Exact Decoupling Research Articles

EXACT DECOUPLING OF A COUPLED SYSTEM OF TWO STATIONARY SCHROEDINGER EQUATIONS

EXACT DECOUPLING OF A COUPLED SYSTEM OF TWO STATIONARY SCHROEDINGER EQUATIONS

Role of interaction-induced tunneling in the dynamics of polar lattice bosons

Intersite dipolar interactions induce, even in absence of disorder, an intriguing nonergodic dynamics for dipolar bosons in an optical lattice. We show that the inherent dipole-induced density-dependent tunneling, typically neglected, plays a crucial role in this dynamics. For shallow-enough lattices, the delocalization stemming from the interaction-induced hopping overcomes the localization induced by intersite interactions. As a result, in stark contrast to the more studied case of hard-core bosons, delocalization is counterintuitively strengthened when the dipolar strength increases. Furthermore, the quasicancellation between bare and interaction-induced tunneling may lead, near a lattice-depth-dependent value of the dipole strength, to an exact decoupling of the Hilbert space between ergodic hard-core states and strongly nonergodic soft-core ones. Our results show that interaction-induced hopping should play a crucial role in future experiments on the dynamics of polar lattice gases.

Quantum dynamics for general time-dependent three coupled oscillators based on an exact decoupling

Quantum dynamics of general time-dependent three coupled oscillators is investigated through an alternative approach based on decoupling of them using the unitary transformation method. From a first unitary transformation, the quantal Hamiltonian of the complicated original system is transformed to an equal but a simple one associated with the three coupled oscillators of which masses are unity. To diagonalize the transformed Hamiltonian, we transform the Hamiltonian once again by introducing a new unitary operator. This transformation corresponds to a three-dimensional rotation parameterized by Euler angles. Through these procedures, the coupled oscillatory subsystems are completely decoupled. The importance of this decouplement is that it enables us to develop exact theory for mechanical treatment of the originally-coupled systems without any restriction in the form of time-varying parameters.

The derivative expansion in asymptotically safe quantum gravity: general setup and quartic order

We present a general framework to systematically study the derivative expansion of asymptotically safe quantum gravity. It is based on an exact decoupling and cancellation of different modes in the Landau limit, and implements a correct mode count as well as a regularisation based on geometrical considerations. It is applicable independent of the truncation order. To illustrate the power of the framework, we discuss the quartic order of the derivative expansion and its fixed point structure as well as physical implications.

Multiport PDN Optimization With the Newton–Hessian Minimization Method

This article proposes an optimization algorithm using the Hessian minimization method, based on the Newton iteration, to evaluate the effectiveness of the placement of multiple decoupling capacitors on a power/ground plane pair. The exact effective decoupling regions are obtained using the Newton iteration method for each decoupling capacitor. The impedance of the IC port is lower than the target impedance no matter where the decoupling capacitor is placed in this region. To optimize specific capacitor placements in this region, the Newton iteration, based on the Hessian matrix, is used to determine the location where the impedance of the IC port is minimized at the antiresonant frequency of the plane pair. This placement optimization algorithm allows for a decoupling design method that can also be applied to a PDN with multiple decoupling capacitors for multiple IC ports. Compared with the method of random selection from within the effective decoupling area, the method proposed here requires fewer decoupling capacitors and less computational time.

General solution of the time evolution of two interacting harmonic oscillators

We study the time evolution of an ideal system composed of two harmonic oscillators coupled through a quadratic Hamiltonian with arbitrary interaction strength. We solve its dynamics analytically by employing tools from symplectic geometry. In particular, we use this result to completely characterize the dynamics of the two oscillators interacting in the ultrastrong coupling regime with additional single-mode squeezing on both oscillators, as well as higher order terms. Furthermore, we compute quantities of interest, such as the average number of excitations and the correlations that are established between the two subsystems due to the evolution. We find that this model predicts a second order phase transition and we compute the critical exponents and the critical value. We also provide an exact decoupling of the time evolution in terms of simple quantum optical operations, which can be used for practical implementations and studies. Finally, we show how our techniques can be extended to include more oscillators and higher order interactions.

Feedback Linearization Based Direct Torque Control for IPMSMs

A feedback linearization based direct torque control (FL-DTC) scheme is designed for interior permanent magnet synchronous motors drives. Several main aspects related to FL are studied thoroughly. A proper choice of two new state variables (the torque and the squared amplitude of stator flux) yields the exact input-output linearization decoupling of the closed-loop system. The FL-DTC frame is obtained after finding a feasible diffeomorphism. For the partially linearized subsystem, i.e., the independent torque and flux loops, two FL-DTC-based controller design solutions are provided. The control problem of internal dynamics is converted to an absolute tracking problem of an equivalent Lur'e system.

Fractional Order Sliding Mode Speed Control of Feedback Linearized Induction Motor

Abstract Induction machines have of late become the most popular workhorses in the industry replacing DC machines because of their advantages, such as reduced cost, reliability and the absence of commutators, which make them adapt to unfavourable conditions with lower maintenance requirement. However, higher-order models of AC machines, nonlinearities in model equations, uncertainties in parameters and load disturbances make induction motors difficult to control than that of DC motors. In this paper, a robust control strategy considering the recent advances in technology is proposed employing input-output feedback linearization for the exact decoupling of electromagnetic torque and rotor flux using fractional-order sliding mode controller (SMC) for the outer speed loop. The sliding mode observer is also designed to extract the rotor flux from the DC input voltage of the inverter and stator current measurements. Finally, a comparison between the commonly used proportional integral and derivative (PID) controller and the proposed fractional-order SMCs is made and the conclusion is reached.

Restoring the iso-orbital limit of the kinetic energy density in relativistic density functional theory.

In contrast to nonrelativistic density functional theory, the ratio between the von Weizsäcker and the Kohn-Sham kinetic energy density, commonly used as iso-orbital indicator t within exchange-correlation functionals beyond the generalized-gradient level, violates the exact iso-orbital limit and the appropriate parameter range, 0 ≤ t ≤ 1, in relativistic density functional theory. Based on the exact decoupling procedure within the infinite-order two-component method and the Cauchy-Schwarz inequality, we present corrections to the relativistic and the picture-change-transformed nonrelativistic kinetic energy density that restores these exact constraints. We discuss the origin of the new correction terms and illustrate the effectiveness of the current approach for several representative cases. The proposed generalized iso-orbital indicator tλ is expected to be a useful ingredient for the development of relativistic exchange-correlation functionals.

A Global Stability of Linearizing Control of Induction Motor for PV Water Pumping Application

This article describes how a field oriented control can provide the same performance as it is achieved by a DC motor. However, this technique requires a mechanic sensor and is very sensitive to the variation of motor parameters which results in an undesirable coupling between the flux and the torque. To solve these problems, this paper proposes a global stability and robust nonlinear controller, applied to induction motor (IM), in order to achieve an exact decoupling between speed and flux for all motor operating conditions. The induction motor is coupled with a centrifugal hydraulic pump, powered by a photovoltaic array speeding system. The proposed system is designed for usage in rural areas or remote electricity needs in absence of the grid network. A nonlinear controller adjusts the motor speed reference to attain the maximum power point (MPPT). In presence of rotor and stator resistances and irradiation disturbance the results obtained by simulations confirm the effectiveness of the proposed method.

An atomic mean-field spin-orbit approach within exact two-component theory for a non-perturbative treatment of spin-orbit coupling.

An atomic mean-field (AMF) spin-orbit (SO) approach within exact two-component theory (X2C) is reported, thereby exploiting the exact decoupling scheme of X2C, the one-electron approximation for the scalar-relativistic contributions, the mean-field approximation for the treatment of the two-electron SO contribution, and the local nature of the SO interactions. The Hamiltonian of the proposed SOX2CAMF scheme comprises the one-electron X2C Hamiltonian, the instantaneous two-electron Coulomb interaction, and an AMF SO term derived from spherically averaged Dirac-Coulomb Hartree-Fock calculations of atoms; no molecular relativistic two-electron integrals are required. Benchmark calculations for bond lengths, harmonic frequencies, dipole moments, and electric-field gradients for a set of diatomic molecules containing elements across the periodic table show that the SOX2CAMF scheme offers a balanced treatment for SO and scalar-relativistic effects and appears to be a promising candidate for applications to heavy-element containing systems. SOX2CAMF coupled-cluster calculations of molecular properties for bismuth compounds (BiN, BiP, BiF, BiCl, and BiI) are also presented and compared with experimental results to further demonstrate the accuracy and applicability of the SOX2CAMF scheme.

Efficient implementation of one- and two-component analytical energy gradients in exact two-component theory.

We present an efficient algorithm for one- and two-component analytical energy gradients with respect to nuclear displacements in the exact two-component decoupling approach to the one-electron Dirac equation (X2C). Our approach is a generalization of the spin-free ansatz by Cheng and Gauss [J. Chem. Phys. 135, 084114 (2011)], where the perturbed one-electron Hamiltonian is calculated by solving a first-order response equation. Computational costs are drastically reduced by applying the diagonal local approximation to the unitary decoupling transformation (DLU) [D. Peng and M. Reiher, J. Chem. Phys. 136, 244108 (2012)] to the X2C Hamiltonian. The introduced error is found to be almost negligible as the mean absolute error of the optimized structures amounts to only 0.01 pm. Our implementation in TURBOMOLE is also available within the finite nucleus model based on a Gaussian charge distribution. For a X2C/DLU gradient calculation, computational effort scales cubically with the molecular size, while storage increases quadratically. The efficiency is demonstrated in calculations of large silver clusters and organometallic iridium complexes.

Dipolar dark matter as an effective field theory

Dipolar Dark Matter (DDM) is an alternative model motivated by the challenges faced by the standard cold dark matter model to describe the right phenomenology at galactic scales. A promising realisation of DDM was recently proposed in the context of massive bigravity theory. The model contains dark matter particles, as well as a vector field coupled to the effective composite metric of bigravity. This model is completely safe in the gravitational sector thanks to the underlying properties of massive bigravity. In this work we investigate the exact decoupling limit of the theory, including the contribution of the matter sector, and prove that it is free of ghosts in this limit. We conclude that the theory is acceptable as an Effective Field Theory below the strong coupling scale.

A novel method for calculating relative free energy of similar molecules in two environments

Calculating relative free energies is a topic of substantial interest and has many applications including solvation and binding free energies, which are used in computational drug discovery. However, there remain the challenges of accuracy, simple implementation, robustness and efficiency, which prevent the calculations from being automated and limit their use. Here we present an exact and complete decoupling analysis in which the partition functions of the compared systems decompose into the partition functions of the common and different subsystems. This decoupling analysis is applicable to submolecules with coupled degrees of freedom such as the methyl group and to any potential function (including the typical dihedral potentials), enabling to remove less terms in the transformation which results in a more efficient calculation. Then we show mathematically, in the context of partition function decoupling, that the two compared systems can be simulated separately, eliminating the need to design a composite system. We demonstrate the decoupling analysis and the separate transformations in a relative free energy calculation using MD simulations for a general force field and compare to another calculation and to experimental results. We present a unified soft-core technique that ensures the monotonicity of the numerically integrated function (analytical proof) which is important for the selection of intermediates. We show mathematically that in this soft-core technique the numerically integrated function can be non-steep only when we transform the systems separately, which can simplify the numerical integration. Finally, we show that when the systems have rugged energy landscape they can be equilibrated without introducing another sampling dimension which can also enable to use the simulation results for other free energy calculations.

Relativistic kinetic-balance condition for explicitly correlated basis functions

This paper presents the derivation of a kinetic-balance condition for explicitly correlated basis functions employed in semi-classical relativistic calculations. Such a condition is important to ensure variational stability in algorithms based on the first-quantized Dirac theory of 1/2-fermions. We demonstrate that the kinetic-balance condition can be obtained from the row reduction process commonly applied to solve systems of linear equations. The resulting form of kinetic balance establishes a relation for the components of the spinor of an N-fermion system to the non-relativistic limit, which is in accordance with recent developments in the field of exact decoupling in relativistic orbital-based many-electron theory.

Observers design for switched discrete-time singular time-delay systems with unknown inputs

This paper addresses the state estimation problem for a class of switched discrete-time singular systems simultaneously subject to state delays, unknown inputs (UIs) and arbitrary switching sequences. A mode-dependent observer is constructed, and, in view of the observation performance and the realizability, two class of design methods are proposed. The first class is based on the principle of exact decoupling, which allows the observer to perfectly reconstruct the system state regardless the presence of UIs under stringent rank conditions. The second one, with the aid of introducing L2-attenuation of UIs to the state estimation error, relaxes the rank constraints required in the first class method while permitting the observer to give an unknown input approximately decoupled state estimation. In both methods, by defining an appropriate switched Lyapunov function and using the linear matrix inequality technique, sufficient conditions for the existence of the observer are given, and the convergence rate of the state estimation error is proved to be exponential. Finally, two numerical examples are given to show the effectiveness of the developed theoretical results.

Local relativistic exact decoupling

We present a systematic hierarchy of approximations for local exact decoupling of four-component quantum chemical Hamiltonians based on the Dirac equation. Our ansatz reaches beyond the trivial local approximation that is based on a unitary transformation of only the atomic block-diagonal part of the Hamiltonian. Systematically, off-diagonal Hamiltonian matrix blocks can be subjected to a unitary transformation to yield relativistically corrected matrix elements. The full hierarchy is investigated with respect to the accuracy reached for the electronic energy and for selected molecular properties on a balanced test molecule set that comprises molecules with heavy elements in different bonding situations. Our atomic (local) assembly of the unitary exact-decoupling transformation--called local approximation to the unitary decoupling transformation (DLU)--provides an excellent local approximation for any relativistic exact-decoupling approach. Its order-N(2) scaling can be further reduced to linear scaling by employing a neighboring-atomic-blocks approximation. Therefore, DLU is an efficient relativistic method well suited for relativistic calculations on large molecules. If a large molecule contains many light atoms (typically hydrogen atoms), the computational costs can be further reduced by employing a well-defined nonrelativistic approximation for these light atoms without significant loss of accuracy. We also demonstrate that the standard and straightforward transformation of only the atomic block-diagonal entries in the Hamiltonian--denoted diagonal local approximation to the Hamiltonian (DLH) in this paper--introduces an error that is on the order of the error of second-order Douglas-Kroll-Hess (i.e., DKH2) when compared with exact-decoupling results. Hence, the local DLH approximation would be pointless in an exact-decoupling framework, but can be efficiently employed in combination with the fast to evaluate DKH2 Hamiltonian in order to speed up calculations for which ultimate accuracy is not the major concern.

Exact decoupling of the relativistic Fock operator

It is generally acknowledged that the inclusion of relativistic effects is crucial for the theoretical description of heavy-element-containing molecules. Four-component Dirac-operator-based methods serve as the relativistic reference for molecules and highly accurate results can be obtained—provided that a suitable approximation for the electronic wave function is employed. However, four-component methods applied in a straightforward manner suffer from high computational cost and the presence of pathologic negative-energy solutions. To remove these drawbacks, a relativistic electron-only theory is desirable for which the relativistic Fock operator needs to be exactly decoupled. Recent developments in the field of relativistic two-component methods demonstrated that exact decoupling can be achieved following different strategies. The theoretical formalism of these exact-decoupling approaches is reviewed in this paper followed by a comparison of efficiency and results.

Solved and unsolved problems in relativistic quantum chemistry

A hierarchy of approximations in relativistic many-electron theory is discussed that starts with the Dirac equation and its expansion in a kinetically balanced basis, via a formulation of non-interacting electrons in Fock space (which is the only consistent way to deal with negative-energy states). The most straightforward approximate Hamiltonian for interacting electrons is derived from quantum electrodynamics (QED) in Coulomb gauge with the neglect of transversal photons. This allows an exact (non-perturbative) decoupling of the electromagnetic field from the fermionic field. The electric interaction of the fermions is non-retarded and non-quantized. The quantization of the fermionic field leads to a polarizable vacuum. The simplest (but somewhat problematic) approximation is a no-pair projected theory with external-field projectors. The Dirac-Coulomb operator in configuration space (first quantization) is not acceptable, even if the Brown–Ravenhall disease is much less virulent than often claimed. Effects of transversal photons, such as the Breit interaction and renormalized self-interaction can be taken care of perturbatively at the end, but there are still many open questions.

Inversion-based residual generation for robust detection and isolation of faults by means of estimation of the inverse dynamics in linear dynamical systems

In this article the idea of inversion-based direct input reconstruction for robust detection, separation and estimation of multiple simultaneous faults in the presence of persistent disturbances which proved to be non-separable by conventional decoupling methods in linear dynamical systems is presented. In particular, it is shown how in a specific filtering structure a residual generator, relying on the inverse representation of the system, by means of estimation of the inverse dynamics can be designed providing robust fault decoupling. In the development of the method three solution approaches are contrasted with each other. In the first approach the traditional H ∞ filter providing the classical disturbance attenuated non-decoupling solution is presented. Then, in a novel solution, an H ∞ detection filter on the inverse system is designed providing detection residuals by exact fault decoupling; the estimation of the inverse dynamics is obtained by means of the optimal filter, thus ensuring H ∞ disturbance attenuation on the residual output. Finally, an observer-based residual generator for the estimation of the inverse dynamics is designed resulting in detection residuals with exact fault and disturbance separation, which characterises this novel idea in its extremes. The feasibility of the solution to problems not solvable earlier is demonstrated based on an aircraft monitoring application example that was originally considered in Douglas and Speyer (1995) and Chung and Speyer (1998).