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DOE OSTI2020

Elimination of MHD current sheets by modifications to the plasma wall in a fixed boundary model

Models of magnetohydrodynamic (MHD) equilibia that for computational convenience assume the existence of a system of nested magnetic flux surfaces tend to exhibit singular current sheets. These sheets are located on resonant flux surfaces that are associated with rational values of the rotational transform. We study the possibility of eliminating these singularities by suitable modifications of the plasma boundary, which we prescribe in a fixed boundary setting. We find that relatively straightforward iterative procedures can be used to eliminate weak current sheets that are generated at resonant flux surfaces by the nonlinear interactions of resonating wall harmonics. These types of procedures may prove useful in the design of fusion devices with configurations that enjoy improved stability and transport properties.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

On the relationship between the multi-region relaxed variational principle and resistive inner-layer theory

We show that the variational energy principle of the multi-region relaxed magnetohydrodynamic (MRxMHD) model can be used to predict finite-pressure linear tearing instabilities. In this model, the plasma volume is sliced into sub-volumes separated by 'ideal interfaces', and in each volume the magnetic field relaxes to a Taylor state, where the pressure gradient $\nabla p = 0$. The MRxMHD model is implemented in the Stepped-Pressure Equilibrium Code (SPEC) so that the equilibrium solution in each region is computed while preserving the force balance across the interfaces. As SPEC computes the Hessian matrix (a discretized stability matrix), the stability of an MRxMHD equilibrium can also be computed with SPEC. In this article, using SPEC, we investigate the effect of local pressure gradients and the $\nabla p = 0$ in the vicinity of the resonant surface of a tearing mode. For low-beta plasma, we have been able to illustrate a relationship between the resistive singular-layer theory (Coppi et al 1966 Nucl. Fusion6 101; Glasser et al 1975 Phys. Fluids18 875–88) and the MRxMHD model. Within the singular layer, the volume-averaged magnetic helicity and the flux-averaged toroidal flux are shown to be the invariants for the linear tearing modes in SPEC simulations. Our technique to compute MRxMHD stability is first tested numerically in a cylindrical tokamak and its application in toroidal geometry is demonstrated. We demonstrate an agreement between the stability boundary obtained with SPEC simulation and the resistive inner-layer theories.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2013

Ten scenarios from early radiation to late time acceleration with a minimally coupled dark energy

We consider General Relativity with matter, radiation and a minimally coupled dark energy defined by an equation of state w. Using dynamical system method, we find the equilibrium points of such a theory assuming an expanding Universe and a positive dark energy density. Two of these points correspond to classical radiation and matter dominated epochs for the Universe. For the other points, dark energy mimics matter, radiation or accelerates Universe expansion. We then look for possible sequences of epochs describing a Universe starting with some radiation dominated epoch(s) (mimicked or not by dark energy), then matter dominated epoch(s) (mimicked or not by dark energy) and ending with an accelerated expansion. We find ten sequences able to follow this Universe history without singular behaviour of w at some saddle points. Most of them are new in dark energy literature. To get more than these ten sequences, w has to be singular at some specific saddle equilibrium points. This is an unusual mathematical property of the equation of state in dark energy literature, whose physical consequences tend to be discarded by observations. This thus distinguishes the ten above sequences from an infinity of ways to describe Universe expansion.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2013

Two field matter bounce cosmology

We re-examine the non-singular Matter Bounce scenario first developed in [20], which starts with a matter-dominated period of contraction and transitions into an Ekpyrotic phase of contraction. We consider both matter fields, the first of which plays the role of regular matter, and the second of which is responsible for the non-singular bounce. Since the dominant matter field is massive, the induced curvature fluctuations are initially not scale-invariant, whereas the fluctuations of the second scalar field (which are initially entropy fluctuations) are scale-invariant. We study the transfer of the initial entropy perturbations into curvature fluctuations in the matter-dominated phase of contraction and show that the latter become nearly scale invariant on large scales but are blue tilted on small scales. We study the evolution of both curvature and entropy fluctuations through the bounce, and show that both have a scale-invariant spectrum which is blue-tilted on small scales. However, we find that the entropy fluctuations have an amplitude that is much smaller than that of the curvature perturbations, due to gravitational amplification of curvature perturbations during the bounce phase.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Neural network-based classification and regression of magnetohydrodynamic modes in tokamaks

We present a machine learning-based magnetohydrodynamic (MHD) classifier and regressor that utilizes real or complex-valued 3D magnetic sensor array data to determine neoclassical tearing mode (NTM) onset times in tokamaks with millisecond accuracy. The input dataset consists of poloidal profiles of complex Fourier amplitudes with an n = 1 toroidal mode number from 144 human-labeled ITER Baseline Scenario discharges in the DIII-D tokamak, spanning both tearing-dominated and sawtooth-dominated regimes. Since m, n = 2,1 NTMs frequently emerge alongside sawteeth at the same frequency in this scenario, the focus is on isolating the m = 1 and m = 2 components of the n = 1 MHD mode near the tearing onset. To improve model regularization and prediction stability, singular value decomposition was applied to balance the sawtooth and tearing datasets. The enriched datasets facilitated training neural networks that learn the key distinguishing features of sawtooth and tearing modes in the poloidal profiles of their magnetic amplitude and phase. When the modes occur independently, the networks achieve perfect classification due to the modes’ distinct characteristics and low measurement noise. In the more experimentally relevant case where both modes coexist, the networks maintain exceptional performance across key metrics. Tests on synthetic data with known ground truth demonstrate the superior accuracy of the neural network trained on complex-valued input compared to models using real amplitude, phase, or pseudo-complex data, achieving both a mean time delay and standard deviation below 1 ms. Notably, standard linear regression methods fitting the dominant singular modes to the data closely match the neural network’s performance. Applying these methods across a broad range of H-mode scenarios will enable future studies to systematically identify dominant NTM triggers as scenario-specific variables, paving the way for more effective tearing mode avoidance strategies in future fusion reactor designs.

machine learning↗

DOE OSTI2014

The positive orthogonal Grassmannian and loop amplitudes of ABJM

Herein we study the combinatorics associated with the positive orthogonal Grassmannian OG k and its connection to ABJM scattering amplitudes. We present a canonical embedding of OG k into the Grassmannian Gr( k , 2 k ), from which we deduce the canonical volume form that is invariant under equivalence moves. Remarkably the canonical forms of all reducible graphs can be converted into reduced ones with products of d log forms. Unlike super Yang–Mills, here the Jacobian plays a crucial role to ensure the d log form of the reduced representation. Furthermore in the positive region, we identify the functional map that arises from the triangle equivalence move as a 3-string scattering S -matrix which satisfies the tetrahedron equations by Zamolodchikov, implying (2 + 1)-dimensional integrability. We discuss the solution to the BCFW recursion relation for loop amplitudes, and demonstrate the presence of all physical singularities as well as the absence of all spurious ones. The on-shell diagram solution to the loop recursion relation exhibits manifest two-site cyclic symmetry and reveals that, to all loop, four and six-point amplitudes only have logarithmic singularities.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Exploring the energy landscape of RBMs: reciprocal space insights into bosons, hierarchical learning and symmetry breaking

Deep generative models have become ubiquitous due to their ability to learn and sample from complex distributions. Despite the proliferation of various frameworks, the relationships among these models remain largely unexplored, a gap that hinders the development of a unified theory of AI learning. In this work, we address two central challenges: clarifying the connections between different deep generative models and deepening our understanding of their learning mechanisms. We focus on Restricted Boltzmann Machines (RBMs), a class of generative models known for their universal approximation capabilities for discrete distributions. By introducing a reciprocal space formulation for RBMs, we reveal a connection between these models, diffusion processes, and systems of coupled bosons. Our analysis shows that at initialization, the RBM operates at a saddle point, where the local curvature is determined by the singular values of the weight matrix, whose distribution follows the Marc̆enko-Pastur law and exhibits rotational symmetry. During training, this rotational symmetry is broken due to hierarchical learning, where different degrees of freedom progressively capture features at multiple levels of abstraction. This leads to a symmetry breaking in the energy landscape, reminiscent of Landau’s theory. This symmetry breaking in the energy landscape is characterized by the singular values and the weight matrix eigenvector matrix. We derive the corresponding free energy in a mean-field approximation. We show that in the limit of infinite size RBM, the reciprocal variables are Gaussian distributed. Our findings indicate that in this regime, there will be some modes for which the diffusion process will not converge to the Boltzmann distribution. To illustrate our results, we trained replicas of RBMs with different hidden layer sizes using the MNIST dataset. Our findings not only bridge the gap between disparate generative frameworks but also shed light on the fundamental processes underpinning learning in deep generative models.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory

Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of Lyness [Math. Comp. 30 (1976), pp. 1–23] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see X. Xing, X. Li, and L. Lin [J. Chem. Theory Comput. 17 (2021), pp. 4733–4745]). In conclusion, our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Kinematics and dynamics of disclination lines in three-dimensional nematics

An exact kinematic law for the motion of disclination lines in nematic liquid crystals as a function of the tensor order parameter Q is derived. Unlike other order parameter fields that become singular at their respective defect cores, the tensor order parameter remains regular. Following earlier experimental and theoretical work, the disclination core is defined to be the line where the uniaxial and biaxial order parameters are equal, or equivalently, where the two largest eigenvalues of Q cross. This allows an exact expression relating the velocity of the line to spatial and temporal derivatives of Q on the line, to be specified by a dynamical model for the evolution of the nematic. By introducing a linear core approximation for Q, analytical results are given for several prototypical configurations, including line interactions and motion, loop annihilation, and the response to external fields and shear flows. Behaviour that follows from topological constraints or defect geometry is highlighted. Finally, the analytic results are shown to be in agreement with three-dimensional numerical calculations based on a singular Maier–Saupe free energy that allows for anisotropic elasticity.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Two-component atomic Fermi superfluid with spin-orbital coupling in thin-spherical-shell geometry

We present a theory of two-component atomic Fermi superfluid with tunable pairing interaction in a thin spherical shell subject to spin-orbit coupling (SOC). By incorporating SOC into the Fermi superfluid in the BCS-Bose Einstein condensation (BEC) crossover, we obtain the energy spectrum and equations of state. While the order parameter and chemical potential are suppressed by SOC on the BCS side, the former remains positive but the latter may be pushed to negative values by SOC.Meanwhile, the compressibility exhibits kinks as the pairing interaction or SOC varies, indicating singular behavior of higher derivatives of the free energy despite the presence of the superfluid order parameter. The minimum of the energy dispersion indicates a decoupling of the energy gap from the order parameter, and the compressibility kinks occur when the energy gap approaches 0. We found the particle-hole mixing in the Fermi superfluid essential for the compressibility kinks since a Fermi gas with the same SOC but no pairing interaction only exhibits complicated dispersions but no singular behavior. Therefore, our results show that a combination of superfluid, SOC, and geometry can lead to interesting phenomena. Here, we also discuss possible experimental realizations and implications.

Fermi gases↗

DOE OSTI2019

Taming convergence in the determinant approach for x-ray excitation spectra

A determinant formalism in combination with ab initio calculations was proposed recently and has paved a way for simulating and interpreting x-ray excitation spectra in condensed-phase systems. The method systematically takes into account many-electron effects in the Mahan-Noziéres-De Dominicis (MND) theory, including core-level excitonic effects, the Fermi-edge singularity, shakeup excitations, and wave function overlap effects such as the orthogonality catastrophe, all within a universal framework using many-electron configurations. A heuristic search algorithm was introduced to search for the configurations that are important for defining the x-ray spectral line shape, instead of enumerating them in a brute-force way. The algorithm has proven to be efficient for calculating O K edges of transition metal oxides, which converge at the second excitation order (denoted as $f^{(n)}$ with n = 2), i.e., the final-state configurations with two e-h pairs (with one hole being the core hole). However, it remains unknown how the determinant x-ray spectra converge for general cases and at which excitation order n should one stop the determinant calculation. Even with the heuristic algorithm, the number of many-electron configurations still grows exponentially with the excitation order n. In this work we prove two theorems that can indicate the order of magnitude of the contribution of the $f^{(n)}$ configurations, so that one can estimate their contribution very quickly without actually calculating their amplitudes. The two theorems are based on singular-value decomposition (SVD) analysis, a method that is widely used to quantify entanglement between two quantum many-body systems. Finally, we examine the K edges of several metallic systems with the determinant formalism up to $f^{(5)}$ to illustrate the usefulness of the theorems.