Transfer Matrix Technique Research Articles

Anderson transition of two-dimensional spinful electrons in the Gaussian unitary ensemble

We study the Anderson localization of disordered two-dimensional electron gases (2DEGs) on a square lattice subject to a perpendicular magnetic field $B$, random scalar potentials, and Rashba spin-orbit interactions. Our focus is on the weak magnetic field region, motivated by the intriguing question of how those extended states, existing in the absence of the magnetic field $(B=0)$ when the 2DEGs are the Gaussian symplectic ensemble, change with $B$. Using highly accurate numerical procedures based on the transfer matrix technique and the level statistics, we found that a metallic phase exists at weak magnetic fields, in contrast to the predictions of the one-parameter scaling theory that all states are localized at weak fields except at zero field, and the metallic phase evolves continuously into those at strong magnetic fields. A schematic phase diagram drawn in the field-energy plane elucidates the occurrence and evolution of extended states.

Massless AdS2 scattering and Bethe ansatz

We first analyse the integrable scattering theory describing the massless excitations of AdS2 × S2 × T6 superstrings in the relativistic limit. The matrix part of the S-matrix is obtained in the BMN limit from the conjectured exact expression, and compared to known S-matrices with mathcal{N}=1 supersymmetry in 1 + 1 dimensions. A dressing factor, yet unknown for the complete theory, is here constructed based on relativistic crossing symmetry. We derive a Bethe-ansatz condition by employing a transfer-matrix technique based on the so-called free-fermion condition. This is known to overcome the problem of lack of a reference state. We then generalise the method to the massless non-relativistic case, and compare the resulting Bethe-ansatz condition with a simple massless limit of the one conjectured by Sorokin, Tseytlin, Wulff and Zarembo.

Ground-state phase diagram, fermionic entanglement and kinetically-induced frustration in a hybrid ladder with localized spins and mobile electrons

We introduce an exactly solvable hybrid spin-ladder model containing localized nodal Ising spins and interstitial mobile electrons, which are allowed to perform a quantum-mechanical hopping between the ladder’s legs. The quantum-mechanical hopping process induces an antiferromagnetic coupling between the ladder’s legs that competes with a direct exchange coupling of the nodal spins. The model is exactly mapped onto the Ising spin ladder with temperature-dependent two- and four-spin interactions, which is subsequently solved using the transfer-matrix technique. We report the ground-state phase diagram and compute the fermionic concurrence to characterize the quantum entanglement between the pair of interstitial mobile electrons. We further provide a detailed analysis of the local spin ordering including the pair and four-spin correlation functions around an elementary plaquette, as well as, the local ordering diagrams. It is shown that a complex sequence of distinct local orderings and frustrated correlations takes place when the model parameters drive the investigated system close to a zero-temperature triple coexistence point.

Thermal entanglement in a spin-1/2 Ising-XYZ distorted diamond chain with the second-neighbor interaction between nodal Ising spins

We consider a spin-1/2 Ising-XYZ distorted diamond chain with the XYZ interaction between the interstitial Heisenberg dimers, the nearest-neighbor Ising coupling between the nodal and interstitial spins, respectively, and the second-neighbor Ising coupling between the nodal spins. The ground-state phase diagram of the spin-1/2 Ising-XYZ distorted diamond chain exhibits several intriguing phases due to the XY anisotropy and the second-neighbor interaction, whereas the model can be exactly solved using the transfer-matrix technique. The quantum entanglement within the Heisenberg spin dimers is studied through a bipartite measure concurrence, which is calculated from a relevant reduced density operator. The concurrence may either show a standard thermal dependence with a monotonous decline with increasing temperature or a more peculiar thermal dependence accompanied with reentrant behavior of the concurrence. Based in the present model, it is conjectured that the bipartite entanglement between the interstitial Heisenberg spin pairs in the natural mineral azurite is quite insensitive to the applied magnetic field and it persists up to approximately 30 K.

Vibration analysis and development of an ultrasonic elliptical vibration tool based on a portal frame structure

An ultrasonic elliptical vibration tool utilizing the coupled resonant vibration modes is the key component in the elliptical vibration cutting/texturing process, which has been successfully applied to ultra-precision machining and surface texturing. In this paper, an analytical approach is proposed to analyze the resonant frequency and mode shapes of the ultrasonic elliptical vibration tool. A new design of ultrasonic elliptical vibration tool based on a portal frame structure is presented and analyzed using the proposed model. The model assumes Euler-Bernoulli beams and utilizes the transfer matrix technique to reduce the order of the system to only six variables. The model can be utilized to provide a systematic approach for an optimal design and be extended to dynamic analysis of the tool for study of machine-tool dynamics. Finite element simulation results as well as experimental data based on the prototype design are presented to verify the model. An application of the proposed tool is also demonstrated in machining micro/nano-structured surfaces for structural coloration.

Optical selection rules of zigzag graphene nanoribbons

We present an analytical tight-binding theory of the optical properties of graphene nanoribbons with zigzag edges. Applying the transfer matrix technique to the nearest-neighbor tight-binding Hamiltonian, we derive analytical expressions for electron wave functions and optical transition matrix elements for incident light polarized along the structure axis. It follows from the obtained results that optical selection rules result from the wave function parity factor ${(\ensuremath{-}1)}^{J}$, where $J$ is the band number. These selection rules are that $\mathrm{\ensuremath{\Delta}}J$ is odd for transitions between valence and conduction subbands and that $\mathrm{\ensuremath{\Delta}}J$ is even for transitions between only valence (conduction) subbands. Although these selection rules are different from those in armchair carbon nanotubes, there is a hidden correlation between absorption spectra of the two structures that should allow one to use them interchangeably in some applications. The correlation originates from the fact that van Hove singularities in the tubes are centered between those in the ribbons if the ribbon width is about a half of the tubes circumference. The analysis of the matrix elements dependence on the electron wave vector for narrow ribbons shows a smooth nonsingular behavior at the Dirac points and the points where the bulk states meet the edge states.

Enhancement of optical gain by controlling waveguide modes in semipolar InGaN quantum well laser diodes

Waveguide modes in semipolar InGaN quantum‐well (QW) laser diodes (LDs) with cleaved‐facet cavity mirrors have been theoretically investigated using a 4 × 4 transfer matrix technique. It is shown that the waveguide mode can be controlled by changing the refractive‐index contrast in the waveguide structures, although previous studies suggest that the waveguide modes are either ordinary or extraordinary due to the birefringence of the material itself, and that optical gain cannot be fully utilized for lasing because the polarization direction which determines optical gain in the devices does not coincide with the polarization direction in the waveguide mode. This behavior can be explained based on the competition between the layer‐structural effect and the material birefringence. From the results of our calculations, it is concluded that the utilization of a high refractive‐index‐contrast waveguide structure is effective in enhancing the optical gain in semipolar LDs with cleaved‐facet cavity mirrors.

Numerical study on mode curvature for damage detection of a drilling riser using transfer matrix technique

In order to prevent marine pipeline accidents caused by corrosion defect, crack or loose contact, the technique of non-destructive testing or examination (NDT or NDE) becomes great important in the flied of structural health monitoring (SHM) in ocean engineering. The objective of this study is to propose a numerical method to localize and size the structure damage of a drilling riser. One theoretical development is a simple and effective algorithm used for ocean risers’ mode extraction and damage detection based on the transfer matrix technique. The merits of the developed method are that damage of risers with different boundary conditions and variable cross-sections can be localized and sized. Additionally, the relationships between damage locations and modes are discussed by comparing mode shape difference, mode curvature and mode curvature difference. Numerical results show that mode curvature values of damaged elements have little influence on those of intact elements. Based on this fact, a damage index method is developed to evaluate damage severities and explore the influence of single and multiple damage severities on damage indexes. More importantly, the outcomes are verified to be accurate enough to indicate the feasibility and reliability of the numerical method by the standard FEM package.

Magnetization Plateaus and Thermal Entanglement of Spin Systems

The geometrically magnetic frustrations and quantum thermal entanglement of antiferromagnetic metal-containing compounds are considered on a diamond chain. We researched the magnetic and thermal properties of the symmetric Hubbard dimers with delocalized interstitial spins and the quantum entanglement states. It is presented magnetization plateaus and negativity in spin-1 Ising-Heisenberg model using transfer matrix technique. Applying the dynamic system approach we study the magnetic curves, Lyapunov exponents and superstable point in the two-dimensional mapping for the partition function of spin-1 classical and Ising-Heisenberg models at T → 0 on a diamond chain.

The modification of low-temperature magnetic properties of the Fe-based double-zigzag single-chain magnet under irradiation

An exact solution for a generalized Ising model describing single-chain magnet [FeIII(Tp*)(CN)3]2FeII(bpmh)}·2H2O with alternating high-spin and low-spin iron ions and double-zigzag magnetic topology has been calculated making use of the transfer-matrix technique. The introduction of a statistical ensemble taking into account the presence of iron ions with high-spin (HS) and low-spin (LS) states made it possible to describe the modification of magnetic susceptibility under optical irradiation. It has been shown that joint implementation of iron ions with photoinduced magnetic states and non-magnetic intersite repulsion ions of these states is caused by the difference of the ionic radii of HSFeII and LSFeII leads to the quantum phase transitions in the system.

Octonacci photonic crystals with negative refraction index materials

We investigate the optical transmission spectra for s-polarized (TE) and p-polarized (TM) waves in one-dimensional photonic quasicrystals on a quasiperiodic multilayer structure made up by alternate layers of SiO2 and metamaterials, organized by following the Octonacci sequence. Maxwell's equations and the transfer-matrix technique are used to derive the transmission spectra for the propagation of normally and obliquely incident optical fields. We assume Drude-Lorentz-type dispersive response for the dielectric permittivity and magnetic permeability of the metamaterials. For normally incident waves, we observe that the spectra does not have self-similar behavior or mirror symmetry and it also features the absence of optical band gap. Also for normally incident waves, we show regions of full transmittance when the incident angle θC = 0° in a particular frequency range.

Effect of intersite repulsion on the correlation functions and thermodynamics of an Ising chain with annealed magnetic disorder

An exact solution for a model describing the equilibrium behavior of an ensemble of Ising chains with nonmagnetic intersite repulsion of nearest neighbors and an equilibrium distribution of nonmagnetic impurities is obtained using the transfer matrix technique. The possibility of exciting quantum phase transitions using the intersite repulsion parameter in a system is demonstrated. Proximity to the critical points of these transitions has a substantial effect on the temperature dependence of a system’s magnetic susceptibility.

Ultrasonic waves in an orthorhombic porous piezo-thermoelastic laminated structure immersed in a fluid

Reflection and transmission of plane elastic waves in porous piezo-thermoelastic laminated plate immersed in a fluid has been studied. A theoretical model is derived which is based on transfer matrix method. The layered structure is considered to be consisting of ‘n’ number of layers of porous piezo-thermoelastic materials immersed in a fluid. All the layers are considered to be 2 mm PPTE materials. The formal solution for the mechanical displacements, electric potential, mechanical stresses, electric displacements, temperature changes and temperature gradient are derived. The transfer matrix technique is used to study the layered materials. The closed form expressions of the elements of transfer matrix are derived. The analytical expressions for the reflection coefficient and transmission coefficient are derived. The effects of frequency, angle of incidence, number of layers, layer’s thickness and porosity on the reflection coefficient and transmission coefficient, reflection loss and transmission loss are investigated numerically for different configurations. Particular cases of the present study have also been obtained for the verification of the results.

Estimate of the Critical Exponent of the Anderson Transition in the Three and Four-Dimensional Unitary Universality Classes

Disordered non-interacting systems are classified into ten symmetry classes, with the unitary class being the most fundamental. The three and four dimensional unitary universality classes are attracting renewed interest because of their relation to three dimensional Weyl semi-metals and four dimensional topological insulators. Determining the critical exponent of the correlation/localistion length for the Anderson transition in these classes is important both theoretically and experimentally. Using the transfer matrix technique, we report numerical estimations of the critical exponent in a U(1) model in three and four dimensions.

Absence of localization in disordered two-dimensional electron gas at weak magnetic field and strong spin-orbit coupling

The one-parameter scaling theory of localization predicts that all states in a disordered two-dimensional system with broken time reversal symmetry are localized even in the presence of strong spin-orbit coupling. While at constant strong magnetic fields this paradigm fails (recall the quantum Hall effect), it is believed to hold at weak magnetic fields. Here we explore the nature of quantum states at weak magnetic field and strongly fluctuating spin-orbit coupling, employing highly accurate numerical procedure based on level spacing distribution and transfer matrix technique combined with one parameter finite-size scaling hypothesis. Remarkably, the metallic phase, (known to exist at zero magnetic field), persists also at finite (albeit weak) magnetic fields, and eventually crosses over into a critical phase, which has already been confirmed at high magnetic fields. A schematic phase diagram drawn in the energy-magnetic field plane elucidates the occurrence of localized, metallic and critical phases. In addition, it is shown that nearest-level statistics is determined solely by the symmetry parameter β and follows the Wigner surmise irrespective of whether states are metallic or critical.

Sol-gel fabrication of one-dimensional photonic crystals with predicted transmission spectra

One-dimensional multilayer structures of periodically alternating low refractive index (silica) and high refractive index (titania) materials have been deposited by sol-gel spincoating. Experimental spectra of the structures are in agreement with spectra calculated by transfer matrix technique. As an example, theoretical and experimental spectra with a stop band corresponding 600 nm-reflection are shown.

Three-dimensional analysis of freely vibrating multilayered piezoelectric plates through adaptive global piecewise-smooth functions

In this article, freely vibrating multilayered piezoelectric plates are analyzed through a set of adaptive global piecewise-smooth functions along with governing differential equations and associated boundary conditions, which are consistently derived from the classical theorem of virtual displacements. The analysis demonstrates the capability of the adaptive global piecewise-smooth functions to treat any multilayered plate as if it were made up of a single layer even in the presence of multiphysics analyses such as piezoelectric layers. The relevant model is essentially two-dimensional because it is based on an expansion through the thickness of the plate aimed at modeling a three-dimensional dynamical behavior. In order to demonstrate the effectiveness of the model, all the results are compared to exact three-dimensional results; these latter are extracted through a three-dimensional model based on a transfer matrix technique whose numerical stability is achieved using scaled electric potentials. The exact graphical results are herein illustrated, thus showing both the effectiveness of using weighted electric potentials and the capability of adaptive global piecewise-smooth functions to converge at exact results through a minimum computational effort.

Structural parameters and their effects on the electronic transport properties in aperiodic superlattice profile

The aperiodic GaAs/AlxGa1−xAs superlattices (SL) with trimer disorder have been studied in this paper. The transfer-matrix technique and the exact Airy function formalism have been used to determine the miniband structure, the transmission coefficient, the resonance energy and resonant tunneling time (RTT). Although the disorder localizes the states on average, our numerical calculations show that the localization length of the states becomes more extended when the disorder is correlated (trimer case). We have also found that the RTT is of the order of several femtoseconds.

Reprint of : Connection between wave transport through disordered 1D waveguides and energy density inside the sample: A maximum-entropy approach

We study the average energy – or particle – density of waves inside disordered 1D multiply-scattering media. We extend the transfer-matrix technique that was used in the past for the calculation of the intensity beyond the sample to study the intensity in the interior of the sample by considering the transfer matrices of the two segments that form the entire waveguide. The statistical properties of the two disordered segments are found using a maximum-entropy ansatz subject to appropriate constraints. The theoretical expressions are shown to be in excellent agreement with 1D transfer-matrix simulations.

Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions

In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-) or many-body localized phase. This approximation, which consists in summing over the amplitudes of only the shortest paths in the locator expansion, is known to over-estimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is $0.9\%$ off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder $h_c=4.0\pm 0.3$ is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent $\nu=1$. In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium.