Showing papers on "Transfer matrix published in 1981"

Simple scheme for surface-band calculations. I

Abstract: A simple, efficient scheme for calculating the electronic structure of a surface is presented. The scheme is applicable to any general Hamiltonian that can be described within a localized-orbital basis. The method is much faster than the current techniques available. The basic concept is that of wave-function matching through a transfer matrix. The eigensolutions of this matrix then provide all the information concerning the projected band structure, surface-state energies, orbital character, and decay lengths. A rather detailed discussion of the formalism is presented for a general surface system. A comprehensive and illustrative example is also presented for readers who are interested in using the scheme but not in the details of the theory.

On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings

Abstract: Various types of transfer matrix factorizations are of interest when designing regulators for generalized types of linear systems (delay differential, 2-D, and families of systems). This paper studies the existence of stable and of stable proper factorizations, in the context of the theory of systems over rings. Factorability is related to stabilizability and detectability properties of realizations of the transfer matrix.

Corner transfer matrices

Abstract: For two-dimensional lattice models with interactions only between nearest (and diagonally nearest) neighbour spins, a well-known concept is the row-to-row transfer matrix. Less well-known is the “corner” transfer matrix (CTM). This has some very useful properties. If it is normalized so that its largest eigenvalue is unity, and the eigenvalues are arranged in numerically decreasing order, then each eigenvalue tends to a limit as the lattice becomes large. For those models which have been solved exactly (notably the Ising, eight-vertex and hard hexagon models), this limiting eigenvalue distribution is very simple, being basically that of a direct product of two-by-two matrices. From it the order parameter can easily be obtained. For all models one can write down formally exact matrix relations for the CTM, but the matrices are of infinite size. If one uses a representation in which the CTM is diagonal, and then truncates these relations to finite size, then one obtains a quite accurate approximation. The larger the size the greater the accuracy. I.G. Enting and I have thereby obtained comparatively long series expansions for the Ising model in a field, and for the hard squares model.

Feynman's Relativistic Chessboard as an Ising Model

Abstract: Feynman has described a chessboard model for a one-dimensional relativistic quantum problem which yields the correct kernel for a free spin-1/2 particle moving in one spatial dimension. This chessboard problem can be solved as an Ising model, using the transfer matrix technique of statistical mechanics. The 2×2 transfer matrix represents the infinitesimal time evolution operator for the two eigenstates of the velocity operator.

New transfer-matrix theory of linear induction machines, taking into account longitudinal and transverse ferromagnetic end effects

Abstract: A new 3-dimensional theory of short-primary linear induction machines is presented. The theory almost completely incorporates both longitudinal and transverse end effects, especially including ferromagnetic end effects caused by the geometry of the primary laminated iron. The theory is based on the idea of considering two different regions of the primary iron and the airspace around it as one fictitious region having an inhomogeneous permeability, with step variations only at the four iron/air boundaries if the primary iron is assumed nonconducting. The 5-layer boundary-value problem including the fictitious layer is solved neatly in a mathematical sense using the new transfer-matrix method developed here. The theory is verified from test data on the double-sided linear induction machine at General Electric Co. The theory can very accurately predict the machine behaviour, including the vertical force, at low computational cost.

Electromagnetic scattering by two bodies of revolution

Abstract: A systematic procedure is presented for computing the electromagnetic scattering by two arbitrarily oriented bodies of revolution when the size of the objects are in the resonance region. The analysis uses a System Transfer Operator (T-matrix) approach and iteratively solves for the scattering of the two-body configuration. A modified unimoment technique is employed to generate the system transfer matrix of each body of revolution. Results are presented and compared to measured data to demonstrate the versatility and accuracy of the analysis. Applications of the two-body analysis stem from extending it to N-bodies. Some potential applications are discussed along with the major problems that need to be addressed.

Finite-lattice calculations for the two-dimensional axial-next-nearest-neighbor Ising model

Abstract: The two-dimensional axial-next-nearest-neighbor Ising model is studied numerically using finite-lattice results. The row-to-row transfer matrix is mapped into a simple one-dimensional spin-$\frac{1}{2}$ quantum Hamiltonian. The leading eigenvalues and the corresponding eigenvectors are calculated numerically for chains of finite lengths. Using a finite-lattice renormalization-group transformation the high- (low-) temperature region is analyzed with the use of the Hamiltonian for the original (dual) model. Although there is no clear evidence for a massless phase, the phase boundaries obtained in this way are a strong indication of the presence of a sinusoidal phase between the paramagnetic and the $〈2〉$ antiphase, in fair agreement with previous analytical results.

Localized states in polymeric molecules. I. The transfer matrix for long range interactions

Abstract: We have extended the transfer matrix technique to determine the Green's function for 1-dimensional systems with long range interactions. The formalism is applied to study the density of states and impurity modes of linear chains; calculations are presented for nearest and next nearest neighbors interactions.

Cluster Expansion and Generalized Transfer Matrices for the Statistical Mechanics of Linear Chains

Abstract: A cluster expansion of the statistical mechanical density operator for a general linear chain model with nearest-neighbor interactions is made. This expansion is then shown to lead to an expansion of a generalized transfer matrix, whose maximum eigenvalue is the per-site partition function. A number of computational features, as well as some illustrative examples, of this approach are described.

Dynamic Response of Tapered Fluid Lines : 1st Report, Transfer Matrix and Frequency Response

Abstract: A transfer matrix equation relating the pressure and volume flux in tapered fluid lines is derived under the assumption that the rate of divergence (or convergence) of the line is comparatively small. The model employed in the analysis is one of an unsteady viscous flow, that is, the frequency-dependent effect of viscosity is taken into consideration. Natural frequencies and frequency response curves of the line systems are calculated from the matrix, and dynamic characteristics of systems are discussed under various pipe-end conditions. The experimental data obtained from frequency response tests are compared with theoretical analysis, the validity of which is thus ascertained.

Application of liouville's theorem to electron and ion optical systems possessing the primary aberrations

Abstract: The method of matrix calculations is widely applied in the theory of electron- and ion-optics, especially in the aberration theory. In the present paper, a rotationally symmetrical electron-optical system and an ion-optical system with crossed toroidal electric and inhomogeneous magnetic fields are treated respestively. The general transfer matrices for the above mentioned electron- and ion-optical systems possessing the primary aberrations are derived. As a direct consequence of Liouville's theorem, we prove that the determinant of the transfer matrix has the value 1 in the approximation up to the primary aberrations. This general conclusion is useful for the electron-and ion-optical a berration theory and the computer-aided design.

Infinite- and finite-spin Ising chains at low temperatures

Abstract: Transfer matrix methods are used to obtain the partition function and correlation length of Ising chains at low temperatures. For the infinite-spin chain, asymptotically exact eigenvalues and eigenvectors are obtained from an iteration scheme, and from a secular equation method. It is also shown how the Ising chain for general spin S (including S to infinity ) may be treated using a 2*2 transfer matrix. For low temperatures (K identical to J/KBT>>1) the inverse correlation length, in units of the spin spacing, is xi -1=4S e-2K+O(e-4K) for finite S, and xi -1=8K e-2K(ln 4K+ gamma +O((lnK)/K)) for infinite spin where gamma is Euler's constant.

Calculation of the transfer matrix T in six dimensions for an rf-deflector element

Abstract: One possible element for funneling two beams together is a deflector with a constant or time-varying electric-field strength. With such an element, arbitrary beams can be brought together and maintained on the axis, if the appropriate combination of deflector parameters is chosen. A parallel beam can be handled only with a time-varying voltage of the deflector. The six-dimensional transfer matrices are calculated for constant or time-varying fields; all the results are correct in first-order approximation.

Direct and Indirect LCAO Recursion Methods for Surfaces

Abstract: In the LCAO representation our direct recursion (transfer matrix, T) method (DRM) applies strictly valid recurrence relations straight to the solution of the Hamiltonian matrix eigenvalue problem for the electronic system in bounded finite or infinite crystals and polymers with perturbed boundaries. We present the different reduced forms of the auxiliary equations of the DRM and introduce the notion of duality for these. Their relations to the equations of cyclic and half-infinite systems are given. The indirect recursion method (IRM) of Fromm and Koutecky (FK) utilize recurrence relations (strictly valid only for half-infinite perfect systems) to derive the bordering blocks of the resolvent (Green) matrix R. We completed their expressions with those of the inner blocks of R. Contrary to their statement, the inversion of matrix block B which describes the interactions between the consecutive sub-systems has similar role in both methods. We proved that the B-1-ree reduced forms of the auxiliary equations of the DRM for semisimple T are identical to the FK equation of the IRM for the corner diagonal block of R multiplied by B’. Algebraic methods are proposed to eliminate the “B-1-difficulty” of the transfer matrix method. The IRM was found generally much more complicated than the DRM is.

Impurity States and the Transfer Matrix Approach: An Eleriientary Example*

Abstract: The single impurity problem for the classical l inear chain is solved exactly using the T ransfer Matrix approach to calculate the densities of states of excitations (phonons). This paper is a didactic pre sentation to illustrate how the formalism works in a simple case, having in mind future extensions to deal with problems on disordered alloys. The same resul ts have been obtained previously by Hori and Asahi (Prog.~heor. Phys. 17, 523 (1957)) and exact calculations for arbitrary dimensions can be found in Maradudin et aZrs paper "Theory of Lattice Dynamics i n the Harmonic Approximat ion" (sol id State Phys Supp. (1 963)).