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DOE OSTI2023

Progenitors and explosion properties of supernova remnants hosting central compact objects: II. A global systematic study with a comparison to nucleosynthesis models

ABSTRACT Core-collapse explosions of massive stars leave behind neutron stars, with a known diversity that includes the ‘Central Compact Objects’ (CCOs). Typified by the neutron star discovered near the centre of the Cas A supernova remnant (SNR), CCOs have been observed to shine only in X-rays. To address their supernova progenitors, we perform a systematic study of SNRs that contain a CCO and display X-ray emission from their shock-heated ejecta. We make use of X-ray data primarily using the Chandra X-ray observatory, complemented with XMM–Newton. This study uses a systematic approach to the analysis of each SNR aimed at addressing the supernova progenitor as well as the explosion properties (energy and ambient density). After fitting for the ejecta abundances estimated from a spatially resolved spectroscopic study, we compare the data to six nucleosynthesis models making predictions on supernova ejecta yields in core-collapse explosions. We find that the explosion models commonly used by the astrophysics community do not match the ejecta yields for any of the SNRs, suggesting additional physics, for example multidimensional explosion models or updated progenitor structures, are required. Overall we find low-mass (≤25 solar masses) progenitors among the massive stars population and low-energy explosions (<1051 ergs). We discuss degeneracies in our model fitting, particularly how altering the explosion energy affects the estimate of the progenitor mass. Our systematic study highlights the need for improving on the theoretical models for nucleosynthesis predictions as well as for sensitive, high-resolution spectroscopy observations to be acquired with next generation X-ray missions.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Secular outflows from 3D MHD hypermassive neutron star accretion disc systems

Magnetized hypermassive neutron stars (HMNSs) have been proposed as a way for neutron star mergers to produce high electron fraction, high-velocity ejecta, as required by kilonova models to explain the observed light curve of GW170817. The HMNS drives outflows through neutrino energy deposition and mechanical oscillations, and raises the electron fraction of outflows through neutrino interactions before collapsing to a black hole (BH). Here, in this study, we perform 3D numerical simulations of HMNS–torus systems in ideal magnetohydrodynamics, using a leakage/absorption scheme for neutrino transport, the nuclear APR equation of state, and Newtonian self-gravity, with a pseudo-Newtonian potential added after BH formation. Due to the uncertainty in the HMNS collapse time, we choose two different parametrized times to induce collapse. We also explore two initial magnetic field geometries in the torus, and evolve the systems until the outflows diminish significantly ($\sim\!\! 1\!\! - \!\!2\ \mathrm{s}$). We find bluer, faster outflows as compared to equivalent BH–torus systems, producing M ∼ 10 −3 M ⊙ of ejecta with Y e ≥ 0.25 and v ≥ 0.25c by the simulation end. Approximately half the outflows are launched in disc winds at times $t\lesssim 500 \ \mathrm{ms}$, with a broad distribution of electron fractions and velocities, depending on the initial condition. The remaining outflows are thermally driven, characterized by lower velocities and electron fractions. Nucleosynthesis with tracer particles shows patterns resembling solar abundances in all models. Although outflows from our simulations do not match those inferred from two-component modelling of the GW170817 kilonova, self-consistent multidimensional detailed kilonova models are required to determine whether our outflows can power the blue kilonova.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

SDSS-IV MaNGA: stellar rotational support in disc galaxies versus central surface density and stellar population age

ABSTRACT We investigate how the stellar rotational support changes as a function of spatially resolved stellar population age ($D_n4000$) and relative central stellar surface density ($\Delta \Sigma _1$) for MaNGA isolated/central disc galaxies. We find that the galaxy rotational support indicator $\lambda _{R_\mathrm{e}}$ varies smoothly as a function of $\Delta \Sigma _1$ and $D_n4000$. $D_n4000$ versus $\Delta \Sigma _1$ follows a ‘J-shape’, with $\lambda _{R_\mathrm{e}}$ contributing to the scatters. In this ‘J-shaped’ pattern rotational support increases with central $D_n4000$ when $\Delta \Sigma _1$ is low but decreases with $\Delta \Sigma _1$ when $\Delta \Sigma _1$ is high. Restricting attention to low-$\Delta \Sigma _1$ (i.e. large-radius) galaxies, we suggest that the trend of increasing rotational support with $D_n4000$ for these objects is produced by a mix of two different processes, a primary trend characterized by growth in $\lambda _{R_\mathrm{e}}$ along with mass through gas accretion, on top of which disturbance episodes are overlaid, which reduce rotational support and trigger increased star formation. An additional finding is that star-forming galaxies with low $\Delta \Sigma _1$ have relatively larger radii than galaxies with higher $\Delta \Sigma _1$ at fixed stellar mass. Assuming that these relative radii rankings are preserved while galaxies are star forming then implies clear evolutionary paths in central $D_n4000$ versus $\Delta \Sigma _1$. The paper closes with comments on the implications that these paths have for the evolution of pseudo-bulges versus classical bulges. The utility of using $\rm D_n4000$–$\Delta \Sigma _1$ to study $\lambda _{R_\mathrm{e}}$ reinforces the notion that galaxy kinematics correlate both with structure and with stellar-population state, and indicates the importance of a multidimensional description for understanding bulge and galaxy evolution.

Wang, Xiaohan (ORCID:0009000283121002)↗

DOE OSTI2024

BOSS quasar outflows traced by C iv

ABSTRACT We investigate possible factors that drive fast quasar outflows using a sample of 39 249 quasars at median redshift 〈z〉 ≈ 2.17. Unique to this study, the quasar redshifts are re-measured based on the Mg ii emission line, and explore unprecedented outflow velocities (>6000 km s−1), while maintaining statistical significance. We measure reliable C iv blueshifts for 1178 quasars with velocities >2500 km s−1. From those, 255(13) quasars have blueshifts above 4000(6000) km s−1, with the highest C iv velocity ≈7000 km s−1. Several correlations are observed, where higher C iv blueshifts in general are in quasars with broader, weaker C iv emission profiles, weak He ii emission, larger Eddington ratios, and bluer ultraviolet (UV) continuum slope across the rest-frame UV to near-infrared. Analysis reveals two primary factors contributing to faster outflows: higher Eddington ratios, and softer far-UV continuum (hν >24.6 eV). We find supporting evidence that radiative line driving may generate extreme outflow velocities, influenced by multiple factors as suggested by the aforementioned correlations. This evidence highlights the importance of considering a multidimensional parameter space in future studies when analyzing large C iv blueshifts to determine the fundamental causes of outflows.

Gillette, Jarred (ORCID:0000000185284839)↗

DOE OSTI2025

Mixing by internal gravity waves in stars: assessing numerical simulations against theory

ABSTRACT Here we present a study of radial chemical mixing in non-rotating massive main-sequence stars driven by internal gravity waves (IGWs), based on multidimensional hydrodynamical simulations with the fully compressible code MUSIC. We examine two proposed mechanisms of material mixing in stars by IGWs that are commonly quoted, relating to thermal diffusion and sub-wavelength shearing. Thermal diffusion provides a non-restorative effect to the waves, leaving material displaced from its previous equilibrium, while shearing arising within the waves drives weak localized flows, mixing the fluid there. Using IGW spectra from the simulations, we evaluate theoretical predictions of mixing rates due to these mechanisms. We show, for $20\, \mathrm{M}_\odot$ main-sequence stars, that neither of these mechanisms are likely to create mixing sufficient to correct inaccuracies in current stellar evolution models. Furthermore, we compare these predictions to results obtained from Lagrangian tracer particles, following a method recently used for global simulations of stellar interiors to measure mixing by IGWs in their radiative zones. We demonstrate that tracer particle methods face significant numerical challenges in measuring the small diffusion coefficients predicted by the aforementioned theories, for which they are prone to yielding artificially enhanced coefficients. Diffusion coefficients based on such methods are currently used with stellar evolution codes for asteroseismic studies, but should be viewed with caution. Finally, in a case where tracer particles do not suffer from numerical artefacts, we suggest that a diffusion model is not suitable for time-scales typically considered by 2D numerical simulations.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2017

Low-energy Population III supernovae and the origin of extremely metal-poor stars

Some ancient, dim, metal-poor stars may have formed in the ashes of the first supernovae (SNe). If their chemical abundances can be reconciled with the elemental yields of specific Population III (Pop III) explosions, they could reveal the properties of primordial stars. But multidimensional simulations of such explosions are needed to predict their yields because dynamical instabilities can dredge material up from deep in the ejecta that would otherwise be predicted to fall back on to the central remnant and be lost in one-dimensional (1D) models. We have performed two-dimensional (2D) numerical simulations of two low-energy Pop III SNe, a 12.4 M ⊙ explosion and a 60 M ⊙ explosion, and find that they produce elemental yields that are a good fit to those measured in the most iron-poor star discovered to date, SMSS J031300.36-670839.3 (J031300). Fallback on to the compact remnant in these weak explosions accounts for the lack of measurable iron in J031300 and its low iron-group abundances in general. Our 2D explosions produce higher abundances of heavy elements (atomic number Z > 20) than their 1D counterparts due to dredge-up by fluid instabilities. Since almost no 56 Ni is ejected by these weak SNe, their low luminosities will prevent their detection in the near-infrared with the James Webb Space Telescope and future 30-m telescopes on the ground. The only evidence that they ever occurred will remain in the fossil abundance record

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2018

Impact of ejecta morphology and composition on the electromagnetic signatures of neutron star mergers

The electromagnetic transients accompanying compact binary mergers (γ-ray bursts, afterglows and ‘macronovae’) are crucial to pinpoint the sky location of gravitational wave sources. Macronovae are caused by the radioactivity from freshly synthesized heavy elements, e.g. from dynamic ejecta and various types of winds. We study macronova signatures by using multidimensional radiative transfer calculations. We then employ the radiative transfer code SUPERNU and state-of-the-art LTE opacities for a few representative elements from the wind and dynamical ejecta (Cr, Pd, Se, Te, Br, Zr, Sm, Ce, Nd, U) to calculate synthetic light curves and spectra for a range of ejecta morphologies. The radioactive power of the resulting macronova is calculated with the detailed input of decay products. We assess the detection prospects for our most complex models, based on the portion of viewing angles that are sufficiently bright, at different cosmological redshifts (z). The brighter emission from the wind is unobscured by the lanthanides (or actinides) in some of the models, permitting non-zero detection probabilities for redshifts up to z = 0.07. We also find that the nuclear mass model and the resulting radioactive heating rate are crucial for the detectability. While for the most pessimistic heating rate (from the finite range droplet model) no reasonable increase in the ejecta mass or velocity, or wind mass or velocity, can possibly make the light curves agree with the observed near-infrared excess after GRB130603B, a more optimistic heating rate (from the Duflo–Zuker model) leads to good agreement. We conclude that future reliable macronova observations would constrain nuclear heating rates, and consequently help constrain nuclear mass models.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2018

On the importance of progenitor asymmetry to shock revival in core-collapse supernovae

The progenitor stars of core-collapse supernovae (CCSNe) are asymmetrically fluctuating due to turbulent convections in the late stages of their lives. The progenitor asymmetry at the pre-supernova stage has recently caught the attention as a new ingredient to facilitate shock revival in the delayed neutrino-heating mechanism. In this paper, we investigate the importance of the progenitor asymmetries to shock revival with a semi-analytical approach. Free parameters were chosen such that the time evolution of shock radii and mass accretion rates are compatible with the results of detailed numerical simulations of CCSNe in spherical symmetry. We first estimate the amplitude of asymmetries required for the shock revival by the impulsive change of pre-shock flows in the context of neutrino-heating mechanism, and then convert the amplitude to the corresponding amplitude in the pre-supernova phase by taking into account the growth of asymmetries during infall. Further, we apply our model to various types of progenitors and find that the requisite amplitude of pre-supernova asymmetry is roughly three times larger than the prediction by current stellar evolution models unless other additional physical ingredients such as multidimensional fluid instabilities and turbulent convections in post-shock flows aid shock revival. We thus conclude that progenitor asymmetries cannot trigger the shock revival by the impulsive way but rather play a supplementary role in reality.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

A comparison of explosion energies for simulated and observed core-collapse supernovae

ABSTRACT There are now 20 multidimensional core-collapse supernova (CCSN) simulations that explode. However, these simulations have explosion energies that are a few times 1050 erg, not 1051 erg. In this manuscript, we compare the inferred explosion energies of these simulations and observations of 40 SN IIP. Assuming a lognormal distribution, the mean explosion energy for these observations is $\mu _{\rm obs} = -0.23^{+0.08}_{-0.12}$ ($\log _{10}(E/10^{51}\, {\rm erg})$) and the width is $\sigma _{\rm obs} = 0.52^{+0.09}_{-0.08}$. Only three CCSN codes have sufficient simulations to compare with observations: CHIMERA, CoCoNuT-FMT, and FORNAX. Currently, FORNAX has the largest sample of simulations. The two-dimensional FORNAX simulations show a correlation between explosion energy and progenitor mass, ranging from linear to quadratic, Esim ∝ M1 − 2; this correlation is consistent with inferences from observations. In addition, we infer the ratio of the observed-to-simulated explosion energies, Δ = log10(Eobs/Esim). For the CHIMERA set, Δ = 0.25 ± 0.07; for CoCoNuT-FMT, Δ = 0.49 ± 0.07; for FORNAX2D, Δ = 0.62 ± 0.06, and for FORNAX3D, Δ = 0.85 ± 0.07. On average, the simulations are less energetic than inferred energies from observations (Δ ≈ 0.6), but we also note that the variation among the simulations [max(Δ) − min(Δ) ≈ 0.6] is as large as this average offset. This suggests that further improvements to the simulations could resolve the discrepancy. Furthermore, both the simulations and observations are heavily biased. In this preliminary comparison, we model these biases, but to more reliably compare the explosion energies, we recommend strategies to unbias both the simulations and observations.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

Comparative Transcriptomics Analyses across Species, Organs, and Developmental Stages Reveal Functionally Constrained lncRNAs

The functionality of long noncoding RNAs (lncRNAs) is disputed. In general, lncRNAs are under weak selective pressures, suggesting that the majority of lncRNAs may be nonfunctional. However, although some surveys showed negligible phenotypic effects upon lncRNA perturbation, key biological roles were demonstrated for individual lncRNAs. Most lncRNAs with proven functions were implicated in gene expression regulation, in pathways related to cellular pluripotency, differentiation, and organ morphogenesis, suggesting that functional lncRNAs may be more abundant in embryonic development, rather than in adult organs. To test this hypothesis, we perform a multidimensional comparative transcriptomics analysis, across five developmental time points (two embryonic stages, newborn, adult, and aged individuals), four organs (brain, kidney, liver, and testes), and three species (mouse, rat, and chicken). We find that, overwhelmingly, lncRNAs are preferentially expressed in adult and aged testes, consistent with the presence of permissive transcription during spermatogenesis. LncRNAs are often differentially expressed among developmental stages and are less abundant in embryos and newborns compared with adult individuals, in agreement with a requirement for tighter expression control and less tolerance for noisy transcription early in development. For differentially expressed lncRNAs, we find that the patterns of expression variation among developmental stages are generally conserved between mouse and rat. Moreover, lncRNAs expressed above noise levels in somatic organs and during development show higher evolutionary conservation, in particular, at their promoter regions. Thus, we show that functionally constrained lncRNA loci are enriched in developing organs, and we suggest that many of these loci may function in an RNA-independent manner.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2022

Multimodal X-ray nano-spectromicroscopy analysis of chemically heterogeneous systems

Abstract Understanding the nanoscale chemical speciation of heterogeneous systems in their native environment is critical for several disciplines such as life and environmental sciences, biogeochemistry, and materials science. Synchrotron-based X-ray spectromicroscopy tools are widely used to understand the chemistry and morphology of complex material systems owing to their high penetration depth and sensitivity. The multidimensional (4D+) structure of spectromicroscopy data poses visualization and data-reduction challenges. This paper reports the strategies for the visualization and analysis of spectromicroscopy data. We created a new graphical user interface and data analysis platform named XMIDAS (X-ray multimodal image data analysis software) to visualize spectromicroscopy data from both image and spectrum representations. The interactive data analysis toolkit combined conventional analysis methods with well-established machine learning classification algorithms (e.g. nonnegative matrix factorization) for data reduction. The data visualization and analysis methodologies were then defined and optimized using a model particle aggregate with known chemical composition. Nanoprobe-based X-ray fluorescence (nano-XRF) and X-ray absorption near edge structure (nano-XANES) spectromicroscopy techniques were used to probe elemental and chemical state information of the aggregate sample. We illustrated the complete chemical speciation methodology of the model particle by using XMIDAS. Next, we demonstrated the application of this approach in detecting and characterizing nanoparticles associated with alveolar macrophages. Our multimodal approach combining nano-XRF, nano-XANES, and differential phase-contrast imaging efficiently visualizes the chemistry of localized nanostructure with the morphology. We believe that the optimized data-reduction strategies and tool development will facilitate the analysis of complex biological and environmental samples using X-ray spectromicroscopy techniques.

36 MATERIALS SCIENCE↗

DOE OSTI2018

Use of computational modeling combined with advanced visualization to develop strategies for the design of crop ideotypes to address food security

Sustainable crop production is a contributing factor to current and future food security. Innovative technologies are needed to design strategies that will achieve higher crop yields on less land and with fewer resources. Computational modeling coupled with advanced scientific visualization enables researchers to explore and interact with complex agriculture, nutrition, and climate data to predict how crops will respond to untested environments. These virtual observations and predictions can direct the development of crop ideotypes designed to meet future yield and nutritional demands. This review surveys modeling strategies for the development of crop ideotypes and scientific visualization technologies that have led to discoveries in “big data” analysis. Combined modeling and visualization approaches have been used to realistically simulate crops and to guide selection that immediately enhances crop quantity and quality under challenging environmental conditions. Lastly, this survey of current and developing technologies indicates that integrative modeling and advanced scientific visualization may help overcome challenges in agriculture and nutrition data as large-scale and multidimensional data become available in these fields.

3D crop modeling↗

DOE OSTI2025

From disorganized data to emergent dynamic models: Questionnaires to partial differential equations

Starting with sets of disorganized observations of spatially varying and temporally evolving systems, obtained at different (also disorganized) sets of parameters, we demonstrate the data-driven derivation of parameter dependent, evolutionary partial differential equation (PDE) models capable of generating the data. This tensor type of data is reminiscent of shuffled (multidimensional) puzzle tiles. The independent variables for the evolution equations (their “space” and “time”) as well as their effective parameters are all emergent , i.e. determined in a data-driven way from our disorganized observations of behavior in them. We use a diffusion map based questionnaire approach to build a smooth parametrization of our emergent space/time/parameter space for the data. This approach iteratively processes the data by successively observing them on the “space,” the “time” and the “parameter” axes of a tensor. Once the data become organized, we use machine learning (here, neural networks) to approximate the operators governing the evolution equations in this emergent space. Our illustrative examples are based (i) on a simple advection–diffusion model; (ii) on a previously developed vertex-plus-signaling model of Drosophila embryonic development; and (iii) on two complex dynamic network models (one neuronal and one coupled oscillator model) for which no obvious smooth embedding geometry is known a priori. This allows us to discuss features of the process like symmetry breaking, translational invariance, and autonomousness of the emergent PDE model, as well as its interpretability.

generative models↗

DOE OSTI2025

Self-consistent microscopic calculations for electron captures on nuclei in core-collapse supernovae

Calculations for electron capture rates on nuclei with atomic numbers between 𝑍 = 20 and 𝑍 = 52 are performed in a self-consistent finite-temperature covariant energy density functional theory within the relativistic quasiparticle random-phase approximation. Electron captures on these nuclei contribute most to reducing the electron fraction during the collapse phase of core-collapse supernovae. The rates include contributions from allowed (Gamow-Teller) and first-forbidden (FF) transitions, and it is shown that the latter become dominant at high stellar densities and temperatures. Temperature-dependent effects such as Pauli unblocking and transitions from thermally excited states are also included. The new rates are implemented in a spherically symmetric one-dimensional simulation of the core-collapse phase. The results indicate that the increase in electron capture rates, due to inclusion of FF transitions, leads to reductions of the electron fraction at nuclear saturation density, the peak neutrino luminosity, and enclosed mass at core bounce. The new rates reaffirm that the most relevant nuclei for the deleptonization situate around the 𝑁 = 50 and 82 shell closures, but, compared to previous simulations, nuclei are less proton rich. Here, the new rates developed in this work are available, and will be of benefit to improve the accuracy of multidimensional supernova simulations.

Electron & muon capture↗

DOE OSTI2026

Boosted decision tree reweighting of simulated neutrino interactions for O ( 1 ) GeV neutrino cross-section measurements

This paper illustrates a generic method for multidimensional reweighting of O ( 1 ) GeV neutrino interaction Monte Carlo samples. The reweighting is based on a boosted decision tree algorithm trained on high-dimensional space in detector final-state observables. This enables one generator’s events to be reweighted so that its reconstructed particle content and kinematics distributions, as well as detector efficiency, match those of a target model. The approach establishes an efficient way to reuse legacy Monte Carlo data, avoiding regeneration. As an example, we test its use in a measurement of transverse kinematic imbalance of the μ - and proton in charged-current quasielastic like ν μ events from the MINERvA experiment.

Lin, Z. [Rochester U.] (ORCID:0009000188903698)↗

DOE OSTI2026

Enhancements in Laser-Direct-Drive Nuclear Performance with Target Radius

Inertial confinement fusion is likely to require significant improvements in technology and design for large targets to implode at high driver energies. Here, to assess the potential for benefits, we report on nuclear performance as a function of target radius 𝑅 t in direct-drive cryogenic implosions as performed on the OMEGA laser. The neutron yield and areal density are found to increase as 𝑅 5.0±0.2 t and 𝑅 1.8±0.2 t , respectively, and exhibit a much stronger dependence on target radius than previous studies have assumed. As this paper demonstrates, these types of sensitivities should be expected when implosions are unstable, and have been degraded by a range of multidimensional effects in 3D. If the imperfections in a target and laser system are fixed in magnitude, it follows that larger implosions have higher relative quality, and approach criteria to ignite much more quickly than commonly appreciated.

inertial confinement fusion↗

DOE OSTI2025

Neural-network extraction of unpolarised transverse-momentum-dependent distributions

We present the first extraction of transverse-momentum-dependent distributions of unpolarized quarks from experimental Drell-Yan data using neural networks to parametrize their nonperturbative part. We show that neural networks outperform traditional parametrizations providing a more accurate description of data. This Letter establishes the feasibility of using neural networks to explore the multidimensional partonic structure of hadrons and paves the way for more accurate determinations based on machine-learning techniques.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Rethinking 𝛼−RuCl 3 : Parameters, models, and phase diagram

RuCl 3 was likely the first ever deliberately synthesized ruthenium compound, following the discovery of the 44 Ru element in 1844. For a long time it was known as an oxidation catalyst, with its physical properties being discrepant and confusing, until a decade ago when its allotropic form 𝛼−RuCl 3 rose to exceptional prominence. This “rediscovery” of 𝛼−RuCl 3 has not only reshaped the hunt for a material manifestation of the Kitaev spin liquid, but it has opened the floodgates of theoretical and experimental research in the many unusual phases and excitations that the anisotropic-exchange magnets as a class of compounds have to offer. Given its importance for the field of Kitaev materials, it is astonishing that the low-energy spin model that describes this compound and its possible proximity to the much-desired spin-liquid state is still a subject of significant debate ten years later. In the present study, we argue that the existing key phenomenological observations put strong natural constraints on the effective microscopic spin model of 𝛼−RuCl 3 , and specifically on its spin-orbit-induced anisotropic-exchange parameters that are responsible for the nontrivial physical properties of this material. These constraints allow one to focus on the relevant region of the multidimensional phase diagram of the 𝛼−RuCl 3 model, suggest an intuitive description of it via a different parametrization of the exchange matrix, offer a unifying view on the earlier assessments of its parameters, and bring closer together several approaches to the derivation of anisotropic-exchange models. We explore extended phase diagrams relevant to the 𝛼−RuCl 3 parameter space using quasiclassical, Luttinger-Tisza, exact diagonalization, and density-matrix renormalization-group methods, demonstrating a remarkably close quantitative accord between them on the general structure and hierarchy of the phases, with the zigzag, ferromagnetic, and incommensurate phases that are proximate to each other. As a result, one of the highlights is the detailed agreement on the nature of the incommensurate phases that realize two distinct counterrotating helical states.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗