Direct Diagonalization Research Articles

Several Ways to Solve the Jaynes-Cummings Model

We go through the several ways that the Jaynes-Cummings model, a cornerstone in the study of light-matter interactions, may be solved. We emphasize two not well known methods (one based on the London phase operator and the other one on the direct diagonalization of the Hamiltonian) considering that they may be of help for solving other systems like the interaction of light with a moving mirror, ion-laser interactions, etc.

Finite-size corrections to the spectrum of regular random graphs: An analytical solution.

We develop a thorough analytical study of the O(1/N) correction to the spectrum of regular random graphs with N→∞ nodes. The finite-size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the O(1/N) correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.

Tunneling splitting in double-proton transfer: direct diagonalization results for porphycene.

Zero-point and excited level splittings due to double-proton tunneling are calculated for porphycene and the results are compared with experiment. The calculation makes use of a multidimensional imaginary-mode Hamiltonian, diagonalized directly by an effective reduction of its dimensionality. Porphycene has a complex potential energy surface with nine stationary configurations that allow a variety of tunneling paths, many of which include classically accessible regions. A symmetry-based approach is used to show that the zero-point level, although located above the cis minimum, corresponds to concerted tunneling along a direct trans - trans path; a corresponding cis - cis path is predicted at higher energy. This supports the conclusion of a previous paper [Z. Smedarchina, W. Siebrand, and A. Fernández-Ramos, J. Chem. Phys. 127, 174513 (2007)] based on the instanton approach to a model Hamiltonian of correlated double-proton transfer. A multidimensional tunneling Hamiltonian is then generated, based on a double-minimum potential along the coordinate of concerted proton motion, which is newly evaluated at the RI-CC2/cc-pVTZ level of theory. To make it suitable for diagonalization, its dimensionality is reduced by treating fast weakly coupled modes in the adiabatic approximation. This results in a coordinate-dependent mass of tunneling, which is included in a unique Hermitian form into the kinetic energy operator. The reduced Hamiltonian contains three symmetric and one antisymmetric mode coupled to the tunneling mode and is diagonalized by a modified Jacobi-Davidson algorithm implemented in the Jadamilu software for sparse matrices. The results are in satisfactory agreement with the observed splitting of the zero-point level and several vibrational fundamentals after a partial reassignment, imposed by recently derived selection rules. They also agree well with instanton calculations based on the same Hamiltonian.

Frequency-dependent attenuation and elasticity in unconsolidated earth materials: Effect of damping

We used the discrete element method (DEM) to understand the underlying attenuation mechanism in granular media, with special applicability to the measurements of the so-called effective mass developed earlier. We considered that the particles interacted via Hertz-Mindlin elastic contact forces and that the damping was describable as a force proportional to the velocity difference of contacting grains. We determined the behavior of the complex-valued normal-mode frequencies using (1) DEM, (2) direct diagonalization of the relevant matrix, and (3) a numerical search for the zeros of the relevant determinant. All three methods were in strong agreement with each other. The real and the imaginary parts of each normal-mode frequency characterized the elastic and the dissipative properties, respectively, of the granular medium. We found that as the interparticle damping [Formula: see text] increased, the normal modes exhibited nearly circular trajectories in the complex frequency plane, and that for a given value of [Formula: see text], they all lay on or near a circle of radius [Formula: see text] centered on the point [Formula: see text] in the complex plane, where [Formula: see text]. We found that each normal mode became critically damped at a value of the damping parameter [Formula: see text], where [Formula: see text] was the (real-valued) frequency when there was no damping. The strong indication was that these conclusions carried over to the properties of real granular media whose dissipation was dominated by the relative motion of contacting grains. For example, P- or S-waves in unconsolidated dry sediments can be expected to become overdamped beyond a critical frequency, depending upon the strength of the intergranular damping constant.

The control of the nonlinear optical response of semiconductor quantum dots

In the present work, we investigate the nonlinear optical properties emerged from excitonic features in an experimentally realized spherical parabolic semiconductor quantum dot (QD). The lowest exciton states together with relevant wave functions are calculated through the expansion method with direct matrix diagonalization method within the effective mass approximation. The effect of the size of QD and confinement potential in exciton state is studied in details. Results show that with increasing the size of the QD the energy of exciton decreases because of decreasing of the effect of coulomb potential. Using the compact density matrix formalism second order nonlinear optical rectification (χ(2)) are obtained. By means of the applied electric and magnetic field we manipulate the exciton states and control the nonlinear optical response in a typical GaAs, InAs, CdSe QDs. Our model system presents a way to control the performance of excitonic optoelectronic devices based on semiconductor nanostructures.

Envelope function method for electrons in slowly-varying inhomogeneously deformed crystals

We develop a new envelope-function formalism to describe electrons in slowly-varying inhomogeneously strained semiconductor crystals. A coordinate transformation is used to map a deformed crystal back to a geometrically undeformed structure with deformed crystal potential. The single-particle Schrödinger equation is solved in the undeformed coordinates using envelope function expansion, wherein electronic wavefunctions are written in terms of strain-parametrized Bloch functions modulated by slowly varying envelope functions. Adopting a local approximation of the electronic structure, the unknown crystal potential in the Schrödinger equation can be replaced by the strain-parametrized Bloch functions and the associated strain-parametrized energy eigenvalues, which can be constructed from unit-cell level ab initio or semi-empirical calculations of homogeneously deformed crystals at a chosen crystal momentum. The Schrödinger equation is then transformed into a coupled differential equation for the envelope functions and solved as a generalized matrix eigenvector problem. As the envelope functions are slowly varying, a coarse spatial or Fourier grid can be used to represent the envelope functions, enabling the method to treat relatively large systems. We demonstrate the effectiveness of this method using a one-dimensional model, where we show that the method can achieve high accuracy in the calculation of energy eigenstates with relatively low cost compared to direct diagonalization of the Hamiltonian. We further derive envelope function equations that allow the method to be used empirically, in which case certain parameters in the envelope function equations will be fitted to experimental data.

Quantum correlations and spatial localization in one-dimensional ultracold bosonic mixtures

We present the complete phase diagram for one-dimensional binary mixtures of bosonic ultracold atomic gases in a harmonic trap. We obtain exact results with direct numerical diagonalization for a small number of atoms, which permits us to quantify quantum many-body correlations. The quantum Monte Carlo method is used to calculate energies and density profiles for larger system sizes. We study the system properties for a wide range of interaction parameters. For the extreme values of these parameters, different correlation limits can be identified, where the correlations are either weak or strong. We investigate in detail how the correlations evolve between the limits. For balanced mixtures in the number of atoms in each species, the transition between the different limits involves sophisticated changes in the one- and two-body correlations. Particularly, we quantify the entanglement between the two components by means of the von Neumann entropy. We show that the limits equally exist when the number of atoms is increased for balanced mixtures. Also, the changes in the correlations along the transitions among these limits are qualitatively similar. We also show that, for imbalanced mixtures, the same limits with similar transitions exist. Finally, for strongly imbalanced systems, only two limits survive, i.e., a miscible limit and a phase-separated one, resembling those expected with a mean-field approach.

Equilibrium currents in a Corbino graphene ring

We address the description of a graphene Corbino disk in the context of a tight binding approach that includes both kinetic and Rashba spin-orbit coupling due to an external out-of-plane electric field. Persistent equilibrium currents are induced by an external magnetic field breaking time reversal symmetry. By direct diagonalization, we compute the spectrum and focus on the dispersion near the $K$ points at the Fermi level. The dispersion keenly reproduces that of a continuum model in spite of the complexity of the boundary conditions. We validate the assumptions of the continuum model in terms of predominant zig-zag boundaries conditions and weak sub-band coupling. The wave functions displaying the lowest transverse modes are obtained, showing the predominance of edge states with charge density at the zig-zag edges. The persistent charge currents, nevertheless, do not follow the traditional argument of current cancellation from levels below the Fermi level, and thus they depart in the tight-binding from those found in the continuum model.

Optical rectification coefficient of a two-dimensional parabolic quantum dot: Effects of hydrogenic impurity, external fields, hydrostatic pressure and temperature

Simultaneous effects of hydrogenic impurity, hydrostatic pressure, temperature and external electric and magnetic fields on the intersubband optical rectification coefficient of a two-dimensional parabolic quantum dot are studied. Energy eigenvalues and eigenvectors are calculated using the direct matrix diagonalization method and optical rectification coefficient is obtained via the compact density matrix approach. The results indicate that the optical rectification coefficient is affected by the hydrogenic impurity, hydrostatic pressure, temperature and external fields.

Effects of Coulomb Impurity in Semiconductor Nanowire

ABSTRACTWe present calculation of electronic structure of impurity in nanowire. Ionization energy of impurities are calculated in dependence on nanowire radius. Direct Hamiltonian matrix diagonalization method with the physically reasonable approximate potential is employed for finding the exact solution of Schrödinger equation in the effective-mass approximation. It is shown that shallow donors are strongly influences by space confinement, which is expressed in sharp increase of ionization energy. Calculations show that effect of space confinement on deep impurities is less pronounced. The obtained results give hope that by selecting optimal value of nanowire radius compensation processes can be suppressed.

Low-temperature hopping dynamics with energy disorder: Renormalization group approach

We formulate a real-space renormalization group (RG) approach for efficient numerical analysis of the low-temperature hopping dynamics in energy-disordered lattices. The approach explicitly relies on the time-scale separation of the trapping/escape dynamics. This time-scale separation allows to treat the hopping dynamics as a hierarchical process, RG step being a transformation between the levels of the hierarchy. We apply the proposed RG approach to analyze hopping dynamics in one- and two-dimensional lattices with varying degrees of energy disorder, and find the approach to be accurate at low temperatures and computationally much faster than the brute-force direct diagonalization. Applicability criteria of the proposed approach with respect to the time-scale separation and the maximum number of hierarchy levels are formulated. RG flows of energy distribution and pre-exponential factors of the Miller-Abrahams model are analyzed.

Recent development of Monte Carlo shell model and its application to no-core calculations

One of the major challenges in nuclear theory is to reproduce and to predict nuclear structure from ab initio calculations with realistic nuclear forces. As the current limitation of direct diagonalization of Hamiltonian matrices by Lanczos iteration method is around the order of matrix dimensionality 1010 in shell-model calculations, it is difficult to access heavier nuclei beyond the p shell with sufficiently large basis spaces. It is possible to overcome this difficulty by utilizing efficient approximate methods to reproduce full ab initio solutions with good precision and quantified uncertainties. Following the major success of the Monte Carlo shell model (MCSM) with an assumed inert core in the sd- and pf-shell regions and also by recent developments in the MCSM algorithm, the no-core MCSM is expected to be one of the most powerful tools to meet these conditions. We have performed benchmark calculations in the p-shell region. Results of energies are compared with those in the full configuration interaction and no-core full configuration methods. These are found to be consistent with each other within quoted uncertainties when they could be quantified. We also compare and discuss the radial density of the helium-4 ground state extracted from the MCSM and FCI many-body wave functions.

Kowalevski top in quantum mechanics

The quantum mechanical Kowalevski top is studied by the direct diagonalization of the Hamiltonian. The spectra show different behaviors depending on the region divided by the bifurcation sets of the classical invariant tori. Some of these spectra are nearly degenerate due to the multiplicity of the invariant tori. The Kowalevski top has several symmetries and symmetry quantum numbers can be assigned to the eigenstates. We have also carried out the semiclassical quantization of the Kowalevski top by the EBK formulation. It is found that the semiclassical spectra are close to the exact values, thus the eigenstates can be also labeled by the integer quantum numbers. The symmetries of the system are shown to have close relations with the semiclassical quantum numbers and the near-degeneracy of the spectra.

GWcalculations using the spectral decomposition of the dielectric matrix: Verification, validation, and comparison of methods

In a recent paper [Nguyen et al., Phys. Rev. B 85, 081101(R) (2012)] we presented an approach to evaluate quasiparticle energies based on the spectral decomposition of the static dielectric matrix. This method does not require the calculation of unoccupied electronic states or the direct diagonalization of large dielectric matrices, and it avoids the use of plasmon-pole models. The numerical accuracy of the approach is controlled by a single parameter, i.e., the number of eigenvectors used in the spectral decomposition of the dielectric matrix. Here we present a comprehensive validation of the method, encompassing calculations of ionization potentials and electron affinities of various molecules and of band gaps for several crystalline and disordered semiconductors. We demonstrate the efficiency of our approach by carrying out $GW$ calculations for systems with several hundred valence electrons.

External electric and magnetic field effects on the optical absorption coefficients and refractive index changes of a hydrogenic impurity confined in a cylindrical quantum wire with convex bottom

In this paper, we have studied the simultaneous effects of external electric and magnetic fields on the optical absorption coefficients and refractive index changes of a hydrogenic impurity confined in a cylindrical quantum wire with convex bottom. Energy eigenvalues and eigenvectors are calculated using the direct matrix diagonalization method and optical properties are obtained using the compact density matrix approach. Our results indicate that, the wire radius, convexity parameter, hydrogenic impurity and external electric and magnetic fields have a great influence on the linear and the third-order nonlinear optical absorption coefficients as well as the refractive index changes of the system.

Description of two-particle transfer in superfluid systems

Exact results of pair transfer probabilities for the Richardson model with equidistant or random level spacing are presented. The results are then compared either to particle-particle random-phase approximation (ppRPA) in the normal phase or quasiparticle random-phase approximation (QRPA) in the superfluid phase. We show that both ppRPA and QRPA are globally well reproducing the exact case although some differences are seen in the superfluid case. In particular, the QRPA overestimates the pair transfer probabilities to excited states in the vicinity of the normal-superfluid phase transition, which might explain the difficulty in detecting collective pairing phenomena as, for example, the giant pairing vibration. The shortcoming of QRPA can be traced back to the breaking of particle number that is used to incorporate pairing. A method based on direct diagonalization of the Hamiltonian in the space of two quasiparticles projected onto good particle number is shown to improve the description of pair transfer probabilities in superfluid systems.

Direct Diagonalisation of the Heisenberg Hamiltonian for a Magnetic Ring within the Two-Deviation Sector by Means of the Chebyshev Polynomials

The eigenproblem for the Heisenberg Hamiltonian for a ring of N nodes with the spin 1/2 and isotropic interaction, solved by the famous Bethe Ansatz, is reconsidered here for the special case of two reversed spins in an another, independent way. In particular, the derivation does not involve the hypothesis of strings. The exact solution for the eigenergy is derived with the use of Chebyshev polynomials, which reproduce the characteristic polynomial of the Hamiltonian. A comparison with the Bethe Ansatz solution is realised as the so called “Inverse Bethe Ansatz” (IBA), which consists in derivation ex post the original Bethe parameters (pseudomomenta, spectral parameters etc.) from known quasi-momentum and energy. The departures from the hypothesis of strings, associated with the change of bound states to scattered ones for odd quasi-momenta and sufficiently large N , accounted by Essler at al., are adequately described in terms of the trigonometric/hyperbolic regime for Chebyshev polynomials.

Linear and nonlinear optical properties of a hydrogenic impurity confined in a two-dimensional quantum dot: Effects of hydrostatic pressure, external electric and magnetic fields

In this work, combined effects of external electric and magnetic fields, and hydrostatic pressure on the refractive index changes and optical absorption coefficients of a hydrogenic impurity confined in a two-dimensional parabolic quantum dot are studied. Energy eigenvalues and eigenvectors are calculated using the direct matrix diagonalization method and optical properties are obtained using the compact density matrix approach. It is found that the confinement potential strength, hydrogenic impurity, hydrostatic pressure, external electric and magnetic fields and the tilt angle θ considerably change the transition energy between the subbands and dipole moment matrix elements. Therefore, these parameters have a great influence on the linear and the third-order nonlinear optical absorption coefficients as well as the refractive index changes of the system.

A high-performance Fortran code to calculate spin- and parity-dependent nuclear level densities

A high-performance Fortran code is developed to calculate the spin- and parity-dependent shell model nuclear level densities. The algorithm is based on the extension of methods of statistical spectroscopy and implies exact calculation of the first and second Hamiltonian moments for different configurations at fixed spin and parity. The proton–neutron formalism is used. We have applied the method for calculating the level densities for a set of nuclei in the sd-, pf-, and pf+g9/2- model spaces. Examples of the calculations for 28Si (in the sd-model space) and 64Ge (in the pf+g9/2-model space) are presented. To illustrate the power of the method we estimate the ground state energy of 64Ge in the larger model space pf+g9/2, which is not accessible to direct shell model diagonalization due to the prohibitively large dimension, by comparing with the nuclear level densities at low excitation energy calculated in the smaller model space pf. Program summaryProgram title: MMCatalogue identifier: AENM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 193181No. of bytes in distributed program, including test data, etc.: 1298585Distribution format: tar.gzProgramming language: Fortran 90, MPI.Computer: Any architecture with a Fortran 90 compiler and MPI.Operating system: Linux.RAM: Proportional to the system size, in our examples, up to 75MbClassification: 17.15.External routines: MPICH2 (http://www.mcs.anl.gov/research/projects/mpich2/)Nature of problem:Calculating of the spin- and parity-dependent nuclear level density.Solution method:The algorithm implies exact calculation of the first and second Hamiltonian moments for different configurations at fixed spin and parity. The code is parallelized using the Message Passing Interface and a master-slaves dynamical load-balancing approach.Restrictions:The program uses two-body interaction in a restricted single-level basis. For example, GXPF1A in the pf-valence space.Running time:Depends on the system size and the number of processors used (from 1 min to several hours).

Rovibrational Dynamics of RbCs on its Lowest 1,3Σ+ Potential Curves Calculated by Coupled Cluster Method with All-Electron Basis Set

Relativistic ab initio potential curves of RbCs lowest (1,3)Σ(+) states are calculated by diagonalizing the Douglas-Kroll-Hess Hamiltonian as implemented in Gaussian09 suite of programs. The ab initio calculations are performed at the CCSD(T) level with UGBS1P+ basis set, a huge all-electron basis set. The rovibrational eigenenergies and eigenfunctions on the lowest (1,3)Σ(+) ab initio potential curves are calculated by direct diagonalization of molecular Hamiltonian in a Fourier grid discrete variable representation. The results agree well with available experimental and theoretical work and the accuracy of theoretical descriptions of RbCs are increased, which is expected to be a good reference for further investigations.