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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Mshock HED Council Request FY24 [Slides]

ICF and HEDP experiments experience complicated loadings including successive shocks, but modeling and analytic work has mostly engaged the simpler shock or shock-reshock case. This is because the co propagating case is difficult to achieve with conventional (non-HED) drivers. Successive shocks will be challenging to model in BHR or modal model interpretations: 1) Ex: Consider the case where long wavelength modes re-invert coherently, but short wavelength nonlinear modes are spun up turbulently. 2) Necessitates more advanced diagnostics and analysis than simple “mixing layers.” High resolution spectral information (concomitant with Ω-EP campaigns) and higher-order moment analyses will be necessary to stress and validate the reduced BHR-type models.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

Many-Body Level Statistics of Single-Particle Quantum Chaos

We consider a noninteracting many-fermion system populating levels of a unitary random matrix ensemble (equivalent to the q = 2 complex Sachdev-Ye-Kitaev model)—a generic model of single-particle quantum chaos. We study the corresponding many-particle level statistics by calculating the spectral form factor analytically using algebraic methods of random matrix theory, and match it with an exact numerical simulation. Despite the integrability of the theory, the many-body spectral rigidity is found to have a surprisingly rich landscape. In particular, we find a residual repulsion of distant many-body levels stemming from single-particle chaos, together with islands of level attraction. These results are encoded in an exponential ramp in the spectral form factor, which we show to be a universal feature of nonergodic many-fermion systems embedded in a chaotic medium.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

DSPS: Differentiable stellar population synthesis

ABSTRACT Models of stellar population synthesis (SPS) are the fundamental tool that relates the physical properties of a galaxy to its spectral energy distribution (SED). In this paper, we present DSPS: a python package for SPS. All of the functionality in DSPS is implemented natively in the JAX library for automatic differentiation, and so our predictions for galaxy photometry are fully differentiable, and directly inherit the performance benefits of JAX, including portability onto GPUs. DSPS also implements several novel features, such as i) a flexible empirical model for stellar metallicity that incorporates correlations with stellar age, ii) support for the Diffstar model that provides a physically-motivated connection between the star formation history of a galaxy (SFH) and the mass assembly of its underlying dark matter halo. We detail a set of theoretical techniques for using autodiff to calculate gradients of predictions for galaxy SEDs with respect to SPS parameters that control a range of physical effects, including SFH, stellar metallicity, nebular emission, and dust attenuation. When forward modelling the colours of a synthetic galaxy population, we find that DSPS can provide a factor of 5 speed-up over standard SPS codes on a CPU, and a factor of 300-400 on a modern GPU. When coupled with gradient-based techniques for optimization and inference, DSPS makes it practical to conduct expansive likelihood analyses of simulation-based models of the galaxy–halo connection that fully forward model galaxy spectra and photometry.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Parallel transport dynamics for mixed quantum states with applications to time-dependent density functional theory

Direct simulation of the von Neumann dynamics for a general (pure or mixed) quantum state can often be expensive. One prominent example is the real-time time-dependent density functional theory (rt-TDDFT), a widely used framework for the first principle description of many-electron dynamics in chemical and materials systems. Practical rt-TDDFT calculations often avoid the direct simulation of the von Neumann equation, and solve instead a set of Schrödinger equations, of which the dynamics is equivalent to that of the von Neumann equation. However, the time step size employed by the Schrödinger dynamics is often much smaller. Here, in order to improve the time step size and the overall efficiency of the simulation, we generalize a recent work of the parallel transport (PT) dynamics for simulating pure states [An, Lin, Multiscale Model. Simul. 18, 612, 2020] to general quantum states. The PT dynamics provides the optimal gauge choice, and can employ a time step size comparable to that of the von Neumann dynamics. Going beyond the linear and near adiabatic regime in previous studies, we find that the error of the PT dynamics can be bounded by certain commutators between Hamiltonians, density matrices, and their derived quantities. Such a commutator structure is not present in the Schrödinger dynamics. We demonstrate that the parallel transport-implicit midpoint (PT-IM) method is a suitable method for simulating the PT dynamics, especially when the spectral radius of the Hamiltonian is large. The commutator structure of the error bound, and numerical results for model rt-TDDFT calculations in both linear and nonlinear regimes, confirm the advantage of the PT dynamics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Cloud droplet diffusional growth in homogeneous isotropic turbulence: bin microphysics versus Lagrangian super-droplet simulations

The increase in the spectral width of an initially monodisperse population of cloud droplets in homogeneous isotropic turbulence is investigated by applying a finite-difference fluid flow model combined with either Eulerian bin microphysics or a Lagrangian particle-based scheme. The turbulence is forced applying a variant of the so-called linear forcing method that maintains the mean turbulent kinetic energy (TKE) and the TKE partitioning between velocity components. The latter is important for maintaining the quasi-steady forcing of the supersaturation fluctuations that drive the increase in the spectral width. We apply a large computational domain (64 3 m 3 ), one of the domains considered in Thomas et al. (2020). The simulations apply 1 m grid length and are in the spirit of the implicit large eddy simulation (ILES), that is, with small-scale dissipation provided by the model numerics. This is in contrast to the scaled-up direct numerical simulation (DNS) applied in Thomas et al. (2020). Two TKE intensities and three different droplet concentrations are considered. Analytic solutions derived in Sardina et al. (2015), valid for the case when the turbulence integral timescale is much larger than the droplet phase relaxation timescale, are used to guide the comparison between the two microphysics simulation techniques. The Lagrangian approach reproduces the scalings relatively well. Representing the spectral width increase in time is more challenging for the bin microphysics because appropriately high resolution in the bin space is needed. The bin width of 0.5 µm is only sufficient for the lowest droplet concentration (26 cm -3 ). For the highest droplet concentration (650 cm -3 ), an order of magnitude smaller bin size is barely sufficient. The scalings are not expected to be valid for the lowest droplet concentration and the high-TKE case, and the two microphysics schemes represent similar departures. Finally, because the fluid flow is the same for all simulations featuring either low or high TKE, one can compare point-by-point simulation results. Such a comparison shows very close temperature and water vapor point-by-point values across the computational domain and larger differences between simulated mean droplet radii and spectral width. The latter are explained by fundamental differences in the two simulation methodologies, numerical diffusion in the Eulerian bin approach and a relatively small number of Lagrangian particles that are used in the particle-based microphysics.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

First Annual Report on Development of Microwave Resonant Cavity Transducer for Fluid Flow Sensing: Development of Sensor Performance Model of Microwave Cavity Flow Meter for Advanced Reactor High Temperature Fluids

We are investigating a microwave cavity-based transducer for in-core high-temperature fluid flow sensing in molten salt cooled reactors (MSCR) and sodium fast reactors (SFR). This sensor is a hollow metallic cylindrical cavity, which can be fabricated from stainless steel, and as such is expected to be resilient to radiation, high temperature and corrosive environment of MSCR and SFR. The principle of sensing consists of making one wall of the cylindrical cavity flexible enough so that dynamic pressure, which is proportional to fluid velocity, will cause membrane deflection. Membrane deflection causes cavity volume change, which leads to a shift in the resonant frequency. Feasibility of the sensor was initially investigated with analytical derivations and with COMSOL RF Module computer simulations of resonant frequency spectral shift due to uniform load. We also investigated the mechanical integrity of the flowmeter’s membrane through analytical modelling and COMSOL Structural Mechanics Module computer simulations. Both the analytic model and COMSOL model showed that maximum stresses on the plate, which are at the radial boundary of the plate, are three orders of magnitude smaller than the material’s yield strength and ultimate tensile strength. This indicates that the sensor is at a low risk of mechanical failure. Using results from models, we have developed an initial design for a microwave K-band sensor. A cylindrical resonator prototype was fabricated from brass for the initial tests. The external dimensions of the cavity are matched to the flange of a standard WR-42 waveguide. Microwave field is coupled into the resonant cavity through a subwavelength-size aperture. A test article was developed consisting of a piping Tee with a bulkhead WR-42 microwave waveguide installed in a leak-proof assembly. A microwave waveguide circulator was installed in the setup to suppress the effect of reflections at the cavity entrance by increasing the isolation between the input and the output port. Preliminary spectral characterization of cavity spectral response was performed with a portable PXIe chassis microwave VNA with a custom GUI. Preliminary dry tests of the transducer response were conducted with a set of calibrated weights. Transducer frequency shift was shown to be monotonically increasing with increasing pressure. The next steps will involve investigation of the transducer performance for water flow sensing.

42 ENGINEERING↗
DOE OSTI2022

Calibration, modeling, parameterization, and verification of the instrument transfer function of an interferometric microscope

Interferometric microscopes are used to measure surface roughness, from which the computed power spectral density function can be used to extract bandwidth-limited values of the various surface properties, such as root-mean-square (rms) height and slope errors. Measurements with a microscope equipped with different objectives that have an overlapping spatial frequency range usually give different rms results over the common frequency bandwidth. This is a result of different instrument transfer functions (ITFs) that attenuate spatial frequencies by different amounts over the overlapping range. We report on the use of binary pseudo-random array (BPRA) standards to characterize the ITF of an interferometric microscope with the various objectives. We use a simple model of a 1D binary Results show that the spectrum for an undersampled array is a cosine function, rather than a straight line, constant white noise spectrum. We have an analytical model of the ITF that includes the effects of an obscured aperture and defocus, in addition to the usual parameters of numerical aperture, wavelength, and sampling period. In addition, the model includes the effect of aliasing of spatial frequency components beyond the Nyquist back into the sub-Nyquist region. We compare the model PSD predictions to the measurements performed with different objectives. Departures of the measured PSDs from the model predictions indicate that there are higher-order ITF corrections yet to be identified and included in the model.

Takacs, PZ↗
DOE OSTI2023

Low-pass spectral analysis of time-resolved serial femtosecond crystallography data

Low-pass spectral analysis (LPSA) is a recently developed dynamics retrieval algorithm showing excellent retrieval properties when applied to model data affected by extreme incompleteness and stochastic weighting. In this work, we apply LPSA to an experimental time-resolved serial femtosecond crystallography (TR-SFX) dataset from the membrane protein bacteriorhodopsin (bR) and analyze its parametric sensitivity. While most dynamical modes are contaminated by nonphysical high-frequency features, we identify two dominant modes, which are little affected by spurious frequencies. The dynamics retrieved using these modes shows an isomerization signal compatible with previous findings. We employ synthetic data with increasing timing uncertainty, increasing incompleteness level, pixel-dependent incompleteness, and photon counting errors to investigate the root cause of the high-frequency contamination of our TR-SFX modes. By testing a range of methods, we show that timing errors comparable to the dynamical periods to be retrieved produce a smearing of dynamical features, hampering dynamics retrieval, but with no introduction of spurious components in the solution, when convergence criteria are met. Using model data, we are able to attribute the high-frequency contamination of low-order dynamical modes to the high levels of noise present in the data. Finally, we propose a method to handle missing observations that produces a substantial dynamics retrieval improvement from synthetic data with a significant static component. Reprocessing of the bR TR-SFX data using the improved method yields dynamical movies with strong isomerization signals compatible with previous findings.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Cross-scale sensing of field-level crop residue cover: Integrating field photos, airborne hyperspectral imaging, and satellite data

Conservation tillage practices can bring benefits to agricultural sustainability. Accurate spatial and temporal resolved information of field-scale crop residue cover, which reflects tillage intensity, is highly valuable for evaluating the outcomes of government conservation programs and voluntary ecosystem service markets, as well as facilitating agroecosystem modeling to quantify cropland biogeochemical processes. Remote sensing has the potential to cost-effectively detect crop residue cover, however, existing regional-scale studies were limited by insufficient ground truth data, scale mismatch between coarse satellite pixels and ground data, and the lack of key spectral data for detecting crop residues. Therefore, this study developed an innovative cross-sensing framework to integrate proximal sensing, airborne hyperspectral imaging, and satellite Earth Observation through deep learning to quantify field-level crop residue cover fractions at the regional scale. Specifically, we have collected intensive ground orthographic photos and conducted airborne hyperspectral surveys at corn and soybean fields of Champaign and nearby counties in Illinois, the heartland of the U.S. Corn Belt. Through semi-automatic labeling aided by ResNet-50 and superpixel image segmentation, we obtained 6719 records of ground residue fractions. With these ground data, we developed the 1-dimensional convolution neural network (CNN) model using airborne hyperspectral reflectance, which has 0.5m spatial resolution and 3–5 nm spectral resolution from 400 to 2400 nm, to predict residue fractions. By applying the CNN model to airborne pixels, we augmented “ground truth” data of crop residues and further combined them with Harmonized Landsat and Sentinel-2 (HLS) satellite data to quantify regional residue fractions at 30 m resolution. Results show that airborne hyperspectral imagery with CNN can accurately detect residue fractions (R 2 = 0.82, relative RMSE = 11.73%) to effectively generate quasi “ground truth” data to support satellite upscaling to all fields. With independent ground data for testing, we found that the ground-airborne-satellite integrative framework achieved better predictions in estimating crop residue cover (R 2 = 0.67, relative RMSE = 17.53%) than the conventional ground-satellite upscaling (R 2 = 0.22, relative RMSE = 32.09%). Here we also found that the shortwave infrared wavelengths, particularly 2100–2300 nm, are vital for predicting crop residue cover. Sentinel-2 and Landsat-8 data have a comparable capability to track residue fractions due to similar shortwave infrared wavelengths. This study highlights the high accuracy of hyperspectral imaging to detect agroecosystem tillage management practices and the advantages of cross-scale sensing to cost-effectively integrate multi-source data to quantify field-level agroecosystem variables across scales.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2025

Mind the crosscap: $τ$-scaling in non-orientable gravity and time-reversal-invariant systems

Spectral statistics of quantum chaotic systems are governed by random matrix universality. In many cases of interest, time-reversal symmetry selects the Gaussian Orthogonal Ensemble (GOE) as the relevant universality class. In holographic CFTs, this is mirrored by the presence of non-orientable geometries in the dual gravitational path integral. In this work, we analyze general properties of these matrix models and their gravitational counterparts. First, we develop a formalism to express the universal level statistics in the canonical ensemble for arbitrary spectral curves, leading to a topological expansion with finite radius of convergence in the late-time $τ$-scaling limit. Then, we focus on topological gravity and study topological recursion on the moduli space of non-orientable surfaces. We find that the Weil-Petersson volumes display non-analytic behaviour multiplying polynomials in the boundary lengths. The volumes give rise to wormholes with late-time divergences, in contrast with the orientable case, which is finite. We identify systematic cancellations among WP volumes implied by the consistency and finiteness of the $τ$-scaling limit. In particular, the cancellation of late-time divergences requires a nontrivial genus resummation. Working in the gravitational microcanonical ensemble, we derive and resum all orders of the topological expansion matching the GOE matrix model in the high-energy regime.

Chaotic Dynamics (nlin.CD)↗
DOE OSTI2020

Phase identification using co‐association matrix ensemble clustering

Calibrating distribution system models to aid in the accuracy of simulations such as hosting capacity analysis is increasingly important in the pursuit of the goal of integrating more distributed energy resources. The recent availability of smart meter data is enabling the use of machine learning tools to automatically achieve model calibration tasks. This research focuses on applying machine learning to the phase identification task, using a co‐association matrix‐based, ensemble spectral clustering approach. The proposed method leverages voltage time series from smart meters and does not require existing or accurate phase labels. This work demonstrates the success of the proposed method on both synthetic and real data, surpassing the accuracy of other phase identification research.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Residual estimation for grid modification in wall-modeled large eddy simulation using unstructured high-order methods

Here, the accuracy and computational cost of a large eddy simulation are highly dependent on the computational grid. Building optimal grids manually from a priori knowledge is not feasible in most practical use cases; instead, solution-adaptive strategies can provide a robust and cost-efficient method to generate a grid with the desired accuracy. We adapt the residual estimation algorithm developed by Toosi and Larsson for Discontinuous Galerkin Spectral Elements Methods (DGSEM) to guide the grid-adaptation process. The core of the method is the computation of the estimated modeling residual using the polynomial basis functions used in DGSEM, and the averaging of the estimated residual over each element. The final method is assessed in multiple channel flow test cases and for the transonic flow over an airfoil, in both cases making use of mortar interfaces between elements with hanging nodes. The method is found to be robust and reliable, and to provide solutions on grids with significantly fewer elements at comparable accuracy compared to when using human-generated grids.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Abundance of Major Cell Wall Components in Natural Variants and Pedigrees of Populus trichocarpa

The rapid analysis of biopolymers including lignin and sugars in lignocellulosic biomass cell walls is essential for the analysis of the large sample populations needed for identifying heritable genetic variation in biomass feedstocks for biofuels and bioproducts. In this study, we reported the analysis of cell wall lignin content, syringyl/guaiacyl (S/G) ratio, as well as glucose and xylose content by high-throughput pyrolysis-molecular beam mass spectrometry (py-MBMS) for >3,600 samples derived from hundreds of accessions of Populus trichocarpa from natural populations, as well as pedigrees constructed from 14 parents (7 × 7). Partial Least Squares (PLS) regression models were built from the samples of known sugar composition previously determined by hydrolysis followed by nuclear magnetic resonance (NMR) analysis. Key spectral features positively correlated with glucose content consisted of m/z 126, 98, and 69, among others, deriving from pyrolyzates such as hydroxymethylfurfural, maltol, and other sugar-derived species. Xylose content positively correlated primarily with many lignin-derived ions and to a lesser degree with m/z 114, deriving from a lactone produced from xylose pyrolysis. Models were capable of predicting glucose and xylose contents with an average error of less than 4%, and accuracy was significantly improved over previously used methods. The differences in the models constructed from the two sample sets varied in training sample number, but the genetic and compositional uniformity of the pedigree set could be a potential driver in the slightly better performance of that model in comparison with the natural variants. Broad-sense heritability of glucose and xylose composition using these data was 0.32 and 0.34, respectively. In summary, we have demonstrated the use of a single high-throughput method to predict sugar and lignin composition in thousands of poplar samples to estimate the heritability and phenotypic plasticity of traits necessary to develop optimized feedstocks for bioenergy applications.

09 BIOMASS FUELS↗
DOE OSTI2020

Discovery of a Nearby Young Brown Dwarf Disk

We report the discovery of the youngest brown dwarf with a disk at 102 pc from the Sun, WISEA J120037.79−784508.3 (W1200−7845), via the Disk Detective citizen science project. We establish that W1200−7845 is located in the 3.7{sub −1.4}{sup +4.6} Myr old ε Cha association. Its spectral energy distribution (SED) exhibits clear evidence of an infrared (IR) excess, indicative of the presence of a warm circumstellar disk. Modeling this warm disk, we find the data are best fit using a power-law description with a slope α = −0.94, which suggests that it is a young, Class II type disk. Using a single blackbody disk fit, we find T{sub eff,disk}=521 K and L{sub IR}/L{sub ∗}=0.14. The near-IR spectrum of W1200−7845 matches a spectral type of M6.0 γ ± 0.5, which corresponds to a low surface gravity object, and lacks distinctive signatures of strong Paβ or Brγ accretion. Both our SED fitting and spectral analysis indicate that the source is cool (T {sub eff} = 2784–2850 K), with a mass of 42–58 M {sub Jup}, well within the brown dwarf regime. The proximity of this young brown dwarf disk makes the system an ideal benchmark for investigating the formation and early evolution of brown dwarfs.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Measurements of Effective Optical Depth in the Lyα Forest from the BOSS DR12 Quasar Sample

We measure the effective optical depth in the Lyα forest using 40,035 quasar spectra from the Twelfth Data Release (DR12) of the Baryon Oscillation Spectroscopic Survey of Sloan Digital Sky Survey IV. A rigorous selection based on spectral index and the equivalent width of the C iv emission line is applied to choose seven uniform samples with minimal intrinsic variations across redshifts. Modeling the redshift evolution of the effective optical depth with a power law, ${\tau }_{\mathrm{eff}}={\tau }_{0}{(1+z)}^{\gamma },$ produces ${\tau }_{0}=(5.54\pm 0.64)\times {10}^{-3}$ and $\gamma =3.182\pm 0.074$. The 2.2% precision estimate on γ is dominated by systematic errors, likely arising from the bias and uncertainties in spectral index estimates. Even after incorporating the systematic errors, this work provides the most precise estimates of optical depth parameters to date. Finally, using the reconstructed Lyα forest continuum to directly measure the transmitted flux ratio as a function of redshift, we find deviations of less than 2.5% from the predictions from the global model and no convincing evidence for signal associated with He ii reionization.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Simulated effect of edge plasma density parameters on lower hybrid wave scattering in EAST

The incorporation of lower hybrid (LH) wave spectrum broadening in the poloidal wavenumber (⁠k θ ) space at the last close field surface (LCFS) is reported to lead to better agreement of the modeled LH wave current profile with the experimental results [Baek et al., Nucl. Fusion 61, 106034 (2021)]. To further understand its underlying mechanism and find the possible influence factors, effects of wave scattering caused by drift-wave type density fluctuation on the probability distribution of the LH wave polar refractive index (⁠N θ ) at the LCFS are studied under density parameters in the scrape-off-layer. According to a scattering model [P. T. Bonoli and E. Ott, Phys Fluids 25(2), 359–375 (1982)], scattering probability and scattering angle distribution are two main factors that determine the degree of spectral broadening. Studies presented here show that the total scattering probability increases first and then decreases as the wave propagates toward a smaller normalized radius of poloidal magnetic flux (⁠ρ⁠). The degree of spectral broadening is found to depend on the density and density fluctuation together by changing the intensity and a proportion of the geometrical optics approximation term and the E×B drift term in the scattering model. Additionally, the fluctuation correlation length can significantly modify the probability distribution of N θ at the LCFS, which is found to significantly change the LH wave current profile.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Bayesian inference for plasmonic nanometrology

Here, we introduce a Bayesian method for the characterization of plasmonic nanoparticles, which is applicable to both near- and far-field problems. Designed to combine data generated from any photon-plasmon interaction experiment with physically motivated theoretical models, our approach leverages state-of-the-art Markov chain Monte Carlo sampling techniques and returns parameter estimates on nanometric scales. Simulated spectral data sets, describing resonant scattering of photons from ellipsoidal and toroidal nanoparticles, are explored as concrete examples of our approach, with the resulting Bayesian estimates showing excellent agreement with the ground truth, even under conditions of high statistical noise. By incorporating Bayes factors into the method as well, we reveal how model selection can determine which one of competing geometric shapes better explains the observed data. Our comprehensive nanometrology procedure can be tailored to a variety of light-particle interaction models, and its reliance on Bayesian inference furnishes automatic uncertainty quantification. In addition to applicability to a host of plasmonic configurations such as nanoparticle dimers, trimers, and array studies, it is proposed that the presented analysis can be extended to the quantum regime, where nonclassical photon statistics may provide additional insight for inference of scatterer properties.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Floquet operator dynamics and orthogonal polynomials on the unit circle

Operator spreading under stroboscopic time evolution under a unitary is studied. An operator Krylov space is constructed and mapped to orthogonal polynomials on a unit circle (OPUC), as well as to the Krylov space of the edge operator of the Floquet transverse field Ising model with inhomogeneous couplings (ITFIM). The Verblunsky coefficients in the OPUC representation are related to the Krylov angles parametrizing the ITFIM. The relations between the OPUC and spectral functions are summarized and several applications are presented. These include derivation of analytic expressions for the OPUC under persistent m-periodic dynamics, and the numerical construction of the OPUC for autocorrelations of the homogeneous Floquet-Ising model as well as the Z 3 clock model. The numerically obtained Krylov angles of the Z 3 clock model with long-lived period tripled autocorrelations show a spatial periodicity of six, and this observation is used to develop an analytically solvable model for the ITFIM that mimics this behavior.

Floquet systems↗