Determinantal Method Research Articles

Complete monotonicity for inverse powers of some combinatorially defined polynomials

We prove the complete monotonicity on (0,∞)n for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab-initio methods for proving that P-β is completely monotone on a convex cone C: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of P-β for some β>0 can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for P, and is also related to the Rayleigh property for matroids.

Determinantal method for locally modified structures. Application to the vibration damping of a space launcher

In order to improve the dynamic behaviour of an existing or already designed structure, local modifications can be performed by taking advantage of the relative displacement between two points of the structure. A stiffner, damper or viscoelastic rod may be added and its effect on the initial structure must be assessed. A new formulation is developed, based on the response of the initial structure at the attachment points of the local modification. A determinantal equation results, whose roots are the eigenvalues of the modified structure. The equation is solved numerically with a dedicated algorithm and it is shown that this is faster than performing an eigenvalue problem reanalysis. The method is able to deal with both undamped and damped systems, and can handle several modifications simultaneously. It is applied on the last stage of a space launcher, along with a double modal synthesis method.

How Can We Explain the Stability of Conjugated Hydrocarbon- and Heterosubstituted Networks by Topological Descriptors?

Correlation between the π-electronic stability (Hückel's Eπ) and several topological descriptors, including Hosoya's Z, Wiener's W, and Randic's 1χ , was compared for the isomeric acyclic conjugated hydrocarbon molecules from hexatriene to decapentaene. From the analysis of the best descriptor Z and mean length of conjugated paths L, the origin of π-electronic stability was logically explained. The effect of heteroatom substitution to polyenes was studied and analyzed by graph-theoretical techniques. The difference in the stability and electron flow for this heteroatom substitution was also quantitatively and easily explained. Discussion was given for the difference in the topological dependency of HOMO level of these networks with even and odd number of skeletal carbon atoms. An efficient determinantal method for calculating the Z-index for tree graphs is introduced.

General formulation for direct evaluation of the local field amplitude and transition amplitude based on the Fredholm determinant

We develop a determinantal method in a quantum scattering system for direct evaluation of various quantities including the local field amplitude and transition amplitude ($S$-matrix element). This method is applicable to multichannel elastic and inelastic scattering systems without rearranging the particles. The underlying principle of our formulation is that requested information can be extracted from a wave operator without solving the wave equation. The wave operator is the master operator maintaining all of the information about a system; the determinant of the inverse wave operator is just the Fredholm determinant. In the 1960s, a similar determinantal method was developed for the $S$-matrix element or equivalently the far-field amplitude of the wave function. Until now, however, no determinantal method for near-field quantities existed, although de Broglie and electromagnetic near fields are observable using scanning probe microscopes. Additionally, we prove that our formula for $S$-matrix element covers the known formula (the Le Couteur-Newton formula).

A model study of short range correlations with a multi determinantal method

Using a simple model of fermions interacting with a strong short range repulsive potential, we discuss how short range correlations can be described with a linear combination of Slater determinants using variational methods. The many-body wavefunction obtained in this way is used to evaluate the two-particle correlation function which shows the typical depletion at distances between particles comparable with the range of the repulsive potential.

A Gessel–Viennot-Type Method for Cycle Systems in a Directed Graph

We introduce a new determinantal method to count cycle systems in a directed graph that generalizes Gessel and Viennot's determinantal method on path systems. The method gives new insight into the enumeration of domino tilings of Aztec diamonds, Aztec pillows, and related regions.

An extension based determinantal method to compute Casimir operators of Lie algebras

We present a method based on degree one extensions of Lie algebras by a derivation to compute the Casimir operator of perfect Lie algebras having only one invariant for the coadjoint representation and an Abelian radical. In particular, the Casimir operator of the special affine Lie algebras sa(n, R) results from the determinant of the commutator matrix of an extension. Examples are given for the case of non-Abelian radicals, and the corresponding generalization of the method for this case is formulated.

Quantum Monte Carlo study of the alternating extended Peierls–Hubbard model applied to the trans-polyacetylene

The one-dimensional alternating Peierls–Hubbard model is especially interesting as nontrivial model for conjugated polymer chains, such as polyacetylene. We study this model for chains of 64 sites using the determinantal method based on Hubbard–Stratonovich transformation. We obtain the first electronic energies and their mean fluctuations at half-filling as a function of the on-site electron–electron interaction (both short and long range U, V coupling are considered). We also study the effect of the electron–electron interaction on the dimerization by investigating some of the important correlation functions, such as spin–spin correlation, on-site charge and the specific heat. The effect of nonlinear excitations is also examined, particularly soliton excitation.

Numerical study of the one-dimensional Hubbard model in determination of trans-polyacetylene properties

Quantum Monte Carlo simulation on one-dimensional Hubbard model is developed to study the effect of electron–electron interaction on trans-polyacetylene (t-PA) properties. The numerical calculation is based on the evaluation of the trace of the partition function using a special version of the so-called determinantal method. We use the Blakenbecler, Scalapino, Sugar and Hirsch (BSSH) algorithm based on the Hubbard–Stratonovich transformation to overcome the original problem involving anticommuting fermion operators. This numerical study allowed us to calculate mean values of different observables of physical interest such as the energies and their fluctuations. The behavior of quantum fluctuations is confirmed and interpreted by determining the specific heat. The system exhibits different ordering depending on different ranges of repulsion energy of Coulomb (U) and the on-site energy (E).

Absorption-edge singularities for a nonequilibrium Fermi sea. III. Determinantal nonperturbative theory.

The nonperturbative solution to the problem of threshold singularities for a (${\mathrm{\ensuremath{\mu}}}_{1}$,${\mathrm{\ensuremath{\mu}}}_{2}$) nonequilibrium Fermi sea is obtained using the determinantal method of Ohtaka and Tanabe. The critical exponents of the absorption power-law behavior we find agree with those estimated from the perturbative treatment of the problem given in papers I and II. A family of possibly diverging singularities is found at energies ${\mathrm{\ensuremath{\mu}}}_{2}$+n(${\mathrm{\ensuremath{\mu}}}_{1}$-${\mathrm{\ensuremath{\mu}}}_{2}$), for n\ensuremath{\ge}1.

Determinantal and worldline quantum Monte Carlo methods for many-body systems.

We examine three different quantum Monte Carlo methods for studying systems of interacting particles. The determinantal quantum Monte Carlo method is compared to two different worldline simulations. The first worldline method consists of a simulation carried out in the real-space basis, while the second method is implemented using as basis the eigenstates of the Hamiltonian on blocks of the two-dimensional lattice. We look, in particular, at the Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 lattice with periodic boundary conditions. The block method is superior to the real-space method in terms of the computational cost of the simulation, but shows a much worse negative sign problem. For larger values of U and away from half-filling it is found that the real-space method can provide results at lower temperatures than the determinantal method. We show that the sign problem in the block method can be slightly improved by an appropriate choice of basis.

Monte Carlo simulations in fermionic systems: The three band Hubbard model case

We present an application of the so-called determinantal method for the Monte Carlo simulation of fermionic systems, for the case of the two-dimensional three band Hubbard model. Our analysis of the conjugate gradient procedure shows that a proper treatment of the linear system of equations is necessary in order to avoid introducing asymmetries which are responsible for numerical instabilities in the algorithm.

HUBCOM: A program for Monte Carlo simulations on generalized Hubbard models

We present here a program for performing Monte Carlo (MC) simulations in discrete fermionic systems. We use a special version of the so-called determinantal method. The algorithm is based in the standard Metropolis method, but makes use of a non-uniform probability distribution for the selection of the transition-candidate states. Detailed balance is preserved by suitably modifying the Metropolis acceptance-rejection test. The method was originally developed and used in a one-dimensional spinless fermion model. We apply it to the case of a two-dimensional multi-band Hubbard model. The program code is self-contained and optimized for use in vectorial computers.

Lack of mirror symmetry between x-ray absorption and emission edges of simple metals.

We have calculated core emission and absorption spectra for a free-electron metal, using a determinantal method. The results indicate that the Mahan--Nozi\`eres--De Dominicis model is accurate near threshold to the extent testable with experimental data. Experimental data however, analyzed using the energy range justified above, indicate that the expected mirror symmetry rarely exists. Furthermore, Na core photoemission line shapes are incompatible with absorption and emission. We suggest a possible explanation for these discrepancies.

Determinantal method and the Littlewood-Richardson rule

Determinantal method is a powerful tool to study invariants and representations of classical groups. Let R, = Z![x,,,],, , = 1, 2 ,,,,, n be the integral coordinate ring of the module of all n x n integral matrices. In [7], Doubilet, Rota, and Stein gave a beautiful theory on combinatorial approach to invariant theory using a nice basis of R, called standard basis whose elements are simply parametrized by means of “standard tableaux.” This standard basis also occurs in the theory of flag manifolds (cf. [12], [17]). As application of this theory, Procesi and DeConcini gave many results on the characteristic free approach to the invariants and represen- tation spaces of classical groups [4], [S J, 161. In particular, a representation space of each finite dimensional irreducible representation of GL(n, C) is realized in R, @I C canonically. This representation space plays a key roll in application to physics, since each weight vector in this representation space is actually an element of standard basis. Now let us consider the tensor product representation of two arbitrarily irreducible representations pi, and pfl of GL(n, @). Tensor product representations have deep relation with the interaction of particles (cf. [Z]). The branching rule of the tensor product representation pA @ pp into its irreducible constituents is described using combinatorial methods on Young diagrams. This rule, the Littlewood-Richardson rule, was first found by Littlewood [14] and proved rather recently (cf. [15]). However, in application to physics, not only the branching rule but also more detailed information of the representation spaces of tensor product representations is required. In case of GL(2, C), the decomposition of pj.@pp is multiplicity free. So in this case, each weight vector of each irreducible constituent of

Overlap integral between two electron clouds at different sites.

The overlap integral between many-electron wave functions perturbed by local contact potentials at different sites is calculated with use of the determinantal method developed for the soft-x-ray problem. The expression for the exponent ${K}_{0}$ in the orthogonality theorem is derived as a function of the phase shift ${\ensuremath{\delta}}_{0}$ and the distance a between impurities and is found to behave differently from the result given by Yamada et al. [Prog. Theor. Phys. 70, 73 (1983)] when ${\ensuremath{\delta}}_{0}$ exceeds \ensuremath{\pi}/2. This implies that ${K}_{0}$ is discontinuous at a=\ensuremath{\infty} for such ${\ensuremath{\delta}}_{0}$, and that 2${K}_{0}$ will never be larger than 1/2 for any value of ${\ensuremath{\delta}}_{0}$.

The determinantal method for bound states and resonances of three-particle systems

Bound state energies, positions and widths of resonances of two-particle systems may be calculated as zeros of an analytic function of the energy, the modified Fredholm determinant of the Lippmann-Schwinger equation. This generates degenerate perturbation theory particularly easily. The method is generalized to the threebody problem. Here too, the bound states and resonances appear as zeros of an analytic function of the energy, the modified Fredholm determinant of a square-integrable kernel. It is proved that the multiplicity of a zero equals the degeneracy of the corresponding eigenvalue.

A comparison between the determinantal method and Padé approximants for theS-matrix

A comparison between the determinantal method and Padé approximants for theS-matrix

Relativistic extension of the Le Couteur-Newton formalism

Assuming T-invariance and extended unitarity, we study in this paper a multichannel relativistic extension of the Le Couteur-Newton formalism (« determinantal method »). We prove that the two above-mentioned assumptions are entirely sufficient to work out a D/D formalism as a particular solution of the relativistic multichannel problem, having the same form as the nonrelativistic solution. We also prove in this case the existence of a new formalism, denoted B/B, for the multichannel reduced-reactance K-matrix. The 2n single-valued meromorphic functions $$B_\lambda \left( s \right)$$ give the multichannel K-matrix a structure, which is similar to the one given to the multichannel S-matrix, by the 2n values of the many-valued meromorphic function D(s). We also find and interpret over-determination relations between the values of the D-function on the various sheets, and similar ones for the $$B_\lambda $$ -functions. Finally, we prove that the D-function exists and we give a simple application of the B/B formalism.

On analytical continuation in the determinantal method

On analytical continuation in the determinantal method