Ansatz Functions Research Articles

Dynamics of the Entanglement Zero Modes in the Haldane Model under a Quantum Quench

We investigate evolution of entanglement spectra of the Haldane model for Chern insulators upon a sudden quench within the same topological phase. In particular, we focus on the location of the entanglement spectrum crossing, which signifies the bulk topology. It is shown that the coplanarity condition for the pseudomagnetic field of the model, which can be used to determine the crossing in the equilibrium case, needs to be relaxed. We analytically derive the non-equilibrium condition with the help of an edge-state wave function ansatz and a dynamically induced length-scale cutoff. With spectral analyses, it is realized that the oscillatory behavior of the crossing is dominated by the interband excitations at the van Hove singularities.

Negative Gilbert damping

Modern ferromagnetism theory is based on the Landau-Lifshitz-Gilbert equation that governs the dynamics of magnetic moment in a ferromagnet. As a deep-rooted notion, the Gilbert damping parameter must be positive because it describes the (energy and angular momentum) dissipation. In this paper, we report a negative Gilbert damping via the magneto-optical interaction of three orthogonal circularly-polarized laser beams with a submicron magnet placed in an optical cavity. We show that the off-resonant coupling between the driving laser and cavity photon in the far-blue detuning can induce a magnetic torque exactly of the Gilbert type with negative damping coefficient. A hyperbolic-tangent function ansatz is found to well describe the time-resolved spin switching as the intrinsic dissipation is overcome. Feasible experiments and materials are discussed to test our theory.

Variational truncated Wigner approximation for weakly interacting Bose fields: Dynamics of coupled condensates

The truncated Wigner approximation is an established approach that describes the dynamics of weakly interacting Bose gases beyond the mean-field level. Although it allows a quantum field to be expressed by a stochastic c-number field, the simulation of the time evolution is still very demanding for most applications. Here, we develop a numerically inexpensive scheme by approximating the c-number field with a variational ansatz. The dynamics of the ansatz function is described by a tractable set of coupled ordinary stochastic differential equations for the respective variational parameters. We investigate the non-equilibrium dynamics of a three-dimensional Bose gas in a one-dimensional optical lattice with a transverse isotropic harmonic confinement. The accuracy and computational inexpensiveness of our method are demonstrated by comparing its predictions to experimental data.

Analytic calculation of the Berry curvature and diagonal Born-Oppenheimer correction for molecular systems in uniform magnetic fields.

The diagonal nonadiabatic term arising from the Born-Oppenheimer wave function ansatz contains contributions from a vector and scalar potential. The former is provably zero when the wave function can be taken to be real valued, and the latter, known as the diagonal Born-Oppenheimer correction (DBOC), is typically small in magnitude. Therefore, unless high accuracy is sought, the diagonal nonadiabatic term is usually neglected when calculating molecular properties. In the presence of a magnetic field, the wave function is generally complex, and the geometric vector potential gives rise to a screening force that is qualitatively important for molecular dynamics. This screening force is written in terms of the Berry curvature and is added to the bare Lorentz force acting on the nuclei in the presence of the field. In this work, we derive analytic expressions for the Berry curvature and DBOC using both first- and second-quantization formalisms for the case of generalized and restricted Hartree-Fock theories in a uniform magnetic field. The Berry curvature and DBOC are calculated as a function of the magnetic field strength and the bond distance for the ground-state singlets of H2, LiH, BH, and CH+. We also examine the stability and time-reversal symmetry of the underlying self-consistent field solutions. The character of the DBOC and Berry curvature is found to depend on the magnetic field and varies between molecules. We also identify instances of broken time-reversal symmetry for the dissociation curves of BH and CH+.

Quantum algorithms for quantum dynamics: A performance study on the spin-boson model

Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter approximation of the time-evolution operator. This approach typically relies on deep circuits and is therefore hampered by the substantial limitations of available noisy and near-term quantum hardware. On the other hand, variational quantum algorithms (VQAs) have become an indispensable alternative, enabling small-scale simulations on present-day hardware. However, despite the recent development of VQAs for quantum dynamics, a detailed assessment of their efficiency and scalability is yet to be presented. To fill this gap, we applied a VQA based on McLachlan's principle to simulate the dynamics of a spin-boson model subject to varying levels of realistic hardware noise as well as in different physical regimes, and discuss the algorithm's accuracy and scaling behavior as a function of system size. We observe a good performance of the variational approach used in combination with a general, physically motivated wave function ansatz, and compare it to the conventional first-order Trotter evolution. Finally, based on this, we make scaling predictions for the simulation of a classically intractable system. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage for the solution of time-dependent problems.

On two simple virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials

The virtual element method allows to revisit the construction of Kirchhoff-Love elements because the C^1-continuity condition is much easier to handle in the VEM framework than in the traditional Finite Elements methodology. Here we study the two most simple VEM elements suitable for Kirchhoff-Love plates as stated in Brezzi and Marini (Comput Methods Appl Mech Eng 253:455–462, 2013). The formulation contains new ideas and different approaches for the stabilisation needed in a virtual element, including classic and energy stabilisations. An efficient stabilisation is crucial in the case of C^1-continuous elements because the rank deficiency of the stiffness matrix associated to the projected part of the ansatz function is larger than for C^0-continuous elements. This paper aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilisation. Different examples and convergence studies discuss and demonstrate the accuracy of the resulting VEM elements. Finally, reduction of virtual plate elements to triangular and quadrilateral elements with 3 and 4 nodes, respectively, yields finite element like plate elements. It will be shown that these C^1-continuous elements can be easily incorporated in legacy codes and demonstrate an efficiency and accuracy that is much higher than provided by traditional finite elements for thin plates.

Conditional Wave Function Theory: A Unified Treatment of Molecular Structure and Nonadiabatic Dynamics.

We demonstrate that a conditional wave function theory enables a unified and efficient treatment of the equilibrium structure and nonadiabatic dynamics of correlated electron–ion systems. The conditional decomposition of the many-body wave function formally recasts the full interacting wave function of a closed system as a set of lower-dimensional (conditional) coupled “slices”. We formulate a variational wave function ansatz based on a set of conditional wave function slices and demonstrate its accuracy by determining the structural and time-dependent response properties of the hydrogen molecule. We then extend this approach to include time-dependent conditional wave functions and address paradigmatic nonequilibrium processes including strong-field molecular ionization, laser-driven proton transfer, and nuclear quantum effects induced by a conical intersection. This work paves the road for the application of conditional wave function theory in equilibrium and out-of-equilibrium ab initio molecular simulations of finite and extended systems.

Separation of scales: dynamical approximations for composite quantum systems* *Contribution to the special collection ‘Koopman methods in classical and quantum-classical mechanics’

We consider composite quantum-dynamical systems that can be partitioned into weakly interacting subsystems, similar to system–bath type situations. Using a factorized wave function ansatz, we mathematically characterize dynamical scale separation. Specifically, we investigate a coupling régime that is partially flat, i.e. slowly varying with respect to one set of variables, for example, those of the bath. Further, we study the situation where one of the sets of variables is semiclassically scaled and derive a quantum–classical formulation. In both situations, we propose two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field). We analyze the error for the wave function and for the action of observables, obtaining comparable estimates for both approaches. The present study is the first step towards a general analysis of scale separation in the context of tensorized wavefunction representations.

Penerapan Program Software Matlab Dalam Memecahkan Permasalahan Rangkaian Listrik: Dinamika Sistem Kapasitor Dan Induktor (Prinsip Nilai Dan Vektor Eigen)

Matlab software was applied in solving physics cases in electronics using value calculation programs and Eigenvectors. The physics case solved is an moving oscillator circuit consisting of five capacitors and four inductors. The purpose of this program is to determine the value and the Eigenvectors that can be obtained. The method used is to utilize the problem of values and Eigenvectors in the Matlab program. In the case of oscillator motion, each capacitor's size and inductor system have been determined according to the system.. The equation of motion for the capacitor and inductor system is reviewed at each point of the inductor and the function used is  called the ansatz function, then differentiated twice concerning time. By using the Matlab program application, the results obtained show that by using eig (Eigen) command the Matlab program can accurately show the results of running (execution) values and Eigenvector.

Penerapan Program Software Matlab Dalam Memecahkan Permasalahan Rangkaian Listrik: Dinamika Sistem Kapasitor Dan Induktor (Prinsip Nilai Dan Vektor Eigen)

Matlab software was applied in solving physics cases in electronics using value calculation programs and Eigenvectors. The physics case solved is an moving oscillator circuit consisting of five capacitors and four inductors. The purpose of this program is to determine the value and the Eigenvectors that can be obtained. The method used is to utilize the problem of values and Eigenvectors in the Matlab program. In the case of oscillator motion, each capacitor's size and inductor system have been determined according to the system.. The equation of motion for the capacitor and inductor system is reviewed at each point of the inductor and the function used is called the ansatz function, then differentiated twice concerning time. By using the Matlab program application, the results obtained show that by using eig (Eigen) command the Matlab program can accurately show the results of running (execution) values and Eigenvector.

Electron holes in a κ distribution background with singularities

The pseudo-potential method is applied to derive diverse propagating electron–hole structures in a nonthermal or κ particle distribution function background. The associated distribution function Ansatz reproduces the Schamel distribution of [H. Schamel, Phys. Plasmas 22, 042301 (2015)] in the Maxwellian (κ→∞) limit, providing a significant generalization of it for plasmas where superthermal electrons are ubiquitous, such as space plasmas. The pseudo-potential and the nonlinear dispersion relation are evaluated. The role of the spectral index κ on the nonlinear dispersion relation is investigated, in what concerns the wave amplitude, for instance. The energy-like first integral from Poisson's equation is applied to analyze the properties of diverse classes of solutions: with the absence of trapped electrons, with a non-analytic distribution of trapped electrons, or with a surplus of trapped electrons. Special attention is, therefore, paid to the non-orthodox case where the electrons distribution function exhibits strong singularities, being discontinuous or non-analytic.

Initial Maximum Overlap Method for Large Systems by the Quantum Mechanics/Extremely Localized Molecular Orbital Embedding Technique.

Quantum chemistry offers a large variety of methods to treat excited states. Many of them are based on a multireference wave function ansatz and are therefore characterized by an intrinsic complexity and high computational costs. To overcome these drawbacks and also some limitations of simpler single-reference approaches (e.g., configuration interaction singles and time-dependent density functional theory), the single-determinant Δself-consistent field-initial maximum overlap method (ΔSCF-IMOM) has been proposed. This strategy substitutes the aufbau principle with a criterion that occupies molecular orbitals at successive SCF iterations on the basis of their maximum overlap with a proper set of guess orbitals for the target excited state. In this way, it prevents the SCF process to collapse to the ground state wave function and provides excited state single Slater determinant solutions to the SCF equations. Here, we propose to extend the applicability of the IMOM to the treatment of localized excited states of large systems. To accomplish this task, we coupled it with the QM/ELMO (quantum mechanics/extremely localized molecular orbitals) strategy, a quantum mechanical embedding method in which the most chemically relevant part of the system is treated with traditional quantum chemical approaches, while the rest is described by extremely localized molecular orbitals transferred from recently constructed libraries or proper model molecules. After presenting the theoretical foundations of the new IMOM/ELMO technique, in this paper, we will show and discuss the results of preliminary test calculations carried out on both model systems (i.e., decanoic acid, decene, decapentaene, and solvated acrolein) and a system of biological interest (flavin mononucleotide in the flavodoxin protein). We observed that, for localized excited states, the new IMOM/ELMO method provides reliable results, and it reproduces the outcomes of fully IMOM calculations within the chemical accuracy threshold (i.e., 0.043 eV) by including only a limited number of atoms in the QM region. Furthermore, the first application of our embedding technique to a larger biological system gave completely plausible results in line with those obtained through more traditional quantum mechanical methods, thus opening the possibility of using the new approach in future investigations of photobiology problems.

Helmholtz Scaling: An Alternative Approach to Calculate Viscosity with the PCP-SAFT Equation of State

We propose an alternative approach to calculate viscosity, based on the Rosenfeld scaling theory, which can be applied to pure components and their mixtures in a wide range of temperatures, pressures, and compositions. The Helmholtz scaling (A-scaling) applies the Chapman–Enskog relation, an Ansatz function, and the PCP-SAFT equation of state to predict the viscosity. Although it was implemented with the PCP-SAFT EoS, A-scaling has potential to be coupled with other EoS. Substances from n-alkanes, alcohols, aromatics, ethers, chlorofluorocarbons, carbon dioxide, and nitrogen were used to obtain five parameters to better correlate the experimental viscosity data. We tested the A-scaling performance of predicting the viscosity of 33 binary mixtures at different temperature and pressure conditions. For pure components, the minimum and maximum average absolute deviations (AADs) were 1.51 and 6.53%, respectively. Meanwhile, for mixtures, 0.70 and 23.95% were the lowest and highest values. By comparing this method with the successful and consolidated entropy scaling, it was found that AADs in most of the pure substances and mixtures were lower for A-scaling. Thus, since it is suitable for polar, nonpolar, and hydrogen-bonding substances, A-scaling covers many substances for industry applications, with no usage of adjustable parameters to predict viscosity for mixtures.

Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations\u2014Part 1: basics and ansatz function choices

In the two parts of the present note we discuss several questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. In the present Part 1, we provide a robust selection of basis functions and collocation points to design the discrete problem. We substantiate a procedure for its numerical solution later in Part 2. Additionally, in Part 1, a number of new error estimates are proven that support some of the design decisions.

Gravitini zero modes riding on top of multiple black holes

We study gravitini zero modes in four dimensional $\mathcal{N}=2$ pure supergravity theory. The gravitini zero modes we study are solutions to the Rarita-Schwinger equations of motion in the background of the purely bosonic Majumdar-Papapetrou background. We start with a very generic ansatz for the gravitini that involves 32 ansatz functions and reduce the Rarita-Schwinger equations to a set of linear coupled partial differential equations on ${\mathbf{R}}^{3}$: the kind familiar from electromagnetism with divergences, curls, etc. We first show how the gravitini zero modes due to broken supersymmetries arise in this setup and how they solve the equations on ${\mathbf{R}}^{3}$. Then we go on and obtain other solutions to these equations: the ``extra'' fermion zero modes, parametrized by four integration constants (one of the integration constants corresponds to the gravitini zero modes due to broken supersymmetries). A careful analysis of pure gauge solutions reveals that we have four independent extra zero modes.

Improved Accuracy on Noisy Devices by Nonunitary Variational Quantum Eigensolver for Chemistry Applications.

We propose a modification of the Variational Quantum Eigensolver algorithm for electronic structure optimization using quantum computers, named nonunitary Variational Quantum Eigensolver (nu-VQE), in which a nonunitary operator is combined with the original system Hamiltonian leading to a new variational problem with a simplified wave function ansatz. In the present work, as nonunitary operator, we use the Jastrow factor, inspired from classical Quantum Monte Carlo techniques for simulation of strongly correlated electrons. The method is applied to prototypical molecular Hamiltonians for which we obtain accurate ground-state energies with shallower circuits, at the cost of an increased number of measurements. Finally, we also show that this method achieves an important error mitigation effect that drastically improves the quality of the results for VQE optimizations on today's noisy quantum computers. The absolute error in the calculated energy within our scheme is 1 order of magnitude smaller than the corresponding result using traditional VQE methods, with the same circuit depth.

Novel analytical cnoidal and solitary wave solutions of the Extended Kawahara equation

In this work, some novel analytic traveling wave solutions including the cnoidal and solitary wave solutions of the planar Extended Kawahara equation are deduced. Four different analytical methods (the Jacobian elliptic function, Weierrtrass elliptic function, the traditional tanh method and the sech-square) are devoted for solving this equation. By means of the Jacobian elliptic function ansatz, the cnoidal and soliary wave solutions are obtained. Also, new cnoidal wave solutions are derived via a new hypothesis in the form of the Weierrtrass elliptic function. Moreover, the standard tanh method is utilized to get a new set of solitary wave solutions for the evolution equation. Over and above, the hyperbolic ansatz method (a new ansatz in the form of squre-sech) is employed to get a new set of solitary wave solutions for the evolution equation. Furthermore, all obtained solutions of the planar Extended Kawahara equation cover the traveling wave solutions of the planar modified Kawahara equation. These solutions maybe useful to many researchers interested in studying the propagation of nonlinear waves in nonlinear dispersion mediums like plasma physics, optical fibers, fluid mechanics, and many different branches of science.

3D mixed virtual element formulation for dynamic elasto-plastic analysis

The virtual element method (VEM) for dynamic analyses of nonlinear elasto-plastic problems undergoing large deformations is outlined within this work. VEM has been applied to various problems in engineering, considering elasto-plasticity, multiphysics, damage, elastodynamics, contact- and fracture mechanics. This work focuses on the extension of VEM formulations towards dynamic elasto-plastic applications. Hereby low-order ansatz functions are employed in three dimensions with elements having arbitrary convex or concave polygonal shapes. The formulations presented in this study are based on minimization of potential function for both the static as well as the dynamic behavior. Additionally, to overcome the volumetric locking phenomena due to elastic and plastic incompressibility conditions, a mixed formulation based on a Hu-Washizu functional is adopted. For the implicit time integration scheme, Newmark method is used. To show the model performance, various numerical examples in 3D are presented.

Investigation of interactional phenomena and multi wave solutions of the quantum hydrodynamic Zakharov–Kuznetsov model

Abstract The symbolic computation with the ansatz function and the logarithmic transformation method are used to obtain a formula for certain exact solutions of the ( 3 + 1 ) \left(3+1) Zakharov–Kuznetsov (Z–K) equation. We use homoclinic breather, three waves method, and double exponential. There is a conflict of results with considerably known results, which indicates the solutions found in this study are new. By selecting appropriate parameter values, 3d representations are plotted to establish W-shaped, multi-peak, and kinky breathers solutions.

NURBS-based geometries: A mapping approach for virtual serendipity elements

A NURBS-based serendipity virtual element method for general (arbitrary) element shapes is outlined in this work. The low-order VEM ansatz function is now extended towards higher-order formulation. In comparison with the already existing serendipity VEM, a general mapping scheme is developed within this contribution allowing to deviate from the assumption of straight edges of virtual elements. A number of numerical examples illustrates the robustness and accuracy of the new mapping methodology. The results are very promising and underline the advantages of the formulations for dealing with arbitrary geometries.