANS CFD to predict jet mean and turbulent intensities in velocity and temperature. These flow quantities serve as the input to the acoustic source models and flow-sound interaction calculations that yield predictions of far-field jet noise. To develop this capability, a catalog of turbulent jet flows has been created with statistics ranging from mean velocity to space-time correlations of Reynolds stresses. The present document aims to document this catalog and to assess the accuracies of the data, e.g. establish uncertainties for the data. This paper covers the following five tasks: Document acquisition and processing procedures used to create the particle image velocimetry (PIV) datasets. Compare PIV data with hotwire and laser Doppler velocimetry (LDV) data published in the open literature. Compare different datasets acquired at roughly the same flow conditions to establish uncertainties. Create a consensus dataset for a range of hot jet flows, including uncertainty bands. Analyze this consensus dataset for self-consistency and compare jet characteristics to those of the open literature. One final objective fulfilled by this work was the demonstration of a universal scaling for the jet flow fields, at least within the region of interest to aeroacoustics. The potential core length and the spread rate of the half-velocity radius were used to collapse of the mean and turbulent velocity fields over the first 20 jet diameters in a highly satisfying manner.

Bridges, James↗

DOE OSTI2026

SCF Framework, HF Stability, and RPA Correlation for Jordan–Wigner-Transformed Spin Hamiltonians on Arbitrary Coupling Topologies

Mapping spins to fermions via the Jordan–Wigner (JW) transformation can render mean-field (Hartree–Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytical orbital Hessian to diagnose HF stability and compute the ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J 1 –J 2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Pitfalls in the n -mode representation of vibrational potentials

Simulations of anharmonic vibrational motion rely on computationally expedient representations of the governing potential energy surface. The n-mode representation (n-MR)—effectively a many-body expansion in the space of molecular vibrations—is a general and efficient approach that is often used for this purpose in vibrational self-consistent field (VSCF) calculations and correlated analogues thereof. In the present analysis, a lack of convergence in many VSCF calculations is shown to originate from negative and unbound potentials at truncated orders of the n-MR expansion. For cases of strong anharmonic coupling between modes, the n-MR can both dip below the true global minimum of the potential surface and lead to effective single-mode potentials in VSCF that do not correspond to bound vibrational problems, even for bound total potentials. The present analysis serves mainly as a pathology report of this issue. Furthermore, this insight into the origin of VSCF non-convergence provides a simple, albeit ad hoc, route to correct the problem by “painting in” the full representation of groups of modes that exhibit these negative potentials at little additional computational cost. Somewhat surprisingly, this approach also reasonably approximates the results of the next-higher n-MR order and identifies groups of modes with particularly strong coupling. The method is shown to identify and correct problematic triples of modes—and restore SCF convergence—in two-mode representations of challenging test systems, including the water dimer and trimer, as well as protonated tropine.

Chemistry↗

DOE OSTI2022

Construction of wave dark matter halos: Numerical algorithm and analytical constraints

Here we present a wave generalization of the classic Schwarzschild method for constructing self-consistent halos—such a halo consists of a suitable superposition of waves instead of particle orbits, chosen to yield a desired mean density profile. As an illustration, the method is applied to spherically symmetric halos. We derive an analytic relation between the particle distribution function and the wave superposition amplitudes and show how it simplifies in the high-energy (WKB) limit. We verify the stability of such constructed halos by numerically evolving the Schrödinger-Poisson system. The algorithm provides an efficient and accurate way to simulate the time-dependent halo substructures from wave interference. We use this method to construct halos with a variety of density profiles, all of which have a core from the ground-state wave function, though the core-halo relation need not be the standard one.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2012

Large-Amplitude Longitudinal Oscillations in a Solar Filament

We have developed the first self-consistent model for the observed large-amplitude oscillations along filament axes that explains the restoring force and damping mechanism. We have investigated the oscillations of multiple threads formed in long, dipped flux tubes through the thermal nonequilibrium process, and found that the oscillation properties predicted by our simulations agree with the observed behavior. We then constructed a model for the large-amplitude longitudinal oscillations that demonstrates that the restoring force is the projected gravity in the tube where the threads oscillate. Although the period is independent of the tube length and the constantly growing mass, the motions are strongly damped by the steady accretion of mass onto the threads by thermal nonequilibrium. The observations and our model suggest that a nearby impulsive event drives the existing prominence threads along their supporting tubes, away from the heating deposition site, without destroying them. The subsequent oscillations occur because the displaced threads reside in magnetic concavities with large radii of curvature. Our model yields a powerful seismological method for constraining the coronal magnetic field and radius of curvature of dips. Furthermore, these results indicate that the magnetic structure is most consistent with the sheared-arcade model for filament channels.

lament axes↗

DOE OSTI2020

Molecular dynamics simulations for glass transition temperature predictions of polyhydroxyalkanoate biopolymers

Polyhydroxyalkanoates (PHAs) represent an emerging class of biosynthetic and biodegradable polyesters that exhibit considerable potential to replace petroleum-based plastics towards a sustainable future. Despite the promise, general structure–property mappings within this class of polymers remain largely unexplored. An efficient exploration of this vast chemical space calls for the development and validation of predictive methods for accurate estimation of a diverse range of properties for PHA-based polymers. Towards this aim, we present and validate in this work the results of our molecular dynamics (MD) simulation based approach aimed at predicting glass transition temperatures (T g ) of PHA-based polymers. Since generally available and widely used polymer forcefields exhibit a relatively poor performance for T g predictions, we have developed a new forcefield by modifying the polymer consistent force field (PCFF) via refining a selected set of torsion potentials of the polymer backbone using accurate density functional theory (DFT) computations. After carefully assessing the dependence of critical simulation parameters, such as, polymer chain length, number of polymer chains, supercell size, and thermal quenching rate used in the simulation, the applicability and transferability of the modified PCFF (mPCFF) is demonstrated by directly comparing the computed T g predictions of various polymers with different chemistries, polymer side chain lengths and functional groups forming the polymer side chains against the respective experimentally measured values. Furthermore, the transport properties such as self-diffusion coefficient and viscosity are computationally determined and their well-known correlation with the target properties is demonstrated. Lastly, we have employed the developed approach to predict T g values for a number of yet-to-be-synthesized PHA-based polymers with a diverse set of functional groups in the polymer side chains. The results are further rationalized by correlating the predicted T g values with the inter-chain H-bond formation tendencies of the different side chain functional groups. This work represents an important first step towards computationally guided design of PHA-based functional polymers and opens up new directions for a systematic investigation of composition- and configuration-dependent structure–property relationships in more complex binary and ternary copolymer systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Elimination of QCD Renormalization Scale and Scheme Ambiguities

The setting of the renormalization scale (μ r ) in the perturbative QCD (pQCD) is one of the crucial problems for achieving precise fixed-order pQCD predictions. The conventional prescription is to take its value as the typical momentum transfer Q in a given process, and theoretical uncertainties are then evaluated by varying it over an arbitrary range. The conventional scale-setting procedure introduces arbitrary scheme-and-scale ambiguities in fixed-order pQCD predictions. The principle of maximum conformality (PMC) provides a systematic way to eliminate the renormalization scheme-and-scale ambiguities. The PMC method has rigorous theoretical foundations; it satisfies the renormalization group invariance (RGI) and all of the self-consistency conditions derived from the renormalization group. The PMC has now been successfully applied to many physical processes. In this paper, we summarize recent PMC applications, including event shape observables and heavy quark pair production near the threshold region in e + e – annihilation and top-quark decay at hadronic colliders. In addition, estimating the contributions related to the uncalculated higher-order terms is also summarized. These results show that the major theoretical uncertainties caused by different choices of μ r are eliminated, and the improved pQCD predictions are thus obtained, demonstrating the generality and applicability of the PMC.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

Anomalous Hall crystals in rhombohedral multilayer graphene. II. General mechanism and a minimal model

Here we propose a minimal "three-patch model"for the anomalous Hall crystal (AHC), a topological electronic state that spontaneously breaks both time-reversal symmetry and continuous translation symmetry. The proposal for this state is inspired by the recently observed integer and fractional quantum Hall states in rhombohedral multilayer graphene at zero magnetic field. There, interaction effects appear to amplify the effects of a weak moiré potential, leading to the formation of stable, isolated Chern bands. It has been further shown that Chern bands are stabilized in mean-field calculations even without a moiré potential, enabling a realization of the AHC state. Our model is built on the dissection of the Brillouin zone into patches centered around high-symmetry points. Within this model, the wave functions at high-symmetry points fully determine the topology and energetics of the state. We extract two quantum geometrical phases of the noninteracting wave functions that control the stability of the topologically nontrivial AHC state. The model predicts that the AHC state wins over the topological trivial Wigner crystal in a wide range of parameters, and agrees very well with the results of full self-consistent Hartree-Fock calculations of the rhombohedral multilayer graphene Hamiltonian.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Gauge constrained algorithm of variational discrete action theory at N = 3 for the multiorbital Hubbard model

The recently developed variational discrete action theory (VDAT) provides a systematic variational approach to the ground state of the quantum many-body problem, where the quality of the solution is controlled by an integer N, and increasing N monotonically approaches the exact solution. VDAT can be exactly evaluated in the d = ∞ multiorbital Hubbard model using the self-consistent canonical discrete action theory (SCDA), which requires a self-consistency condition for the integer time Green's functions. Previous work demonstrates that N = 3 accurately captures multiorbital Mott/Hund physics at a cost similar to the Gutzwiller approximation. Here we employ a gauge constraint to automatically satisfy the self-consistency condition of the SCDA at N = 3, yielding an even more efficient algorithm with enhanced numerical stability. We derive closed form expressions of the gauge constrained algorithm for the multiorbital Hubbard model with general density-density interactions, allowing VDAT at N = 3 to be straightforwardly applied to the seven-orbital Hubbard model. We present results and a performance analysis using N = 2 and N = 3 for the SU⁡(2⁢N orb ) Hubbard model in d = ∞ with N orb = 2–8, and compare to numerically exact dynamical mean-field theory solutions where available. Finally, the developments in this work will greatly facilitate the application of VDAT at N = 3 to strongly correlated electron materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Anomalous Hall Crystals in Rhombohedral Multilayer Graphene. I. Interaction-Driven Chern Bands and Fractional Quantum Hall States at Zero Magnetic Field

Recent experiments on rhombohedral pentalayer graphene with a substrate-induced moiré potential have identified both Chern insulators and fractional quantum Hall states at zero magnetic field. Surprisingly, these states are observed in strong displacement fields where the effects of the moiré lattice are weak, and seem to be readily accessed without fine-tuning. To address these experimental puzzles, we study a model of interacting electrons in this geometry. Within self-consistent Hartree-Fock (SCHF) calculations, we find an isolated Chern band with small bandwidth and good quantum geometry. Exact diagonalization and density-matrix renormalization group calculations both confirm the band hosts fractional quantum Hall states without a magnetic field. Remarkably, the Chern band is stable at a wide range of angles, at four through six rhombohedral layers, at varying rhombohedral hopping parameters, and—most strikingly—survives in SCHF calculations when the moiré potential vanishes. In this limit, the state spontaneously breaks time-reversal and translation symmetry simultaneously, giving a topological crystalline state that we term the “anomalous Hall crystal.” We argue this is a general mechanism to create stable Chern bands in rhombohedral multilayer graphene, opening the door to studying the interplay between electronic topology, fractionalization, and spontaneous translation symmetry breaking.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Modeling fission dynamics at the barrier in a discrete-basis formalism

A configuration-interaction model is presented for the barrier region of induced fission. The configuration space is composed of seniority-zero configurations constructed from self-consistent mean-field wave functions. The Hamiltonian matrix elements between configurations include diabatic and pairing interactions between particles. Other aspects of the Hamiltonian are treated statistically, guided by phenomenological input of compound-nucleus transmission coefficients. In this exploratory study the configuration space is restricted to neutron excitations only. A key observable calculated in the model is the fission-to-capture branching ratio. We find that both pairing and diabatic interactions are important for achieving large branching to the fission channels. In accordance with the transition-state theory of fission, the calculated branching ratio is found to be quite insensitive to the fission decay widths of the pre-scission configurations. Furthermore, the barrier-top dynamics appear to be quite different from transition-state theory in that the transport is distributed over many excited configurations at the barrier top.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Multiband superconductivity in strongly hybridized 1 T ' - WTe 2 / NbSe 2 heterostructures

The interplay of topology and superconductivity has become a subject of intense research in condensed-matter physics for the pursuit of topologically nontrivial forms of superconducting pairing. An intrinsically normal-conducting material can inherit superconductivity via electrical contact to a parent superconductor via the proximity effect, usually understood as Andreev reflection at the interface between the distinct electronic structures of two separate conductors. However, at high interface transparency, strong coupling inevitably leads to changes in the band structure, locally, owing to hybridization of electronic states. Here, we investigate such strongly proximity-coupled heterostructures of monolayer $1T$' - $\mathrm{WTe_2}$ grown on NbSe 2 by van der Waals epitaxy. The superconducting local density of states, resolved in scanning tunneling spectroscopy down to 500 mK, reflects a hybrid electronic structure well described by a multiband framework based on the McMillan equations which captures the multiband superconductivity inherent to the NbSe 2 substrate and that is induced by proximity to $\mathrm{WTe_2}$ self-consistently. Our material-specific tight-binding model captures the hybridized heterostructure quantitatively and confirms that strong interlayer hopping gives rise to a semimetallic density of states in the two-dimensional $\mathrm{WTe_2}$ bulk, even for nominally band-insulating crystals. The model further accurately predicts the measured order parameter Δ ≅ 0.6 meV induced in the $\mathrm{WTe_2}$ monolayer bulk, stable beyond a 2 T magnetic field. So we believe that our detailed multiband analysis of the hybrid electronic structure provides a useful tool for sensitive spatial mapping of induced order parameters in proximitized atomically thin topological materials.

36 MATERIALS SCIENCE↗