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DOE OSTI2016

Towards developing a compact model for magnetization switching in straintronics magnetic random access memory devices

Strain-mediated magnetization switching in a magnetic tunneling junction (MTJ) by exploiting a combination of piezoelectricity and magnetostriction has been proposed as an energy efficient alternative to spin transfer torque (STT) and field induced magnetization switching methods in MTJ-based magnetic random access memories (MRAM). Theoretical studies have shown the inherent advantages of strain-assisted switching, and the dynamic response of the magnetization has been modeled using the Landau-Lifshitz-Gilbert (LLG) equation. However, an attempt to use LLG for simulating dynamics of individual elements in large-scale simulations of multi-megabyte straintronics MRAM leads to extremely time-consuming calculations. Hence, a compact analytical solution, predicting the flipping delay of the magnetization vector in the nanomagnet under stress, combined with a liberal approximation of the LLG dynamics in the straintronics MTJ, can lead to a simplified model of the device suited for fast large-scale simulations of multi-megabyte straintronics MRAMs. In this work, a tensor-based approach is developed to study the dynamic behavior of the stressed nanomagnet. First, using the developed method, the effect of stress on the switching behavior of the magnetization is investigated to realize the margins between the underdamped and overdamped regimes. The latter helps the designer realize the oscillatory behavior of the magnetization when settling along the minor axis, and the dependency of oscillations on the stress level and the damping factor. Next, a theoretical model to predict the flipping delay of the magnetization vector is developed and tested against LLG-based numerical simulations to confirm the accuracy of findings. Lastly, the obtained delay is incorporated into the approximate solutions of the LLG dynamics, in order to create a compact model to liberally and quickly simulate the magnetization dynamics of the MTJ under stress. Using the developed delay equation, the efficiency of the straintronics switching over the STT method is highlighted by analytically investigating the energy-delay trade-off of both methodologies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

A Polarization Tensor for Neutron Optics

A rank 2 dimension 3 tensor formulation for polarized neutron transport in a magnetic field is derived. Using this formulation, the rotating magnetic field problem is revisited to produce a single 3 × 3 vector-transformation matrix that describes the polarization transport. That result is then used to fashion a new strategy for a numerical approximation of polarization transport in an arbitrary magnetic field using time-dependent perturbation theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2002

Definition of Contravariant Velocity Components

This is an old issue in computational fluid dynamics (CFD). What is the so-called contravariant velocity or contravariant velocity component? In the article, we review the basics of tensor analysis and give the contravariant velocity component a rigorous explanation. For a given coordinate system, there exist two uniquely determined sets of base vector systems - one is the covariant and another is the contravariant base vector system. The two base vector systems are reciprocal. The so-called contravariant velocity component is really the contravariant component of a velocity vector for a time-independent coordinate system, or the contravariant component of a relative velocity between fluid and coordinates, for a time-dependent coordinate system. The contravariant velocity components are not physical quantities of the velocity vector. Their magnitudes, dimensions, and associated directions are controlled by their corresponding covariant base vectors. Several 2-D (two-dimensional) linear examples and 2-D mass-conservation equation are used to illustrate the details of expressing a vector with respect to the covariant and contravariant base vector systems, respectively.

Hung, Ching-Mao↗

DOE OSTI2018

Towards space from Hilbert space: finding lattice structure in finite-dimensional quantum systems

We report that field theories place one or more degrees of freedom at every point in space. Hilbert spaces describing quantum field theories, or their finite-dimensional discretizations on lattices, therefore have large amounts of structure: they are isomorphic to the tensor product of a smaller Hilbert space for each lattice site or point in space. Local field theories respecting this structure have interactions which preferentially couple nearby points. The emergence of classicality through decoherence relies on this framework of tensor-product decomposition and local interactions. We explore the emergence of such lattice structure from Hilbert-space considerations alone. We point out that the vast majority of finite-dimensional Hilbert spaces cannot be isomorphic to the tensor product of Hilbert-space subfactors that describes a lattice theory. A generic Hilbert space can only be split into a direct sum corresponding to a basis of state vectors spanning the Hilbert space, we consider setups in which the direct sum is naturally decomposed into two pieces. We define a notion of direct-sum locality which characterizes states and decompositions compatible with Hamiltonian time evolution. Finally, we illustrate these notions for a toy model that is the finite-dimensional discretization of the quantum-mechanical double-well potential. We discuss their relevance in cosmology and field theory, especially for theories which describe a landscape of vacua with different spacetime geometries.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Higgs boson production cross-section measurements and their EFT interpretation in the $4\ell $ decay channel at $\sqrt{s}=$13 TeV with the ATLAS detector

Higgs boson properties are studied in the four-lepton decay channel (where lepton = e, μ) using 139 fb -1 of proton–proton collision data recorded at $\sqrt{s}=$13 TeV by the ATLAS experiment at the Large Hadron Collider. The inclusive cross-section times branching ratio for H→ZZ* decay is measured to be 1.34±0.12 pb for a Higgs boson with absolute rapidity below 2.5, in good agreement with the Standard Model prediction of 1.33±0.08 pb. Cross-sections times branching ratio are measured for the main Higgs boson production modes in several exclusive phase-space regions. The measurements are interpreted in terms of coupling modifiers and of the tensor structure of Higgs boson interactions using an effective field theory approach. Exclusion limits are set on the CP-even and CP-odd ‘beyond the Standard Model’ couplings of the Higgs boson to vector bosons, gluons and top quarks.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Faster Johnson–Lindenstrauss transforms via Kronecker products

The Kronecker product is an important matrix operation with a wide range of applications in signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast Johnson–Lindenstrauss projection for embedding vectors with Kronecker product structure, the Kronecker fast Johnson–Lindenstrauss transform (KFJLT). The KFJLT reduces the embedding cost by an exponential factor of the standard fast Johnson–Lindenstrauss transform’s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. Here, we prove that this computational gain comes with only a small price in embedding power: consider a finite set of $p$ points in a tensor product of $d$ constituent Euclidean spaces $\bigotimes _{k=d}^{1}{\mathbb{R}}^{n_k}$, and let $N = \prod _{k=1}^{d}n_k$. With high probability, a random KFJLT matrix of dimension $m \times N$ embeds the set of points up to multiplicative distortion $(1\pm \varepsilon )$ provided $m \gtrsim \varepsilon ^{-2} \, \log ^{2d - 1} (p) \, \log N$. We conclude by describing a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems for fitting the CP tensor decomposition.

Kronecker structure↗

DOE OSTI2021

Intrinsic Anomalous Hall Conductivity in a Nonuniform Electric Field

We study how the intrinsic anomalous Hall conductivity is modified in two-dimensional crystals with broken time-reversal symmetry due to weak inhomogeneity of the applied electric field. Focusing on a clean noninteracting two-band system without band crossings, we derive the general expression for the Hall conductivity at small finite wave vector q to order q 2 , which governs the Hall response to the second gradient of the electric field. Using the Kubo formula, we show that the answer can be expressed through the Berry curvature, Fubini-Study quantum metric, and the rank-3 symmetric tensor which is related to the quantum geometric connection and physically corresponds to the gauge-invariant part of the third cumulant of the position operator. We further compare our results with the predictions made within the semiclassical approach. By deriving the semiclassical equations of motion, we reproduce the result obtained from the Kubo formula in some limits. We also find, however, that the conventional semiclassical description in terms of the definite position and momentum of the electron is not fully consistent because of singular terms originating from the Heisenberg uncertainty principle. We thus present a clear example of a case when the semiclassical approach inherently suffers from the uncertainty principle, implying that it should be applied to systems in nonuniform fields with extra care.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Scalable learning of potentials to predict time-dependent Hartree–Fock dynamics

We propose a framework to learn the time-dependent Hartree–Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Although the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange–correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to 60 × 60 × 60 × 60. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Constraints on quasinormal modes and bounds for critical points from pole-skipping

We consider a holographic thermal state and perturb it by a scalar operator whose associated real-time Green’s function has only gapped poles. These gapped poles correspond to the non-hydrodynamic quasinormal modes of a massive scalar perturbation around a Schwarzschild black brane. Relations between pole-skipping points, critical points and quasinormal modes in general emerge when the mass of the scalar and hence the dual operator dimension is varied. First, this novel analysis reveals a relation between the location of a mode in the infinite tower of quasinormal modes and the number of pole-skipping points constraining its dispersion relation at imaginary momenta. Second, for the first time, we consider the radii of convergence of the derivative expansions about the gapped quasinormal modes. These convergence radii turn out to be bounded from above by the set of all pole-skipping points. Furthermore, a transition between two distinct classes of critical points occurs at a particular value for the conformal dimension, implying close relations between critical points and pole-skipping points in one of those two classes. We show numerically that all of our results are also true for gapped modes of vector and tensor operators.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Planck 2015 results: XVII. Constraints on primordial non-Gaussianity

We report that the Planck full mission cosmic microwave background (CMB) temperature and E-mode polarization maps are analysed to obtain constraints on primordial non-Gaussianity (NG). Using three classes of optimal bispectrum estimators – separable template-fitting (KSW), binned, and modal – we obtain consistent values for the primordial local, equilateral, and orthogonal bispectrum amplitudes, quoting as our final result from temperature alone ƒ local NL = 2.5 ± 5.7, ƒ equil NL = -16 ± 70, , and ƒ ortho NL = -34 ± 32 (68% CL, statistical). Combining temperature and polarization data we obtain ƒ local NL = 0.8 ± 5.0, ƒ equil NL = -4 ± 43, and ƒ ortho NL = -26 ± 21 (68% CL, statistical). The results are based on comprehensive cross-validation of these estimators on Gaussian and non-Gaussian simulations, are stable across component separation techniques, pass an extensive suite of tests, and are consistent with estimators based on measuring the Minkowski functionals of the CMB. The effect of time-domain de-glitching systematics on the bispectrum is negligible. In spite of these test outcomes we conservatively label the results including polarization data as preliminary, owing to a known mismatch of the noise model in simulations and the data. Beyond estimates of individual shape amplitudes, we present model-independent, three-dimensional reconstructions of the Planck CMB bispectrum and derive constraints on early universe scenarios that generate primordial NG, including general single-field models of inflation, axion inflation, initial state modifications, models producing parity-violating tensor bispectra, and directionally dependent vector models. We present a wide survey of scale-dependent feature and resonance models, accounting for the “look elsewhere” effect in estimating the statistical significance of features. We also look for isocurvature NG, and find no signal, but we obtain constraints that improve significantly with the inclusion of polarization. The primordial trispectrum amplitude in the local model is constrained to be g local NL = (-0.9 ± 7.7 ) X 10 4 (68% CL statistical), and we perform an analysis of trispectrum shapes beyond the local case. The global picture that emerges is one of consistency with the premises of the ΛCDM cosmology, namely that the structure we observe today was sourced by adiabatic, passive, Gaussian, and primordial seed perturbations.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2014

Derivation of general analytic gradient expressions for density-fitted post-Hartree-Fock methods: An efficient implementation for the density-fitted second-order Møller–Plesset perturbation theory

General analytic gradient expressions (with the frozen-core approximation) are presented for density-fitted post-HF methods. An efficient implementation of frozen-core analytic gradients for the second-order Møller–Plesset perturbation theory (MP2) with the density-fitting (DF) approximation (applying to both reference and correlation energies), which is denoted as DF-MP2, is reported. The DF-MP2 method is applied to a set of alkanes, conjugated dienes, and noncovalent interaction complexes to compare the computational cost of single point analytic gradients with MP2 with the resolution of the identity approach (RI-MP2) [F. Weigend and M. Häser, Theor. Chem. Acc. 97, 331 (1997); R. A. Distasio, R. P. Steele, Y. M. Rhee, Y. Shao, and M. Head-Gordon, J. Comput. Chem. 28, 839 (2007)]. In the RI-MP2 method, the DF approach is used only for the correlation energy. Our results demonstrate that the DF-MP2 method substantially accelerate the RI-MP2 method for analytic gradient computations due to the reduced input/output (I/O) time. Because in the DF-MP2 method the DF approach is used for both reference and correlation energies, the storage of 4-index electron repulsion integrals (ERIs) are avoided, 3-index ERI tensors are employed instead. Further, as in case of integrals, our gradient equation is completely avoid construction or storage of the 4-index two-particle density matrix (TPDM), instead we use 2- and 3-index TPDMs. Hence, the I/O bottleneck of a gradient computation is significantly overcome. Therefore, the cost of the generalized-Fock matrix (GFM), TPDM, solution of Z-vector equations, the back transformation of TPDM, and integral derivatives are substantially reduced when the DF approach is used for the entire energy expression. Further application results show that the DF approach introduce negligible errors for closed-shell reaction energies and equilibrium bond lengths.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS2002

An Introduction to Tensors for Students of Physics and Engineering

Tensor analysis is the type of subject that can make even the best of students shudder. My own post-graduate instructor in the subject took away much of the fear by speaking of an implicit rhythm in the peculiar notation traditionally used, and helped us to see how this rhythm plays its way throughout the various formalisms. Prior to taking that class, I had spent many years "playing" on my own with tensors. I found the going to be tremendously difficult but was able, over time, to back out some physical and geometrical considerations that helped to make the subject a little more transparent. Today, it is sometimes hard not to think in terms of tensors and their associated concepts. This article, prompted and greatly enhanced by Marlos Jacob, whom I've met only by e-mail, is an attempt to record those early notions concerning tensors. It is intended to serve as a bridge from the point where most undergraduate students "leave off" in their studies of mathematics to the place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, and physics is assumed. A semi-intuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and higher vector products. The reader must be prepared to do some mathematics and to think. For those students who wish to go beyond this humble start, I can only recommend my professor's wisdom: find the rhythm in the mathematics and you will fare pretty well.

Kolecki, Joseph C.↗

DOE OSTI2024

An Incremental Tensor Train Decomposition Algorithm

We present a new algorithm for incrementally updating the tensor train decomposition of a stream of tensor data. This new algorithm, called the tensor train incremental core expansion (TT-ICE) improves upon the current state-of-the-art algorithms for compressing in tensor train format by developing a new adaptive approach that incurs significantly slower rank growth and guarantees compression accuracy. This capability is achieved by limiting the number of new vectors appended to the TT-cores of an existing accumulation tensor after each data increment. These vectors represent directions orthogonal to the span of existing cores and are limited to those needed to represent a newly arrived tensor to a target accuracy. We provide two versions of the algorithm: TT-ICE and TT-ICE accelerated with heuristics (TT-ICE*). Here, we provide a proof of correctness for TT-ICE and empirically demonstrate the performance of the algorithms in compressing large-scale video and scientific simulation datasets. Compared to existing approaches that also use rank adaptation, TT-ICE* achieves 57× higher compression and up to 95% reduction in computational time.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2010

Introduction to Vector Field Visualization

Vector field visualization techniques are essential to help us understand the complex dynamics of flow fields. These can be found in a wide range of applications such as study of flows around an aircraft, the blood flow in our heart chambers, ocean circulation models, and severe weather predictions. The vector fields from these various applications can be visually depicted using a number of techniques such as particle traces and advecting textures. In this tutorial, we present several fundamental algorithms in flow visualization including particle integration, particle tracking in time-dependent flows, and seeding strategies. For flows near surfaces, a wide variety of synthetic texture-based algorithms have been developed to depict near-body flow features. The most common approach is based on the Line Integral Convolution (LIC) algorithm. There also exist extensions of LIC to support more flexible texture generations for 3D flow data. This tutorial reviews these algorithms. Tensor fields are found in several real-world applications and also require the aid of visualization to help users understand their data sets. Examples where one can find tensor fields include mechanics to see how material respond to external forces, civil engineering and geomechanics of roads and bridges, and the study of neural pathway via diffusion tensor imaging. This tutorial will provide an overview of the different tensor field visualization techniques, discuss basic tensor decompositions, and go into detail on glyph based methods, deformation based methods, and streamline based methods. Practical examples will be used when presenting the methods; and applications from some case studies will be used as part of the motivation.

Kao, David↗

NASA NTRS1995

Global Dynamic Modeling of Space-Geodetic Data

The proposal had outlined a year for program conversion, a year for testing and debugging, and two years for numerical experiments. We kept to that schedule. In first (partial) year, author designed a finite element for isostatic thin-shell deformation on a sphere, derived all of its algebraic and stiffness properties, and embedded it in a new finite element code which derives its basic solution strategy (and some critical subroutines) from earlier flat-Earth codes. Also designed and programmed a new fault element to represent faults along plate boundaries. Wrote a preliminary version of a spherical graphics program for the display of output. Tested this new code for accuracy on individual model plates. Made estimates of the computer-time/cost efficiency of the code for whole-earth grids, which were reasonable. Finally, converted an interactive graphical grid-designer program from Cartesian to spherical geometry to permit the beginning of serious modeling. For reasons of cost efficiency, models are isostatic, and do not consider the local effects of unsupported loads or bending stresses. The requirements are: (1) ability to represent rigid rotation on a sphere; (2) ability to represent a spatially uniform strain-rate tensor in the limit of small elements; and (3) continuity of velocity across all element boundaries. Author designed a 3-node triangle shell element which has two different sets of basis functions to represent (vector) velocity and all other (scalar) variables. Such elements can be shown to converge to the formulas for plane triangles in the limit of small size, but can also applied to cover any area smaller than a hemisphere. The difficult volume integrals involved in computing the stiffness of such elements are performed numerically using 7 Gauss integration points on the surface of the sphere, beneath each of which a vertical integral is performed using about 100 points.

Bird, Peter↗

DOE OSTI2021

Regional Moment Tensor Inversion Using Rotational Observations

There are benefits from the addition of rotational motions to translational displacements for moment tensor (MT) inversions. The rotational radiation pattern is orthogonal to the shear radiation pattern, thus incorporating rotations is equivalent to gaining another observation point on the focal sphere. We demonstrated this by simulating curl and displacement wavefields for a regional distance station. Thus, one 6-C station (3-Component translational + 3-Component rotational) gathers the same information on radiation pattern as two 3-C stations at 90° azimuth from one another along the focal plane axis, which is sometimes difficult to obtain when restricted to surface sensors. We added rotational Green's functions to a regional MT inversion scheme (long-period, time-domain, and linear inversion) by computing spatial gradients from f-κ reflectivity synthetics. For rotational data, we used Array Derived Rotations (ADR) from Piñon Flats Observatory Array and Golay array deployed during IRIS Community Wavefield Demonstration Experiment. The hope is to ultimately use compact and field deployable broadband rotational seismometers instead of ADRs. Rotational motions were predicted from well-constrained deviatoric MT solutions of nine earthquakes recorded by the arrays and other seismic networks. Additionally, we formed three station sparse datasets with (3-C and 6-C) and without rotational ground motions (3-C only) for eight earthquakes testing the benefits of including rotational waveforms in MT inversions when they are equally weighted with translational waveforms. Adding rotational motions improved the double-couple components and reduced compensated-linear-vector-dipole and isotropic components in full-MT solutions using sparse datasets with poor coverage.

58 GEOSCIENCES↗

DOE OSTI2025

Beyond-classical computation in quantum simulation

Quantum computers hold the promise of solving certain problems that lie beyond the reach of conventional computers. However, establishing this capability, especially for impactful and meaningful problems, remains a central challenge. Here, we show that superconducting quantum annealing processors can rapidly generate samples in close agreement with solutions of the Schrödinger equation. We demonstrate area-law scaling of entanglement in the model quench dynamics of two-, three-, and infinite-dimensional spin glasses, supporting the observed stretched-exponential scaling of effort for matrix-product-state approaches. We show that several leading approximate methods based on tensor networks and neural networks cannot achieve the same accuracy as the quantum annealer within a reasonable time frame. Thus, quantum annealers can answer questions of practical importance that may remain out of reach for classical computation.

King, Andrew D. [D-Wave Quantum Inc., Burnaby, BC ↗

NASA NTRS2014

The Primordial Inflation Polarization Explorer (PIPER)

The Primordial Inflation Polarization ExploreR (Piper) is a balloon-borne cosmic microwave background (CMB) polarimeter designed to search for evidence of inflation by measuring the large-angular scale CMB polarization signal. Bicep2 recently reported a detection of B-mode power corresponding to the tensor-to-scalar ratio r = 0.2 on approximately 2 degree scales. If the Bicep2 signal is caused by inflationary gravitational waves (IGWs), then there should be a corresponding increase in B-mode power on angular scales larger than 18 degrees. Piper is currently the only suborbital instrument capable of fully testing and extending the Bicep2 results by measuring the B-mode power spectrum on angular scales theta θ = approximately 0.6 deg to 90 deg, covering both the reionization bump and recombination peak, with sensitivity to measure the tensor-to-scalar ratio down to r = 0.007, and four frequency bands to distinguish foregrounds. Piper will accomplish this by mapping 85% of the sky in four frequency bands (200, 270, 350, 600 GHz) over a series of 8 conventional balloon flights from the northern and southern hemispheres. The instrument has background-limited sensitivity provided by fully cryogenic (1.5 K) optics focusing the sky signal onto four 32×40-pixel arrays of time-domain multiplexed Transition-Edge Sensor (TES) bolometers held at 140 milli-Kelvin. Polarization sensitivity and systematic control are provided by front-end Variabledelay Polarization Modulators (VPMs), which rapidly modulate only the polarized sky signal at 3 Hz and allow Piper to instantaneously measure the full Stokes vector (I,Q,U,0V) for each pointing. We describe the Piper instrument and progress towards its first flight.

polarimeter↗