Transfer matrix

About: Transfer matrix is a research topic. Over the lifetime, 3516 publications have been published within this topic receiving 64660 citations.

The general problem of elastic wave propagation in multilayered anisotropic media

Abstract: Exact analytical treatment of the interaction of harmonic elastic waves with n-layered anisotropic plates is presented. Each layer of the plate can possess up to as low as monoclinic symmetry and thus allowing results for higher symmetry materials such as orthotropic, transversely isotropic, cubic, and isotropic to be obtained as special cases. The wave is allowed to propagate along an arbitrary angle from the normal to the plate as well as along any azimuthal angle. Solutions are obtained by using the transfer matrix method. According to this method formal solutions for each layer are derived and expressed in terms of wave amplitudes. By eliminating these amplitudes the stresses and displacements on one side of the layer are related to those of the other side. By satisfying appropriate continuity conditions at interlayer interfaces a global transfer matrix can be constructed which relates the displacements and stresses on one side of the plate to those on the other. Invoking appropriate boundary conditions on the plates outer boundaries a large variety of important problems can be solved. Of these mention is made of the propagation of free waves on the plate and the propagation of waves in a periodic media consisting of a periodic repetition of the plate. Confidence is the approach and results are confirmed by comparisons with whatever is available from specialized solutions. A variety of numerical illustrations are included.

Quantitative determination of molecular structure in multilayered thin films of biaxial and lower symmetry from photon spectroscopies. I. Reflection infrared vibrational spectroscopy

Abstract: A semitheoretical formalism based on classical electromagnetic wave theory has been developed for application to the quantitative treatment of reflection spectra from multilayered anisotropic films on both metallic and nonmetallic substrates. Both internal and external reflection experiments as well as transmission can be handled. The theory is valid for all wavelengths and is appropriate, therefore, for such experiments as x‐ray reflectivity, uv–visible spectroscopic ellipsometry, and infrared reflection spectroscopy. Further, the theory is applicable to multilayered film structures of variable number of layers, each with any degree of anisotropy up to and including full biaxial symmetry. The reflectivities (and transmissivities) are obtained at each frequency by solving the wave propagation equations using a rigorous 4×4 transfer matrix method developed by Yeh in which the optical functions of each medium are described in the form of second rank (3×3) tensors. In order to obtain optical tensors for materials not readily available in single crystal form, a method has been developed to evaluate tensor elements from the complex scalar optical functions (n) obtained from the isotropic material with the limitations that the molecular excitations are well characterized and obey photon–dipole selection rules.This method is intended primarily for infrared vibrational spectroscopy and involves quantitative decomposition of the isotropic imaginary optical function (k) spectrum into a sum of contributions from fundamental modes, the assignment of a direction in molecular coordinates to the transition dipole matrix elements for each mode, the appropriate scaling of each k vector component in surface coordinates according to a selected surface orientation of the molecule to give a diagonal im(n) tensor, and the calculation of the real(n) spectrum tensor elements by the Kramers–Kronig transformation. Tensors for other surface orientations are generated by an appropriate rotation matrix operation. To test the viability of this approach, three sets of experimentally derived infrared spectra of oriented monolayer assemblies on quite distinctively different substrates were chosen for simulation: (1) n‐alkanethiols self‐ assembled onto gold, (2) n‐alkanoic acid salt Langmuir–Blodgett (LB) monolayers on carbon, and (3) n‐alkanoic acid salt LB monolayers on silica glass. The formalism developed was used to simulate the spectral response and to derive structural features of the monolayers. Good agreement was found where comparisons with independent studies could be made and, in general, the method appears quite useful for structural studies of highly organized thin films.

Corner Transfer Matrix Renormalization Group Method

Abstract: We propose a new fast numerical renormalization group method – the corner transfer matrix renormalization group (CTMRG) method – which is based on a unified scheme involving Baxter's corner transfer matrix method and White's density matrix renormalization group method. The key point is that a product of four corner transfer matrices coincides with the density matrix. We formulate CTMRG as a renormalization group for 2D classical models.

Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field

Abstract: Using the algebraic Bethe ansatz method, and the solution of the quantum inverse scattering problem for local spins, we obtain multiple integral representations of the $n$-point correlation functions of the XXZ Heisenberg spin-$1 \over 2$ chain in a constant magnetic field. For zero magnetic field, this result agrees, in both the massless and massive (anti-ferromagnetic) regimes, with the one obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra ${\cal U}_q (\hat{sl}_2)$ together with the corner transfer matrix approach (massive regime).

Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction

Abstract: An extension of the projected entangled-pair states (PEPS) algorithm to infinite systems, known as the iPEPS algorithm, was recently proposed to compute the ground state of quantum systems on an infinite two dimensional lattice. Here we investigate a modification of the iPEPS algorithm, where the environment is computed using the corner transfer matrix renormalization group (CTMRG) method, instead of using one-dimensional transfer matrix methods as in the original proposal. We describe a variant of the CTMRG that addresses different directions of the lattice independently, and use it combined with imaginary time evolution to compute the ground state of the two dimensional quantum Ising model. Near criticality, the modified iPEPS algorithm is seen to provide a better estimation of the order parameter and correlators.