State Eigenvectors Research Articles

Floating Block Method for Quantum MonteCarlo Simulations.

Quantum MonteCarlo simulations are powerful and versatile tools for the quantum many-body problem. In addition to the usual calculations of energies and eigenstate observables, quantum MonteCarlo simulations can in principle be used to build fast and accurate many-body emulators using eigenvector continuation or design time-dependent Hamiltonians for adiabatic quantum computing. These new applications require something that is missing from the published literature, an efficient quantum MonteCarlo scheme for computing the inner product of ground state eigenvectors corresponding to different Hamiltonians. In this work, we introduce an algorithm called the floating block method, which solves the problem by performing Euclidean time evolution with two different Hamiltonians and interleaving the corresponding time blocks. We use the floating block method and nuclear lattice simulations to build eigenvector continuation emulators for energies of ^{4}He, ^{8}Be, ^{12}C, and ^{16}O nuclei over a range of local and nonlocal interaction couplings. From the emulator data, we identify the quantum phase transition line from a Bose gas of alpha particles to a nuclear liquid.

Absolute excited state molecular geometries revealed by resonance Raman signals

Ultrafast reactions activated by light absorption are governed by multidimensional excited-state (ES) potential energy surfaces (PESs), which describe how the molecular potential varies with the nuclear coordinates. ES PESs ad-hoc displaced with respect to the ground state can drive subtle structural rearrangements, accompanying molecular biological activity and regulating physical/chemical properties. Such displacements are encoded in the Franck-Condon overlap integrals, which in turn determine the resonant Raman response. Conventional spectroscopic approaches only access their absolute value, and hence cannot determine the sense of ES displacements. Here, we introduce a two-color broadband impulsive Raman experimental scheme, to directly measure complex Raman excitation profiles along desired normal modes. The key to achieve this task is in the signal linear dependence on the Frank-Condon overlaps, brought about by non-degenerate resonant probe and off-resonant pump pulses, which ultimately enables time-domain sensitivity to the phase of the stimulated vibrational coherences. Our results provide the tool to determine the magnitude and the sensed direction of ES displacements, unambiguously relating them to the ground state eigenvectors reference frame.

Unified Mathematical Model of Gear Train Analysis Based on State Space Method

The state space method of gear train analysis and the unified mathematical model of state space based gear train analysis are proposed in this paper. Firstly, the concept of basic gear train unit is defined, and the state transformation equation of basic gear train unit is established, which is applied to express the transformation rule of the basic gear train unit and the adjacency relationship between the units. Then, the gear train state space is formed based on the dual vectors including input state eigenvector and output state eigenvector, which describe the state characteristics of the gear train unit. The state space is established based on the mapping relationship among the input-output dual vectors, the state transformation matrix, and the gear train unit. Then the dual vector operation laws and mathematical operation methods in gear train state space are generated. Therefrom, a unified mathematical model based on the kinematic, statics, and structural characteristics of the gear train in the state space is established. Finally, the digital identification and automatic analysis of any complex gear train is realized. The fast comparison and analysis of a large number of gear train schemes is achievable in the scheme design stage.

Regularization in nonperturbative extensions of effective field theory

The process of renormalization in nonperturbative Hamiltonian effective field theory (HEFT) is examined in the $\mathrm{\ensuremath{\Delta}}$-resonance scattering channel. As an extension of effective field theory incorporating the L\"uscher formalism, HEFT provides a bridge between the infinite-volume scattering data of experiment and the finite-volume spectrum of energy eigenstates in lattice QCD. HEFT also provides phenomenological insight into the basis-state composition of the finite-volume eigenstates via the state eigenvectors. The Hamiltonian matrix is made finite through the introduction of finite-range regularization. The extent to which the established features of this regularization scheme survive in HEFT is examined. In a single-channel $\ensuremath{\pi}N$ analysis, fits to experimental phase shifts withstand large variations in the regularization parameter $\mathrm{\ensuremath{\Lambda}}$, providing an opportunity to explore the sensitivity of the finite-volume spectrum and state composition on the regulator. While the L\"uscher formalism ensures the eigenvalues are insensitive to $\mathrm{\ensuremath{\Lambda}}$ variation in the single-channel case, the eigenstate composition varies with $\mathrm{\ensuremath{\Lambda}}$; the admission of short-distance interactions diminishes single-particle contributions to the states. In the two-channel $\ensuremath{\pi}N$, $\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$ analysis, $\mathrm{\ensuremath{\Lambda}}$ is restricted to a small range by the experimental data. Here the inelasticity is particularly sensitive to variations in $\mathrm{\ensuremath{\Lambda}}$ and its associated parameter set. This sensitivity is also manifest in the finite-volume spectrum for states near the opening of the $\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$ scattering channel. Future high-quality lattice QCD results will be able to discriminate $\mathrm{\ensuremath{\Lambda}}$, describe the inelasticity, and constrain a description of the basis-state composition of the energy eigenstates. Finally, HEFT has the unique ability to describe the quark-mass dependence of the finite-volume eigenstates. The robust nature of this capability is presented and used to confront current state-of-the-art lattice QCD calculations.

Numerical Solution Strategies in Permeation Processes

In this work, the strategy for numerical solutions in transport processes is investigated. Permeation problems can be solved analytically or numerically by means of the Finite Difference Method (FDM), while choosing the Euler forward explicit or Euler backwards implicit formalism. The first method is the easiest and most commonly used, while the Euler backwards implicit is not yet well established and needs further development. Hereafter, a possible solution of the Crank-Nicolson algorithm is presented, which makes use of matrix multiplication and inversion, instead of the step-by-step FDM formalism. If one considers the one-dimensional diffusion case, the concentration of the elements can be expressed as a time dependent vector, which also contains the boundary conditions. The numerically stable matrix inversion is performed by the Branch and Bound (B&B) algorithm [2]. Furthermore, the paper will investigate, whether a larger time step can be used for speeding up the simulations. The stability range is investigated by eigenvalue estimation of the Euler forward and Euler backward. In addition, a third solver is considered, referred to as Combined Solver, that is made up of the last two ones. Finally, the Crank-Nicolson solver [9] is investigated. All these results are compared with the analytical solution. The solver stability is analyzed by means of the Steady State Eigenvector (SSEV), a mathematical entity which was developed ad hoc in the present work. In addition, the obtained results will be compared with the analytical solution by Daynes [6,7].

Ground states and associated path measures in the renormalized Nelson model

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson’s operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.

Running Status Diagnosis of S700K Turnout Based on VMD-KPCA and Fuzzy Clustering

S700K turnout is the key equipment of railway line conversion. The diagnosis of S700K turnout in a normal, sub-health, and fault running state is the primary premise to ensure the safe operation of the railway. Aiming at the consistency between the characteristics of the power curve of S700K turnout and its state information, this paper proposes a new algorithm based on variational mode decomposition (VMD) and kernel principal component analysis (KPCA) to extract the characteristics of the power curve of S700K turnout. It uses fuzzy clustering analysis to diagnose the running state of S700K turnout. First, to extract the detailed components of the action power curve, it is decomposed into intrinsic mode function with limited bandwidth (BIMF) by VMD. Secondly, the multi-scale permutation entropy (MPE) is used to characterize the signal complexity of the power curve and different BIMF components, which are taken as the running state feature set. After KPCA analysis, eigenvalues with a contribution rate greater than 95% are selected as the state eigenvector. The experimental results show that the diagnosis algorithm can effectively identify the running state of S700K turnout, meet the characteristics of fewer fault samples of S700K turnout, and do not need to train in advance, which is of great significance for field guidance.

Transition density matrices of Richardson-Gaudin states.

Recently, ground state eigenvectors of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian, Richardson-Gaudin (RG) states, have been employed as a wavefunction ansatz for strong correlation. This wavefunction physically represents a mean-field of pairs of electrons (geminals) with a constant pairing strength. To move beyond the mean-field, one must develop the wavefunction on the basis of all the RG states. This requires both practical expressions for transition density matrices and an idea of which states are most important in the expansion. In this contribution, we present expressions for the transition density matrix elements and calculate them numerically for half-filled picket-fence models (reduced BCS models with constant energy spacing). There are no Slater-Condon rules for RG states, though an analog of the aufbau principle proves to be useful in choosing which states are important.

Two examples of recent advances in sensitivity calculations

This article reviews two recently established methods to compute sensitivities of some core parameters to basic nuclear data. First, perturbation theory offers an efficient way to compute sensitivities to nuclear parameters in continuous energy transport simulations: making use of the Iterated Fission Probability method, and by coupling the Monte Carlo code TRIPOLI-4® to the nuclear evaluation code CONRAD, we were able to compute the sensitivity of core reactivity to nuclear parameters for simple ICSBEP benchmarks. Second, using a multipoint description of a nuclear system and deterministic transport calculations the sensitivity of the state eigenvector of the system to multigroup nuclear data is computed using simple and fast partial importance calculations.

Richardson-Gaudin mean-field for strong correlation in quantum chemistry.

Ground state eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian are employed as a wavefunction Ansatz to model strong electron correlation in quantum chemistry. This wavefunction is a product of weakly interacting pairs of electrons. While other geminal wavefunctions may only be employed in a projected Schrödinger equation, the present approach may be solved variationally with polynomial cost. The resulting wavefunctions are used to compute expectation values of Coulomb Hamiltonians, and we present results for atoms and dissociation curves that are in agreement with doubly occupied configuration interaction data. The present approach will serve as the starting point for a many-body theory of pairs, much as Hartree-Fock is the starting point for weakly correlated electrons.

Strict deformation quantization of the state space of Mk(ℂ) with applications to the Curie–Weiss model

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

Rolling Bearing Fault Prediction Method Based on QPSO-BP Neural Network and Dempster–Shafer Evidence Theory

To effectively predict the rolling bearing fault under different working conditions, a rolling bearing fault prediction method based on quantum particle swarm optimization (QPSO) backpropagation (BP) neural network and Dempster–Shafer evidence theory is proposed. First, the original vibration signals of rolling bearing are decomposed by three-layer wavelet packet, and the eigenvectors of different states of rolling bearing are constructed as input data of BP neural network. Second, the optimal number of hidden-layer nodes of BP neural network is automatically found by the dichotomy method to improve the efficiency of selecting the number of hidden-layer nodes. Third, the initial weights and thresholds of BP neural network are optimized by QPSO algorithm, which can improve the convergence speed and classification accuracy of BP neural network. Finally, the fault classification results of multiple QPSO-BP neural networks are fused by Dempster–Shafer evidence theory, and the final rolling bearing fault prediction model is obtained. The experiments demonstrate that different types of rolling bearing fault can be effectively and efficiently predicted under various working conditions.

Dirac delta -function potential in quasiposition representation of a minimal-length scenario

A minimal-length scenario can be considered as an effective description of quantum gravity effects. In quantum mechanics the introduction of a minimal length can be accomplished through a generalization of Heisenberg’s uncertainty principle. In this scenario, state eigenvectors of the position operator are no longer physical states and the representation in momentum space or a representation in a quasiposition space must be used. In this work, we solve the Schroedinger equation with a Dirac delta -function potential in quasiposition space. We calculate the bound state energy and the coefficients of reflection and transmission for the scattering states. We show that leading corrections are of order of the minimal length ({ O}(sqrt{beta })) and the coefficients of reflection and transmission are no longer the same for the Dirac delta well and barrier as in ordinary quantum mechanics. Furthermore, assuming that the equivalence of the 1s state energy of the hydrogen atom and the bound state energy of the Dirac {{delta }}-function potential in the one-dimensional case is kept in a minimal-length scenario, we also find that the leading correction term for the ground state energy of the hydrogen atom is of the order of the minimal length and varDelta x_{min } le 10^{-25} m.

The dilute Temperley–Lieb O(n = 1) loop model on a semi infinite strip: the sum rule

This is the second part of our study of the ground state eigenvector of the transfer matrix of the dilute Temperley–Lieb loop model with the loop weight n = 1 on a semi infinite strip of width L (Garbali and Nienhuis 2017 J. Stat. Mech. 043108). We focus here on the computation of the normalization (otherwise called the sum rule) ZL of the ground state eigenvector, which is also the partition function of the critical site percolation model. The normalization ZL is a symmetric polynomial in the inhomogeneities of the lattice . This polynomial satisfies several recurrence relations which we solve independently in terms of Jacobi–Trudi like determinants. Thus we provide a few determinant expressions for the normalization ZL.

The dilute Temperley–Lieb O(n = 1) loop model on a semi infinite strip: the ground state

We consider the integrable dilute Temperley–Lieb (dTL) O(n = 1) loop model on a semi-infinite strip of finite width L. In the analogy with the Temperley–Lieb (TL) O(n = 1) loop model the ground state eigenvector of the transfer matrix is studied by means of a set of q-difference equations, sometimes called the qKZ equations. We compute some ground state components of the transfer matrix of the dTL model, and show that all ground state components can be recovered for arbitrary L using the qKZ equation and certain recurrence relation. The computations are done for generic open boundary conditions.

Functional brain mapping using specific sensory-circuit stimulation and a theoretical graph network analysis in mice with neuropathic allodynia.

Allodynia, a form of neuropathic pain, is defined as pain in response to a non-nociceptive stimulus. The brain regions responsible for pain, which are not normally activated, can be activated in allodynic mice by providing a suitable stimulus to Aβ-fibers, which transmit signals from tactile sensory fibers. Functional MRI (fMRI) can be used to objectively observe abnormal brain activation. In the present study, fMRI was conducted to investigate allodynia in mice; allodynia was generated by surgical injury at the L4 spinal nerve root, thus selectively stimulating sensory nerve fibers. In intact mice, only the primary somatosensory cortex (S1) was activated by stimulation of Aβ-fibers. Meanwhile, allodynic mice showed significantly higher BOLD signals in the anterior cingulate area (ACA) and thalamus. Using resting state fMRI, both degree and eigenvector centrality were significantly decreased in the contralateral S1, clustering coefficient and local efficiency were significantly increased in the ACA, and betweenness centrality was significantly higher in the ventral posterolateral nucleus of the thalamus. These results suggest that the observed abnormal BOLD activation is associated with defects in Aβ-fibers when Aβ-fibers in allodynic mice are selectively stimulated. The objective approach enabled by fMRI can improve our understanding of pathophysiological mechanisms and therapeutic efficacy.

A new approach for the development of diabatic potential energy surfaces: Hybrid block-diagonalization and diabatization by ansatz.

A new diabatization method is presented, which is suitable for the development of accurate high-dimensional coupled potential energy surfaces for use in quantum dynamics studies. The method is based on the simultaneous use of adiabatic wave function and energy data, respectively, and combines block-diagonalization and diabatization by ansatz approaches. It thus is called hybrid diabatization. The adiabatic wave functions of suitable ab initio calculations are projected onto a diabatic state space and the resulting vectors are orthonormalized like in standard block-diagonalization. A parametrized diabatic model Hamiltonian is set up as an ansatz for which the block-diagonalization data can be utilized to find the optimal model. Finally, the parameters are optimized with respect to the ab initio reference data such that the deviations between adiabatic energies and eigenvalues of the model as well as projected state vectors and eigenvectors of the model are minimized. This approach is particularly advantageous for problems with a complicated electronic structure where the diabatic state space must be of higher dimension than the number of calculated adiabatic states. This is an efficient way to handle problems with intruder states, which are very common for reactive systems. The use of wave function information also increases the information content for each data point without additional cost, which is beneficial in handling the undersampling problem for high-dimensional systems. The new method and its performance are demonstrated by application to three prototypical systems, ozone (O3), methyl iodide (CH3I), and propargyl (H2CCCH).

Continuity properties of the semi-group and its integral kernel in non-relativistic QED

Employing recent results on stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics by B. Güneysu, J. S. Møller, and the present author, we study the continuity of the corresponding semi-group between weighted vector-valued [Formula: see text]-spaces, continuity properties of elements in the range of the semi-group, and the pointwise continuity of an operator-valued semi-group kernel. We further discuss the continuous dependence of the semi-group and its integral kernel on model parameters. All these results are obtained for Kato decomposable electrostatic potentials and the actual assumptions on the model are general enough to cover the Nelson model as well. As a corollary, we obtain some new pointwise exponential decay and continuity results on elements of low-energetic spectral subspaces of atoms or molecules that also take spin into account. In a simpler situation where spin is neglected, we explain how to verify the joint continuity of positive ground state eigenvectors with respect to spatial coordinates and model parameters. There are no smallness assumptions imposed on any model parameter.

Aposteriorianalysis of a nonlinear Gross–Pitaevskii-type eigenvalue problem

In this paper, we provide a first full {\it a posteriori} error analysis for variational approximations of the ground state eigenvector of a non-linear elliptic problems of the Gross-Pitaevskii type, more precisely of the form $-\Delta u + Vu + u^3 = \lambda u$, $\|u\|_{L^2}=1$, with periodic boundary conditions in one dimension. Denoting by $(u_N,\lambda_N)$ the variational approximation of the ground state eigenpair $(u,\lambda)$ based on a Fourier spectral approximation and $(u_N^k,\lambda_N^k)$ the approximate solution at the $k^{th}$ iteration of an algorithm used to solve the non-linear problem, we first provide a precised \textit{a priori} analysis of the convergence rates of $\|u-u_N\|_{H^1}$, $\|u-u_N\|_{L^2}$, $|\lambda-\lambda_N|$ and then present original \textit{a posteriori} estimates in the convergence rates of $\|u_-u_N^k\|_{H^1}$ when $N$ and $k$ go to infinity. We introduce a residue standing for the global error $R_N^k=-\Delta u_N^k+Vu_N^k+(u_N^k)^3-\lambda_N^ku_N^k$ and we divide it into two residues characterizing respectively the error due to the discretization of the space and the finite number of iterations when solving the problem numerically. We show that the numerical results are coherent with this \textit{a posteriori} analysis.

Simple model of a coherent molecular photocell.

Electron transport in molecular electronic devices is often dominated by a coherent mechanism in which the wave function extends from the left contact over the molecule to the right contact. If the device is exposed to light, photon absorption in the molecule might occur, turning the device into a molecular photocell. The photon absorption promotes an electron to higher energy levels and thus modifies the electron transmission probability through the device. A model for such a molecular photocell is presented that minimizes the complexity of the problem while providing a non-trivial description of the device mechanism. In particular, the role of the molecule in the photocell is investigated. It is described within the Hückel method and the source-sink potential approach [F. Goyer, M. Ernzerhof, and M. Zhuang, J. Chem. Phys. 126, 144104 (2007)] is used to eliminate the contacts in favor of complex-valued potentials. Furthermore, the photons are explicitly incorporated into the model through a second-quantized field. This facilitates the description of the photon absorption process with a stationary state calculation, where eigenvalues and eigenvectors are determined. The model developed is applied to various generic molecular photocells.