Matrix Elements Of Generators Research Articles

Differentiable Matrix Elements with MadJax

MadJax is a tool for generating and evaluating differentiable matrix elements of high energy scattering processes. As such, it is a step towards a differentiable programming paradigm in high energy physics that facilitates the incorporation of high energy physics domain knowledge, encoded in simulation software, into gradient based learning and optimization pipelines. MadJax comprises two components: (a) a plugin to the general purpose matrix element generator MadGraph that integrates matrix element and phase space sampling code with the JAX differentiable programming framework, and (b) a standalone wrapping code interface for accessing the matrix element code and its gradients, which are computed with automatic differentiation. The MadJax implementation and example applications of simulation based inference and normalizing flow based matrix element modeling, with capabilities enabled uniquely with differentiable matrix elements, are presented.

Invariants and reduced Wigner coefficients for quasi-triangular Hopf superalgebras

We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup U q [gl(m|n)]. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients.

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969 Zh. Eksp. Teor. Fiz. 56 1375–90) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be derived in a different form, expressed by the classical Gegenbauer polynomials. This new expression makes it possible to determine coherent and incoherent superpositions of these matrix elements in closed analytic forms. As an application, we describe multiphoton transitions in the system ‘charged particle + electromagnetic radiation’, induced by a (strong) coherent field or by a black-body radiation component (with a Planck–Bose photon number distribution). The exact results are compared with the semi-classical ones. We will show that in case of interaction with a thermal field, the semi-classical result (with a Gaussian stochastic field amplitude) yields an acceptable approximation only in the Rayleigh–Jeans limit, however, in the Wien limit it completely fails.

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

Abstract We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be derived in an different form, expressed by the classical Gegenbauer polynomials. This new expression makes it possible to determine coherent and incoherent superpositions of these matrix elements in closed analytic forms. As an application, we describe multiphoton transitions in the system “charged particle + electromagnetic radiation”, induced by a (strong) coherent field or by a black-body radiation component (with a Planck-Bose photon number distribution). The exact results are compared with the semi-classical ones. We will show that in case of interaction with a thermal field, the semi-classical result (with a Gaussian stochastic field amplitude) yields an acceptable approximation only in the Rayleigh-Jeans limit, however, in the Wien limit it completely fails.

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of {W}_{0,{mathfrak{q}}^{pm 2}} contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in {mathfrak{q}}^{pm 2} . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

Analysis of the Propagation in High-Speed Interconnects for MIMICs by Means of the Method of Analytical Preconditioning: A New Highly Efficient Evaluation of the Coefficient Matrix

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

Matrix elements of unitary group generators in many-fermion correlation problem. III. Green-Gould approach

The third part of our survey series concerning the evaluation of matrix elements (MEs) of the unitary group generators and of their products in the electronic Gel’fand–Tsetlin basis of the two-column irreps of U(n)—which are essential in the unitary group approach (UGA) to the many-electron correlation problem as handled by the configuration interaction (CI) and the coupled cluster (CC) approaches—relies on what we refer to as the Green-Gould (G-G) approach. In addition to the CI and CC methods, the G-G formalism proved to be very helpful in a number of other tasks, particularly in handling of the spin-dependent operators, the density matrices, or partitioned basis sets adapted to a chosen group chain.

Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.

Matrix elements of unitary group generators in many-fermion correlation problem. II. Graphical methods of spin algebras

This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital U(2n) generators, in the basis of the electronic Gel’fand–Tsetlin (G–T) [or Gel’fand–Paldus (G–P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the U(n), $${\mathcal {S}}_N$$ , and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group $${\mathcal {S}}_N$$ . The key is a close relationship between the P-states of UGA and the Yamanouchi–Kotani (Y–K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent U(2n) generators in the UGA P-basis.

Application of Sodium Dodecyl Sulfate/Activated Carbon onto the Preconcentration of Cadmium Ions in Solid-Phase Extraction Flow System

In the present study, activated carbon (AC) surface modified with sodium dodecyl sulfate (SDS), written as SDS/AC, was applied as an adsorbent for preconcentration and determination of trace amount of cadmium ions in environmental sample waters. The SDS modification on AC was performed at the same time, while cadmium ions were concentrated in the flow system as solid-phase extraction. After the separation and preconcentration steps, cadmium retained on SDS/AC was eluted with HNO3 and was subsequently determined by flame atomic absorption spectrometry (FAAS). The analytical parameters that influence the quantitative determination of trace cadmium, such as SDS concentration, pH and volume of sample solution, eluent conditions, and interference, were optimized. At the optimum conditions, the general matrix elements had little interference on the proposed procedure. The detection limits was 17 ng·L−1, and the relative standard deviation (RSD) for 12 experiments at 10 µg·L−1 cadmium solutions was 2.8%. The developed method was applied into the analysis of environmental samples spiked cadmium.

Invariants and matrix elements of the quantum group revisited

In a previous paper the generator matrix elements and (dual) vector reduced Wigner coefficients (RWCs) were evaluated via the polynomial identities satisfied by a certain matrix constructed from the R-matrix R and its twisted counterpart . Here we provide an alternative evaluation utilising the R-matrix . This provides a new direct derivation of the vector RWCs obtained indirectly in earlier work via a symmetry relation. This approach has the advantage that it generalises to the Lie superalgebra case, which will be investigated elsewhere.

An extended likelihood framework for modelling discretely observed credit rating transitions

The estimation of the parameters of a continuous-time Markov chain from discrete-time observations, also known as the embedding problem for Markov chains, plays in particular an important role for the modeling of credit rating transitions. This missing data problem boils down to a latent variable setting and thus, maximum likelihood estimation is usually conducted using the expectation-maximization (EM) algorithm. We illustrate that the EM algorithm is likely to get stuck in local maxima of the likelihood function in this specific problem setting and adapt a stochastic approximation simulated annealing scheme (SASEM) as well as a genetic algorithm (GA) to combat this issue. Above that, our main contribution is to extend our method GA by a rejection sampling scheme, which allows one to derive stochastic monotone maximum likelihood estimates in order to obtain proper (non-crossing) multi-year probabilities of default. We advocate the use of this procedure as direct constrained optimization (of the likelihood function) will not be numerically stable due to the large number of side conditions. Furthermore, the monotonicity constraint enables one to combine structural knowledge of the ordinality of credit ratings with real-life data into a statistical estimator, which has a stabilizing effect on far off-diagonal generator matrix elements. We illustrate our methods by Standard and Poor’s credit rating data as well as a simulation study and benchmark our novel procedure against an already existing smoothing algorithm.

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

AbstractWe present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S2 eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.

The Direct Effect of Toroidal Magnetic Fields on Stellar Oscillations: An Analytical Expression for the General Matrix Element

Abstract Where is the solar dynamo located and what is its modus operandi? These are still open questions in solar physics. Helio- and asteroseismology can help answer them by enabling us to study solar and stellar internal structures through global oscillations. The properties of solar and stellar acoustic modes are changing with the level of magnetic activity. However, until now, the inference on subsurface magnetic fields with seismic measures has been very limited. The aim of this paper is to develop a formalism to calculate the effect of large-scale toroidal magnetic fields on solar and stellar global oscillation eigenfunctions and eigenfrequencies. If the Lorentz force is added to the equilibrium equation of motion, stellar eigenmodes can couple. In quasi-degenerate perturbation theory, this coupling, also known as the direct effect, can be quantified by the general matrix element. We present the analytical expression of the matrix element for a superposition of subsurface zonal toroidal magnetic field configurations. The matrix element is important for forward calculations of perturbed solar and stellar eigenfunctions and frequency perturbations. The results presented here will help to ascertain solar and stellar large-scale subsurface magnetic fields, and their geometric configuration, strength, and change over the course of activity cycles.

Monte Carlo study of the measurement of the Michel parameters in the radiative decay of the τ at Belle

Michel parameters are kinematic values defined as bilinear combinations of the coupling terms in the general matrix element of the Fermi interaction where all possible Lorentz-invariant scalar, vector and tensor terms are included. The leptonic τ decays provide an excellent laboratory in which to measure the Michel parameters. Any deviation of these parameters from the Standard Model expectation would indicate New Physics. Two Michel parameters, η¯ and ξκ, can only be measured in the radiative leptonic decay. These parameters have previously been measured in the radiative decay of the muon but as yet have not been measured in τ decays. We report the results of a feasibility study of the measurement of η¯ and ξκ in radiative leptonic τ decays at Belle and Belle II.

Cubic-scaling iterative solution of the Bethe-Salpeter equation for finite systems

The Bethe-Salpeter equation (BSE) is currently the state of the art in the description of neutral electron excitations in both solids and large finite systems. It is capable of accurately treating charge-transfer excitations that present difficulties for simpler approaches. We present a local basis set formulation of the BSE for molecules where the optical spectrum is computed with the iterative Haydock recursion scheme, leading to a low computational complexity and memory footprint. Using a variant of the algorithm we can go beyond the Tamm-Dancoff approximation (TDA). We rederive the recursion relations for general matrix elements of a resolvent, show how they translate into continued fractions, and study the convergence of the method with the number of recursion coefficients and the role of different terminators. Due to the locality of the basis functions the computational cost of each iteration scales asymptotically as $O(N^3)$ with the number of atoms, while the number of iterations is typically much lower than the size of the underlying electron-hole basis. In practice we see that , even for systems with thousands of orbitals, the runtime will be dominated by the $O(N^2)$ operation of applying the Coulomb kernel in the atomic orbital representation

Unitary group approach to the many-electron correlation problem: spin-dependent operators

Following a brief overview of the unitary group approach (UGA) to the many-electron correlation problem, focusing in particular on Shavitt’s contribution via his graphical unitary group approach, we present a short review of our earlier results for the evaluation of matrix elements (MEs) of unitary group generators or products of generators in the electronic Gel’fand–Tsetlin basis with the help of spin-adapted second-quantization-like creation and annihilation vector operators at the unitary group level. This formalism is then extended to a spin-dependent case that is required when accounting for relativistic effects by developing explicit expressions for MEs of spin-orbital creation and annihilation operators in terms of the standard spin-adapted UGA basis. This leads naturally to a segmentation of these MEs and enables the evaluation of spin-dependent one-body operators while relying largely on the segment values of the standard spin-independent UGA.

Nonstandard Higgs couplings from angular distributions in [Formula: see text

We compute the fully differential rate for the Higgs-boson decay hrightarrow Zell ^+ell ^-, with Zrightarrow ell ^{'+}ell ^{'-}. For these processes we assume the most general matrix elements within an effective Lagrangian framework. The electroweak chiral Lagrangian we employ assumes minimal particle content and Standard Model gauge symmetries, but it is otherwise completely general. We discuss how information on new physics in the decay form factors may be obtained that is inaccessible in the dilepton-mass spectrum integrated over angular variables. The form factors are related to the coefficients of the effective Lagrangian, which are used to estimate the potential size of new-physics effects.

Circular dichroism in angle‐resolved photoemission spectroscopy of topological insulators

Abstractmagnified imageTopological insulators are a new phase of matter that exhibits exotic surface electronic properties. Determining the spin texture of this class of material is of paramount importance for both fundamental understanding of its topological order and future spin‐based applications. In this article, we review the recent experimental and theoretical studies on the differential coupling of left‐circularly versus right‐circularly polarized light to the topological surface states in angle‐resolved photoemission spectroscopy. These studies have shown that the polarization of light and the experimental geometry play a veryimportant role in both photocurrent intensity and spin polarization of photoelectrons emitted from the topological surface states. A general photoemission matrix element calculation can quantitatively explain the observations and is also applicable to topologically trivial systems with strong spin–orbit coupling. These experimental and theoretical investigations suggest that optical excitation with circularly polarized light is a promising route towards mapping the spin–orbit texture and manipulating the spin orientation in topological and other spin–orbit coupled materials.magnified imageThe circular‐dichroic angle‐resolved photoemission spectrum of Bi2Se3 as obtained by the difference between spectra taken with left‐circularly vs. right‐circularly polarized light using a time‐of‐flight electron spectrometer.(© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Invariants and reduced matrix elements associated with the Lie superalgebra gl(m|n)

We construct explicit formulae for the eigenvalues of certain invariants of the Lie superalgebra gl(m|n) using characteristic identities. We discuss how such eigenvalues are related to reduced Wigner coefficients and the reduced matrix elements of generators, and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra.