Real-space Method Research Articles

A simple real-space scheme for periodic Dirac operators

We address in this work the question of the discretization of two-dimensional periodic Dirac Hamiltonians. Standard finite differences methods on rectangular grids are plagued with the so-called Fermion doubling problem, which creates spurious unphysical modes. The classical way around the difficulty used in the physics community is to work in the Fourier space, with the inconvenience of having to compute the Fourier decomposition of the coefficients in the Hamiltonian and related convolutions. We propose in this work a simple real-space method immune to the Fermion doubling problem and applicable to all two-dimensional periodic lattices. The method is based on spectral differentiation techniques. We apply our numerical scheme to the study of flat bands in graphene subject to periodic magnetic fields and in twisted bilayer graphene.

High resolution atomic scale characterization of dislocations in high entropy alloys: Critical assessment of template matching and geometric phase analysis

The paper assesses the applicability of advanced atomic resolution displacement measurement techniques to characterize dislocation character in metallic materials using simulated images derived from anisotropic elasticity and actual measurements in high entropy alloys. We draw attention to two techniques: the real space method of template matching (TeMA) and the reciprocal space method of geometric phase analysis (GPA) and provide a critical assessment. These techniques have limitations for direct evaluation of full dislocations Burgers vector or when local displacements are exceeding 50% lattice spacing. This is clearly illustrated with simulated arctangent displacement profiles reminiscent of dislocation cores. An approach for circumventing this limitation is suggested in the form of a nearest neighbor correction. Additionally, a methodology for determination of the Burgers vector is introduced on the basis of a vectorial rendering of the displacement field upon consideration of two zone axis measurements and applied to TeMA and GPA. The experimental results conform to the Burgers vector of a full lattice dislocation in the FCC crystal structure of the High-Entropy Alloy (HEA). The comparison of simulated and experimental images proves the efficacy of the HR-TEM (High Resolution Transmission Electron Microscopy) displacement mapping techniques while pointing to the need for caution in case of large displacements.

Dynamic crystallization in a quantum Ising chain

The topological degeneracy of ground states in transverse field Ising chain cannot be removed by local perturbation and allows it to be a promising candidate for topological computation. We study the dynamic processes of crystallization and dissolution for the gapped ground states in an Ising chain. For this purpose, the real-space renormalization method is employed to build an effective Hamiltonian that captures the low-energy physics of a given system. We show that the ground state and the first-excited state of an $ \left( N+1\right) $-site chain can be generated from that of the $N$-site one by adding a spin adiabatically and vice versa. Numerical simulation shows that the robust quasidegenerate ground states of finite-size chain can be prepared with high fidelity from a set of noninteracting spins by a quasiadiabatic process. As an application, we propose a scheme for entanglement transfer between a pair of spins and two separable Ising chains as macroscopic topological qubits.

Derivation and analysis of computational methods for fractional Laplacian equations with absorbing layers

This paper is devoted to the derivation and analysis of accurate and efficient perfectly matched layers (PMLs) or efficient absorbing layers for solving fractional Laplacian equations within initial boundary value problems (IBVP). Two main approaches are derived: we first propose a Fourier-based pseudospectral method, and then present a real space method based on an efficient computation of the fractional Laplacian with PML. Some numerical experiments and analytical results are proposed along the paper to illustrate the presented methods.

Emergent flat band lattices in spatially periodic magnetic fields

Motivated by the recent discovery of Mott insulating phase and unconventional superconductivity due to the flat bands in twisted bilayer graphene, we propose more generic ways of getting two-dimensional (2D) emergent flat band lattices using either 2D Dirac materials or ordinary electron gas (2DEG) subject to moderate periodic orbital magnetic fields with zero spatial average. Employing both momentum-space and real-space numerical methods to solve the eigenvalue problems, we find stark contrast between Schr\"{o}dinger and Dirac electrons, i.e., the former show recurring "magic" values of the magnetic field when the lowest band becomes flat, while for the latter the zero-energy bands are asymptotically flat without magicness. By examining the Wannier functions localized by the smooth periodic magnetic fields, we are able to explain these nontrivial behaviors using minimal tight-binding models on a square lattice. The two cases can be interpolated by varying the $g$-factor or effective mass of a 2DEG and by taking into account the Zeeman coupling, which also leads to flat bands with nonzero Chern numbers for each spin. Our work provides flexible platforms for exploring interaction-driven phases in 2D systems with on-demand superlattice symmetries.

Ensemble averaged Madelung energies of finite volumes and surfaces

Exact expressions for ensemble averaged Madelung energies of finite volumes are derived. The extrapolation to the thermodynamic limit converges unconditionally and can be used as a parameter-free real-space summation method of Madelung constants. In the large volume limit, the surface term of the ensemble averaged Madelung energy has a universal form, independent of the crystal structure. The scaling of the Madelung energy with system size provides a simple explanation for the structural phase transition observed in cesium halide clusters.

QMCPACK: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion quantum Monte Carlo.

We review recent advances in the capabilities of the open source ab initio Quantum Monte Carlo (QMC) package QMCPACK and the workflow tool Nexus used for greater efficiency and reproducibility. The auxiliary field QMC (AFQMC) implementation has been greatly expanded to include k-point symmetries, tensor-hypercontraction, and accelerated graphical processing unit (GPU) support. These scaling and memory reductions greatly increase the number of orbitals that can practically be included in AFQMC calculations, increasing the accuracy. Advances in real space methods include techniques for accurate computation of bandgaps and for systematically improving the nodal surface of ground state wavefunctions. Results of these calculations can be used to validate application of more approximate electronic structure methods, including GW and density functional based techniques. To provide an improved foundation for these calculations, we utilize a new set of correlation-consistent effective core potentials (pseudopotentials) that are more accurate than previous sets; these can also be applied in quantum-chemical and other many-body applications, not only QMC. These advances increase the efficiency, accuracy, and range of properties that can be studied in both molecules and materials with QMC and QMCPACK.

Joint Iterative Fast Projection Matching for Fully Automatic Marker-free Alignment of Nano-tomography Reconstructions

Highly accurate, fully automatic marker-free image alignment plays an important role in nano-tomographic reconstruction, particularly in cases where the spatial resolution of the tomographic system is on the nanometer scale. However, highly accurate marker-free methods such as the projection matching method are computationally complex and time-consuming. Achieving alignment accuracy with reduced computational complexity remains a challenging problem. In this study, we propose an efficient method to achieve marker-free fully automatic alignment. Our method implements three main alignment procedures. First, the frequency-domain common line alignment method is used to correct the in-plane rotational errors of each projection. Second, real-space common line alignment method is used to correct the vertical errors of the projections. Finally, a single layer joint-iterative reconstruction and re-projection method is used to correct the horizontal projection errors. This combined alignment approach significantly reduces the computational complexity of the classical projection matching method, and increases the rate of convergence towards determining the accurate alignment. The total processing time can be reduced by up to 4 orders of magnitude as compared to the classical projection matching method. This suggests that the algorithm can be used to process image alignment of nano-tomographic reconstructions on a conventional personal computer in a reasonable time-frame.

Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

Structural defects are inherent in solids at a finite temperature, because they diminish free energies by growing entropy. The arrangement of these defects may display long-range orders, as occurring in quasicrystals, whose hidden structural symmetry could greatly modify the transport of excitations. Moreover, the presence of such defects breaks the translational symmetry and collapses the reciprocal lattice, which has been a standard technique in solid-state physics. An alternative to address such a structural disorder is the real space theory. Nonetheless, solving 1023 coupled Schrödinger equations requires unavailable yottabytes (YB) of memory just for recording the atomic positions. In contrast, the real-space renormalization method (RSRM) uses an iterative procedure with a small number of effective sites in each step, and exponentially lessens the degrees of freedom, but keeps their participation in the final results. In this article, we review aperiodic atomic arrangements with hierarchical symmetry investigated by means of RSRM, as well as their consequences in measurable physical properties, such as electrical and thermal conductivities.

A Real-Space Method to Measure the Persistence Length of Dynamic Microtubules

Microtubules play important roles in regulating the mechanical properties of the cytoskeleton filament system. While microtubules in cells often appear highly curved, microtubules grown in vitro tend to be straight. Therefore, a basic understanding of the mechanical properties of microtubules is essential to understand how they respond to extracellular forces and the binding of microtubule associated proteins and motors. The main techniques used to determine the persistence length of microtubules are either passive, relying on thermal fluctuations, or active, where external forces are applied via optical traps or fluid flow. Many of these studies used drug-stabilized microtubules that are not dynamic. However, stabilizing agents impact microtubule material properties. Existing methods to measure the persistence length of dynamic microtubules require long microtubules (over ∼30 um) and the use of non-intuitive ‘pseudo-modes’ to describe microtubule fluctuations. We present a simple real-space method to infer the persistence length of dynamic microtubules from their thermal fluctuations. We test this method on simulated dynamic microtubules and then apply it to determine the persistence length of dynamic, label-free microtubules imaged via darkfield microscopy. The combination of label-free imaging and a simple real-space analysis method is a powerful tool to investigate the mechanical properties of dynamic microtubules.

An introduction to neutrons for biology

The overlap of biology and neutron scattering remains a relatively narrow domain of research. This is partly due to the a priori maladjustment between real space problems and methods based on spatial and temporal correlations. In addition, some major assets of neutron scattering, such as isotopic substitution, can be tricky with biological molecules. More generally, a mutual lack of knowledge of the two concerned communities precluded potential rich interactions in early times. However, the situation changed to the point that, today, biology represents a substantial part of the research activity at neutron facilities. The purpose of this introduction is not to present one more overview of the subject of “neutron scattering” (excellent comprehensive articles are easily accessible to the interested readers [1–4]), but rather to facilitate the reading of the present book by introducing a few neutron scattering notions that may be useful for the community of biologists eventually less familiar with this technique.

Solving a Structure in the Reciprocal Space, Real Space and Both by Using the EXPO Software

The solution of crystal structures from X-ray powder diffraction data has undergone an intense development in the last 25 years. Overlapping, background estimate, preferred orientation are the main difficulties met in the process of determining the crystal structure from the analysis of the one-dimensional powder diffraction pattern. EXPO is a well known computer program that, designed for solving structures, organic, inorganic, as well as metal-organic by powder diffraction data, employs the two most widely used kinds of solution methods: Direct Methods proceeding in the reciprocal space and Simulated Annealing proceeding in the real space. EXPO allows also to suitably combine these two approaches for validating the structure solution. In this paper, we give examples of structure characterization by EXPO with the aim of suggesting a solution strategy leading towards the application of reciprocal-space methods or real-space methods or both.

On preconditioning the self-consistent field iteration in real-space Density Functional Theory

We present a real-space formulation for isotropic Fourier-space preconditioners used to accelerate the self-consistent field iteration in Density Functional Theory calculations. Specifically, after approximating the preconditioner in Fourier space using a rational function, we express its real-space application in terms of the solution of sparse Helmholtz-type systems. Using the truncated-Kerker and Resta preconditioners as representative examples, we show that the proposed real-space method is both accurate and efficient, requiring the solution of a single linear system, while accelerating self-consistency to the same extent as its exact Fourier-space counterpart.

Real-space renormalization-group methods for hierarchical spin glasses

We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: a sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method (Angelini M C, Parisi G and Ricci-Tersenghi F 2013 Ensemble renormalization group for disordered systems Phys. Rev. B 87 134201) in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical exponent related to the divergence of the correlation length, while it does not predict the correct qualitative behavior of below the upper critical dimension. On the other hand, the ERG procedure has been claimed to predict the correct behavior of both above and below the upper critical dimension. Here, we straighten out the reasons for the discrepancy between the two methods above, by demonstrating that the ERG method predicts a marginally stable critical fixed point, thus implying a prediction for the critical exponent given by . This prediction disagrees, on a qualitative and quantitative level, both with the mean-field value of above the critical dimension, and with numerical estimates of below the upper critical dimension. Therefore, our results show that finding a real-space RG method for spin glasses which yields the correct prediction for universal quantities below the upper critical dimension is still an open problem, for which our analysis may provide some general guidance for future studies.

Real-Space Based Benchmark of G0W0 Calculations on GW100: Effects of Semicore Orbitals and Orbital Reordering.

Using an implementation based on real-space wave functions, we perform G0W0 calculations of the HOMO and LUMO energies of molecules and atoms in the GW100 set. Our main conclusions are as follows: (1) Different implementations of the G0W0 approximation show much better agreements for HOMO (highest occupied molecular orbital) energies than for LUMO (lowest unoccupied molecular orbital) energies. The mean absolute differences between the results calculated with different pseudopotential codes range from 100 to 200 meV. For delocalized LUMOs, all-electron codes that use local orbital basis tend to predict much higher energies than those calculated with plane-wave basis or real-space methods. (2) The effects of semicore electrons in pseudopotentials can explain some of the large discrepancies between results calculated with different GW implementations. For molecules or atoms that include I, Xe, and Ga, pseudopotential-based calculations that exclude semicore electrons produce results that agree better with all-electron calculations. For polar molecules such as NaCl and BrK, however, it is necessary to include semicore electrons for alkaline metal elements to get correct LUMO energies. (3) More than 20 molecules show rearrangement of the order between LUMO (or HOMO) with other orbitals due to GW corrections. Such orbitals that switch order with LUMO are unbound and delocalized in space. The predicted LUMO GW levels (or electron affinity) can be corrected by 2.0 eV if we consider the rearrangement of orbitals in GW calculations. In all, our work clarifies some of the discrepancies between different GW codes and sets a benchmark for real-space implementations of the G0W0 approximation.

Thermoelectric transport in poly(G)-poly(C) double chains

Electronic and phononic transport in DNA systems with macroscopic length are studied by means of a real-space renormalization method within the Boltzmann formalism, where the poly(G)-poly(C) base-pair segments arranged following periodic and Fibonacci sequences are comparatively analyzed. The fishbone model and the two-site coarse grain model based on the Born potential including central and non-central interactions are respectively used for the calculation of electrical and lattice thermal conductivities of these DNA systems connected to two reservoirs at their ends. The results show the appearance of gaps in phononic transmittance spectra of segmented poly(G)-poly(C) double chains, which leads to a better thermoelectric figure of merit (ZT) than that of corresponding non-segmented systems. Such ZT can be further improved by introducing a long-range quasiperiodic order, which avoids the thermal transport of numerous low-frequency phonons responsible of the lattice thermal conduction at low temperature. Finally, the influence of reservoirs on ZT is also investigated.

Bulk spectrum and K-theory for infinite-area topological quasicrystals

The bulk spectrum of a possible Chern insulator on a quasicrystalline lattice is examined. The effect of being a 2D insulator seems to override any fractal properties in the spectrum. We compute that the spectrum is either two continuous bands, or that any gaps other than the main gap are small. After making estimates on the spectrum, we deduce a finite system size, above which the K-theory must coincide with the K-theory of the infinite system. Knowledge of the spectrum and K-theory of the infinite-area system will control the spectrum and K-theory of sufficiently large finite systems. The relation between finite volume K-theory and infinite volume Chern numbers is only proven to begin, for the model under investigation here, for systems on Hilbert space of dimension around 17 × 106. The real-space method based on the Clifford spectrum allows for computing Chern numbers for systems on Hilbert space of dimension around 2.7 × 106. New techniques in numerical K-theory are used to equate the K-theory of systems of different sizes.

The computation of local stress in ab initio molecular simulations

Motivated by the increasingly more important role of ab initio molecular dynamics models in material simulations, this work focuses on the definition of local stress, when the forces are determined from quantum-mechanical (QM) descriptions. Two types of ab initio models, including the Born–Oppenheimer and Ehrenfest dynamics, are considered. In addition, formulas are derived for both tight-binding and real-space methods for the approximations of the QM models. The formulas are examined via comparisons with full ab initio molecular simulations.

Scalable GW software for quasiparticle properties using OpenAtom

The GW method, which can describe electronic excitations accurately, is a powerful ab initio electronic structure technique in molecular, condensed matter and materials physics. However, the applications of the GW method to large systems require supercomputers and efficient fine-grained parallel software to overcome the computational complexity and expense of the method which scales as O(N4). Here, we develop fine-grained parallel GW software for the plane-wave basis. We proceed by revisiting the standard GW formulae in order to discern the optimal approaches for each phase of the GW calculation on present day supercomputers and to take into account recent methodological advances to inform our choices. These best numerical practices are implemented in the OpenAtom software which is written on top of the Charm++ parallel framework. We then evaluate the performance of the new software for a wide range of system sizes. Our GW software shows significantly improved parallel scaling compared to publicly available GW software on the Mira and Blue Waters supercomputers, two of most powerful computing platforms in the world. Program summaryProgram title: OpenAtom-GWProgram files doi:http://dx.doi.org/10.17632/pvj723crcx.1Licensing provisions: BSD 3-clauseProgramming language: C++Nature of problem: The GW method describes electronic excitations in materials: the energetics and quantum wave behavior when electrons are added or removed from a material. The GW method scales as O(N4) where N is the number of atoms in the simulation cell. The input variables for GW method are single-particle electronic states computed from mean-field calculations such as Density Functional Theory (DFT). The GW method, which is a proper quasiparticle approach, provides corrections to the mean-field electronic state information that include physics beyond DFT to treat excitations. We employ a plane-wave basis set as is common in condensed matter and materials physics.Solution method: The first step is to construct the static polarizability matrix (P) which represents the response of the electronic density with respect to changes in the electronic potential, and whose construction is the most expensive part of a GW computation. The P matrix calculation requires a large dense spatial grid (real space) and has a double summation over filled and empty electronic states requiring outer product operations. A large number of Fast Fourier Transforms (FFTs) must also be performed. Next, the polarizability is employed to compute the self-energy which represents the dynamic potential acting upon moving electrons. We employ a real-space representation of static polarizability matrix because the real-space method reduces the number of FFTs and the size of the real-space grids compared to working solely in the Fourier (g-space) representation. Parallelization is carried out using the Charm++ parallel middleware which through overdecomposition and virtualization, supports and enables highly scalable fine-grained parallel applications.Additional comments including restrictions and unusual features: Users need to run a DFT calculation using, e.g., the Quantum Espresso software or OpenAtom itself, to obtain the input electronic states (i.e., single particle wavefunctions and energies). Since the OpenAtom-GW software is written on top of Charm++ parallel middleware, the user must install Charm++; however, Charm++ is an increasingly widely used framework in scientific computing and this requirement does not present a bottleneck to usage.

Deterministic Creation and Braiding of Chiral Edge Vortices.

Majorana zero modes in a superconductor are midgap states localized in the core of a vortex or bound to the end of a nanowire. They are anyons with non-Abelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a real-space alternative, we propose to use the chiral motion along the boundary of the superconductor to braid a mobile vortex in the edge channel with an immobile vortex in the bulk. The measurement scheme is fully electrical and deterministic: edge vortices (π-phase domain walls) are created on demand by a voltage pulse at a Josephson junction and the braiding with a Majorana zero mode in the bulk is detected by the charge produced upon their fusion at a second Josephson junction.