Hamiltonian Matrix Research Articles

Equivalent model for band-to-band tunneling simulation of direct-gap III–V semiconductor nanowires

We have proposed an equivalent model for use in the simulation of tunnel devices. The equivalent model reproduces not only the real band-structure but also the complex band-structure of a target system. It has the features of adjustable spatial resolution and high spatial symmetry. We have tested it by calculating the band-to-band transmission functions and the maximum current of direct-gap III–V semiconductor nanowires, and confirmed that it correctly reproduces the transport characteristics of the target system. The equivalent model enables us to substantially reduce the Hamiltonian matrix size, leading to speed-up the quantum transport simulation of tunnel devices, such as tunnel field-effect transistors.

Complex ground-state and excitation energies in coupled-cluster theory

Since in coupled-cluster (CC) theory, ground-state and excitation energies are eigenvalues of a non-Hermitian matrix, these energies can in principle take on complex values. In this paper, we discuss the appearance of complex energy values in CC calculations from a mathematical perspective. We analyse the behaviour of the eigenvalues of Hermitian matrices that are perturbed (in a non-Hermitian manner) by a real parameter. Based on these results, we show that for CC calculations with real-valued Hamiltonian matrices the ground-state energy generally takes a real value. Furthermore, we show that in the case of real-valued Hamiltonian matrices complex excitation energies only occur in the context of conical intersections. In such a case, unphysical consequences are encountered such as a wrong dimension of the intersection seam, large numerical deviations from full configuration-interaction (FCI) results close to the conical intersection, and the square-root-like behaviour of the potential surfaces near the conical intersection. In the case of CC calculations with complex-valued Hamiltonian matrix elements, it turns out that complex energy values are to be expected for ground and excited states when no symmetry is present. We confirm the occurrence of complex energies by sample calculations using a six-state model and by CC calculations for the molecule in a strong magnetic field. We furthermore show that symmetry can prevent the occurrence of complex energy values. Lastly, we demonstrate that in case of complex Hamiltonians the real part of the complex energy value provides a very good approximation to the FCI energy.

Tinie – a software package for electronic transport through two-dimensional cavities in a magnetic field

Quantum transport has far-reaching applications in modern electronics as it enables the control of currents in nanoscale systems such as quantum dots. In this paper we introduce tinie: a state-of-the-art quantum transport simulation framework, which can efficiently perform first-principle calculations based on the Landauer-Büttiker formalism. The computational repertoire of tinie includes calculations of transmission, conductivity, and currents running through arbitrary multi-terminal two-dimensional transport devices, with additional tools that enable the computation of the local density of states. The generality of tinie ranges from wide-band approximation calculations to investigating systems subject to an external magnetic field. The future prospects of tinie include the simulation of, e.g., two-dimensional cavities, quantum dots, or molecular junctions. The package is written in Python 3.6, and its well-documented modular structure is designed with an intent to create a platform suited for continuous expansion and development. With tinie it is possible to obtain specific information about the effects of impurities and imperfections in quantum devices, particularly between ballistic and diffusive transport regimes. Program summaryProgram title:tinieCPC Library link to program files:https://doi.org/10.17632/7487cpj9hm.1Developer's repository link:https://gitlab.com/compphys-public/tinieLicensing provisions: MIT LicenseProgramming language: Python 3.6Nature of problem: Numerical calculation of the properties of a two-dimensional nanoscale electron transport system in a uniform magnetic field (zero or non-zero), specifically the currents running through the reservoirs (leads) coupled to a quantum dot (central region) and the corresponding transmission coefficients.Solution method: The problem solution is split into two stages. The first stage (tinie_prepare stage) prepares the transport system data for the main transport calculation. This data comprise Hamiltonian matrices of the uncoupled reservoirs and quantum dot regions, their respective sets of eigenfunctions and the coupling matrices between the quantum dot and the reservoirs. The second stage (tinie stage) performs the transport calculation for the given system using the embedding self-energy technique.Additional comments including restrictions and unusual features: The code is restricted to the non-interacting equilibrium transport problems.The code is modular in structure, allowing for easy extension and introduction of different reservoir/quantum dot/coupling types. Additionally, tinie is compatible with systems in a non-zero uniform magnetic field.The source code is available at https://gitlab.com/compphys-public/tinie and Python package in https://pypi.org/project/tinie/. An extensive documentation of the code functionality can be found in the README.md file accompanying the code.

Thermal fluctuations and carrier localization induced by dynamic disorder in MAPbI3 described by first-principles based tight-binding model

Halide perovskites are strongly influenced by large amplitude anharmonic lattice fluctuations at room temperature. We develop a tight-binding model for dynamically disordered ${\mathrm{MAPbI}}_{3}$ based on density functional theory calculations to calculate electronic structure for finite temperature crystal structures at the length scale of thermal disorder and carrier localization. The model predicts individual Hamiltonian matrix elements and band structures with high accuracy, owing to the inclusion of additional matrix elements and descriptors for non-Coulombic interactions. We apply this model to electronic structure at length and timescales inaccessible to first-principles methods, finding an increase in band gap, carrier mass, and the subpicosecond fluctuations in these quantities with increasing temperature as well as the onset of carrier localization in large supercells induced by thermal disorder at 300 K. We identify the length scale ${L}^{*}=5$ nm as the onset of localization in the electronic structure, associated with decreasing band edge fluctuations, increasing carrier mass, and Rashba splitting approaching zero.

Using Diffusion Monte Carlo Wave Functions to Analyze the Vibrational Spectra of H7O3+ and H9O4.

An approach for evaluating spectra from ground state probability amplitudes (GSPA) obtained from diffusion Monte Carlo (DMC) simulations is extended to improve the description of excited state energies and allow for coupling among vibrational excited states. This approach is applied to studies of the protonated water trimer and tetramer, and their deuterated analogs. These ions provide models for solvated hydronium, and analysis of these spectra provides insights into spectral signatures of proton transfer in aqueous environments. In this approach, we obtain a separable set of internal coordinates from the DMC ground state probability amplitude. A basis is then developed from products of the DMC ground state wave function and low-order polynomials in these internal coordinates. This approach provides a compact basis in which the Hamiltonian and dipole moment matrix are evaluated and used to obtain the spectrum. The resulting spectra are in good agreement with experiment and in many cases provide comparable agreement to the results obtained using much larger basis sets. In addition, the compact basis allows for interpretation of the spectral features and how they evolve with cluster size and deuteration.

Observation of topological properties of non-Hermitian crystal systems with diversified coupled resonators chains

Non-Hermiticity extends the topological phase beyond the given Hermitian structure. Whereas the phases of non-Hermitian topological systems derived from Hermitian components have been extensively explored, the topological properties of an acoustic crystal that occur purely due to non-Hermiticity require further investigation. In this letter, we describe the development of an acoustic crystal with an adjustable loss that is composed of a chain of one-dimensional, coupled acoustic resonators. Each unit cell can contain three or six resonators, which are equivalent to 3 × 3 or 6 × 6 non-Hermitian Hamiltonian matrices, respectively. The topological properties of the crystal were verified by calculating the defined topological invariant, and the states of the edge and interface of the acoustic crystal were obtained by using a practical model. We obtained the states of the edges and the interface for both odd and even numbers of resonators in each unit cell and found that the location of the inductive loss had an important effect on the topological properties. This results here can guide research on advanced wave control for sensing and communication applications.

Berry curvature and quantum metric in N -band systems: An eigenprojector approach

The eigenvalues of a parameter-dependent Hamiltonian matrix form a band structure in parameter space. In such $N$-band systems, the quantum geometric tensor (QGT), consisting of the Berry curvature and quantum metric tensors, is usually computed from numerically obtained energy eigenstates. Here, an alternative approach to the QGT based on eigenprojectors and (generalized) Bloch vectors is exposed. It offers more analytical insight than the eigenstate approach. In particular, the full QGT of each band can be obtained without computing eigenstates, using only the Hamiltonian matrix and the respective band energy. Most saliently, the well-known two-band formula for the Berry curvature in terms of the Hamiltonian vector is generalized to arbitrary $N$. The formalism is illustrated using three- and four-band multifold fermion models that have very different geometrical and topological properties despite an identical band structure. From a broader perspective, the methodology used in this work can be applied to compute any physical quantity or to study the quantum dynamics of any observable without the explicit construction of energy eigenstates.

Global symplectic Lanczos method with application to matrix exponential approximation

It is well-known that the symplectic Lanczos method is an efficient tool for computing a few eigenvalues of large and sparse Hamiltonian matrices. A variety of block Krylov subspace methods were introduced by Lopez and Simoncini to compute an approximation of $exp(M)V$ for a given large square Hamiltonian matrix $M$ and a tall and skinny matrix $V$ that preserves the geometric property of $V$. For the same purpose, in this paper, we have proposed a new method based on a global version of the symplectic Lanczos algorithm, called the global $J$-Lanczos method ($GJ$-Lanczos). To the best of our knowledge, this is probably the first adaptation of the symplectic Lanczos method in the global case. Numerical examples are given to illustrate the effectiveness of the proposed approach.

Charmonium Properties Using the Discrete Variable Representation (DVR) Method

The Schrödinger equation is solved numerically for charmonium using the discrete variable representation (DVR) method. The Hamiltonian matrix is constructed and diagonalized to obtain the eigenvalues and eigenfunctions. Using these eigenvalues and eigenfunctions, spectra and various decay widths are calculated. The obtained results are in good agreement with other numerical methods and with experiments.

Quantum Particle on Dual Weight Lattice in Weyl Alcove

Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and Neumann walls on the borders of the Weyl alcoves. The amplitudes of the particle’s propagation to neighbouring positions are determined by a complex-valued dual-weight hopping function of finite support. The discrete dual-weight Hamiltonians are obtained as the sum of specifically constructed dual-weight hopping operators. By utilising the generalised dual-weight Fourier–Weyl transforms, the solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are exactly resolved. The matrix Hamiltonians, stationary states and eigenenergies of the discrete models are exemplified for the rank two cases C2 and G2.

Exploring artificial neural networks to model interatomic and intermolecular potential energy surfaces

We demonstrate how a back-propagation artificial neural network can be trained to represent a potential energy surface (PES) in a formless manner with limited data points and exploited to predict interaction energies for configurations not included in the training set. A similar exercise is undertaken for predicting the eigenvalues and eigenvectors of a model Hamiltonian matrix that delicately depends on parameters and exhibits crossing of eigen values.

An iterative algorithm for generalized Hamiltonian solution of a class of generalized coupled Sylvester-conjugate matrix equations

In this work, we present an iterative algorithm to solve a class of generalized coupled Sylvester-conjugate matrix equations over the generalized Hamiltonian matrices. We show that if the equations are consistent, a generalized Hamiltonian solution can be obtained within finite iteration steps in the absence of round-off errors for any initial generalized Hamiltonian matrix by the proposed iterative algorithm. Furthermore, we can obtain the minimum-norm generalized Hamiltonian solution by choosing the special initial matrices. Finally, numerical examples show that the iterative algorithm is effective.

SpectrumSDT: A program for parallel calculation of coupled rotational-vibrational energies and lifetimes of bound states and scattering resonances in triatomic systems

We present SpectrumSDT – a program for calculations of energies and lifetimes of bound rotational-vibrational states below and scattering resonances above the dissociation threshold on a global potential energy surface of a triatomic system, which may include stable molecules, weekly-bound van-der-Waals complexes, and unbound atom + diatom scattering systems. Large-amplitude vibrational motion is treated explicitly using hyper-spherical coordinates. Three options for the rotational-vibrational interaction are supported: uncoupled (symmetric top rotor), partially coupled (to include interaction between several nearest states only) and full-coupled (vibrating asymmetric-top rotor). In addition to energies and lifetimes, SpectrumSDT is able to integrate ro-vibrational wave functions over the user-defined regions of potential energy surface, which helps to classify these states. In this release of the code, SpectrumSDT is limited to ABA-type molecules with wave functions that do not extend into the regions near Eckart singularities. Program summaryProgram title: SpectrumSDTCPC Library link to program files:https://doi.org/10.17632/9gftxjs4yk.1Developer's repository link:https://github.com/IgorGayday/SpectrumSDTLicensing provisions: GNU General Public License 3 (GPL)Programming language: FortranNature of problem: Calculations of energies and lifetimes of bound rotational-vibrational states below and scattering resonances above the dissociation threshold on a global potential energy surface of a triatomic system, which may include stable molecules, weekly-bound van-der-Waals complexes, and unbound atom + diatom scattering systems.Solution method: A Hamiltonian matrix is built in APH coordinates using an optimal 2D basis set, adjusted for a given problem. A complex absorbing potential (CAP) is added to define a boundary condition for calculation of scattering resonances above the dissociation threshold. The eigenstates of the Hamiltonain matrix are found using a state-of-the-art iterative eigensolver.Restrictions: The present version is restricted to ABA-molecules and wave functions that do not extend into linear and equilateral triangle configurations.Additional comments including restriction and unusual features: Probabilities and lifetimes of the wave functions can be calculated in user-defined regions on the PES, which allows to analyze their localization properties (i.e. automatic isotopomer assignment) and obtain channel-specific lifetimes for scattering resonances.

Structured strong linearizations of structured rational matrices

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational matrices have been introduced recently for computing poles, eigenvalues, eigenvectors, minimal bases and minimal indices of rational matrices. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. With a view to constructing structure-preserving linearizations of structured rational matrices, we propose a family of Fiedler-like pencils and show that the family of Fiedler-like pencils is a rich source of structure-preserving strong linearizations of structured rational matrices. We construct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrix when has the same structure. We also describe recovery of eigenvectors, minimal bases and minimal indices of from those of the linearizations of and show that the recovery is operation-free.

Efficient Method for an Approximate Treatment of the Coriolis Effect in Calculations of Quantum Dynamics and Spectroscopy, with Application to Scattering Resonances in Ozone.

A numerical approach is developed to capture the effect of rotation-vibration coupling in a practically affordable way. In this approach only a limited number of adjacent rotational components are considered to be coupled, while the couplings to other rotational components are neglected. This partially coupled (PC) approach permits to reduce the size of Hamiltonian matrix significantly, which enables the calculations of ro-vibrational states above dissociation threshold (scattering resonances) for large values of total angular momentum. This method is employed here to reveal the role of the Coriolis effect in the ozone formation reaction at room temperature, dominated by large values of total angular momentum states, on the order of J = 24 and 28. We found that, overall, the effect of ro-vibrational coupling is not minor for large J. Compared to the results of symmetric top rotor approximation, where the ro-vibrational coupling is neglected, we found that the widths of scattering resonances, responsible for the lifetimes of metastable ozone states, remain nearly the same (on average), but the number of these states increases by as much as 20%. We also found that these changes are nearly the same in symmetric and asymmetric ozone isotopomers 16O18O16O and 16O16O18O. Therefore, based on the results of these calculations, the Coriolis coupling does not seem to favor the formation of asymmetric ozone molecules and thus cannot be responsible for symmetry-driven mass-independent fractionation of oxygen isotopes.

Calculation of vibrational eigenenergies on a quantum computer: Application to the Fermi resonance in CO2

We apply a modified version of the multistate contracted variational quantum eigensolver method to calculate vibrational eigenstates of ${\mathrm{CO}}_{2}$ on a quantum computer. A two-mode model of ${\mathrm{CO}}_{2}$ is employed, and the vibrational wave function is expanded using three harmonic-oscillator basis functions for each mode. The wave functions are mapped to four qubits by a compact mapping method. The Hamiltonian matrix elements are evaluated on a simulator including noise and on a quantum computer available at IBM Quantum, while the Hamiltonian matrix is diagonalized on a classical computer. We propose an error mitigation method by which the shift of the numerical values of the matrix elements originating from the noise can be corrected, and examine the dependence of the statistical uncertainties on the number of executions of each quantum circuit. We find that, at about $8\ifmmode\times\else\texttimes\fi{}{10}^{6}$ executions, the energy eigenvalues of the Fermi resonance states in ${\mathrm{CO}}_{2}$ can be obtained with an uncertainty within 1 ${\mathrm{cm}}^{\ensuremath{-}1}$.

A Hamiltonian approach for obtaining irreducible projective representations and the k ⋅ p perturbation for anti-unitary symmetry groups

As is known, the irreducible projective representations (Reps) of anti-unitary groups contain three different situations, namely, the real, the complex and quaternionic types with torsion number 1, 2, 4 respectively. This subtlety increases the complexity in obtaining irreducible projective Reps of anti-unitary groups. In the present work, a physical approach is introduced to derive the condition of irreducibility for projective Reps of anti-unitary groups. Then a practical procedure is provided to reduce an arbitrary projective Rep into direct sum of irreducible ones. The central idea is to construct a Hermitian Hamiltonian matrix which commutes with the representation of every group element g ∈ G, such that each of its eigenspaces forms an irreducible representation space of the group G. Thus the Rep is completely reduced in the eigenspaces of the Hamiltonian. This approach is applied in the k ⋅ p effective theory at the high symmetry points (HSPs) of the Brillouin zone for quasi-particle excitations in magnetic materials. After giving the criterion to judge the power of single-particle dispersion around an HSP, we then provide a systematic procedure to construct the k ⋅ p effective model.

An alternative formulation of vibrational heat-bath configuration interaction

Recently, a selected vibrational configuration interaction method was introduced where the selection of the vibrational basis was performed using the heat-bath algorithm (VHCI). This method was adapted from the heat-bath configuration interaction (HCI) method used to solve the electronic Schrödinger equation. The selection algorithm in electronic HCI exploits the fact that most nonzero Hamiltonian matrix elements correspond to a double electronic excitation from one determinant to another and have values equal to the two-electron molecular orbital integrals such that they can be computed once, sorted, stored, and easily accessed later using a dictionary data structure. However, the Hamiltonian matrix elements in VHCI are more complicated than their electronic HCI counterparts, which lead to the possibility of different VHCI implementations. We explore one such implementation. The most significant differences compared to the original VHCI implementation is that we (i) explicitly compute all one- and two-quanta excitations, (ii) include all occupation-number contributions to the Hamiltonian matrix elements, and (iii) add individual Hartree product basis functions to the variational space rather than multiple basis functions as determined by operator products. We apply the new VHCI algorithm to ethylene oxide and naphthalene and achieve good accuracy relative to previous results at a modest computational cost.

Space-Filling Curves for Real-Space Electronic Structure Calculations

Hamiltonian matrices for Kohn-Sham calculations implemented in real space are often large (millions by millions) but very sparse. This poses challenges and opportunities for iterative eigensolvers, which often require a large number of matrix-vector multiplications. As a consequence, an efficient parallel sparse matrix-vector multiplication algorithm is desired. Here, we investigate the benefits of using Hilbert space-filling curves (SFCs) in domain partitioning. We show that the use of Hilbert SFCs in grid-point partitioning brings better locality of the grid points, improves balance of communication, and reduces communication overhead. We also demonstrate an extension of Hilbert SFCs coupled with blockwise operations. The use of blockwise operations helps exploit the vector-processing units in contemporary computational platforms. We illustrate speedup and scalability improvements for an iterative eigensolver based on the Chebyshev-filtered subspace iteration method. Using blockwise Hilbert SFCs, we solve the Kohn-Sham problem for silicon nanocrystals up to 10 nm in diameter, which contain over 26,000 atoms. We illustrate how the density of states of silicon nanocrystals evolves to the bulk limit, where Van Hove singularities are clearly apparent.

Many-core acceleration of the first-principles all-electron quantum perturbation calculations

The first-principles quantum perturbation theory, also called density-functional perturbation theory (DFPT), is the state-of-the-art formalism to directly link the experimental response properties of the materials with the quantum modeling of the electrons. Here in this work, we present an implementation of all-electron DFPT for massively parallel Sunway many-core architectures to accelerate DFPT calculation. We have paid special attention to the calculation of the response density matrix, the real-space integration of the response density as well as the response Hamiltonian matrix. We also employ the fast and massively parallel linear scaling scheme together with the load balance algorithm for the DFPT calculations to improve the scalability. Using the above approaches, the accurate first-principles quantum perturbation calculations can be extended over millions of cores.