Session P22: General Theory/Computational Physics

First Principles Computation of G-Factors in Bi and Bi$_2$Se$_3$

In this talk, we propose a first principles computation method for g-factor tensor,which not only gives comparable results with experiments but also establishes a clear physical picture of Zeeman effect in materials. In our method, the Hilbert space of the electronic states is treated as a direct product of "inner" space and "orbital" space, which are spanned by Bloch wave-functions and envelope functions respectively. Correspondingly, vector-potential is divided into a periodic part acting only in inner space and a non-periodic part acting only in orbital space. With the above flamework we define the g-factors as coupling coefficients between inner space and magnetic field. By the method we developed, we have further computed the g-factors of bismuth and Bi$_2$Se$_3$ and get satisfactory results, which are in good agreement with the experimental data.   

Adaptively truncated Hilbert spaces for Hamiltonian-based impurity solvers

We investigate truncations of the exponentially large Hilbert space in the exact diagonalization (ED) as an impurity solver for the dynamical mean-field theory (DMFT). A key issue is to maintain the high degree of numerical accuracy required in the construction of Greens functions. We test various truncation schemes with similar number of Slater determinants in both Hilbert spaces for the ground state and a particle- or a hole-excited state, and show that the excited states play an important role in accurate computation as well as the ground state. Appropriate truncation for both spaces enables us to compute the accurate self-energy of the impurity Hamiltonian with up to eight correlated orbitals hybridized with a sufficient number of bath orbitals to obtain converged solutions of the self-consistent equation in the DMFT, which is not solvable by the original ED. Application to spin-orbit coupled multi-orbital models and the one-dimensional Hubbard model and comparison to results from exact diagonalization and the configuration interaction based impurity solvers demonstrate the power of the method.   

Equilibration properties of a disordered interacting open quantum system

The central question in the field of many body localization is if a closed interacting quantum system effectively thermalizes in the presence of disorder. However, any experimental test necessarily involves the opening of the ideally closed quantum system. Both from a fundamental point of view as well as for concrete experimental investigations of many body localization phenomena, a solid understanding of the effect of an attached bath is of significant importance. We study the equilibration properties of disordered interacting open quantum systems. On the one hand we consider the equilibration of such a many body localized system by coupling baths to the ends of a 1D spin chain. We find non-monotonous behaviour of the slowest relaxation time towards equilibrium. On the other hand, we take the bath itself to be a disordered interacting open quantum system and investigate the dephasing of a single qubit coupled to it. The model for the bath has a many body localization transition, affecting the dephasing of the single qubit.   

Accurate, efficient, and scalable parallel simulation of mesoscale electrostatic/magnetostatic problems accelerated by a fast multipole method.

Boundary integrals arise in many electrostatic and magnetostatic problems. In computational modeling of these problems, although the integral is performed only on the boundary of a domain, its direct evaluation needs O(N$^{\mathrm{2}})$ operations, where N is number of unknowns on the boundary. The O(N$^{\mathrm{2}})$ scaling impedes a wider usage of the boundary integral method in scientific and engineering communities. We have developed a parallel computational approach that utilize the Fast Multipole Method to evaluate the boundary integral in O(N) operations. To demonstrate the accuracy, efficiency, and scalability of our approach, we consider two test cases. In the first case, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space using a hybrid finite element-boundary integral method. In the second case, we solve an electrostatic problem involving the polarization of dielectric objects in free space using the boundary element method. The results from test cases show that our parallel approach can enable highly efficient and accurate simulations of mesoscale electrostatic/magnetostatic problems.   

Computational algorithms dealing with the classical and statistical mechanics of celestial scale polymers in space elevator technology

Prospects to build Space Elevator (SE) systems have become realistic~with~ultra-strong materials such as carbon nano-tubes and diamond nano-threads. At cosmic length-scales, space elevators can be modeled as polymer like floppy strings of tethered mass beads. A new venue in SE science has emerged with the introduction of the Rotating Space Elevator (RSE) concept [1,2] supported by~novel~algorithms~discussed in this presentation. An RSE is a loopy string reaching into outer space. Unlike the classical geostationary SE concepts of Tsiolkovsky, Artsutanov, and Pearson, our RSE exhibits an internal rotation. Thanks to this, objects sliding along the RSE loop spontaneously oscillate between two turning points, one of which is close to the Earth whereas the other one is in outer space. The RSE concept thus solves a major problem in SE technology which is how to supply energy to the climbers moving along space elevator strings. The investigation of the classical and statistical mechanics of a floppy string interacting with objects sliding along it required development of~subtle computational algorithms described in this presentation.~[1] L. Golubovic and S. Knudsen, Europhys. Lett. 86, 34001 (2009); [2] S. Knudsen and L. Golubovic, Eur. Phys. J. Plus 129, 242 (2014).   

Convex Lower Bounds for Free Energy Minimization

We construct lower bounds on free energy with convex relaxations from the nonlinear minimization over probabilities to linear programs over expectation values. Finite-temperature expectation values are further resolved into distributions over energy. A superset of valid expectation values is delineated by an incomplete set of linear constraints. Free energy bounds can be improved systematically by adding constraints, which also increases their computational cost. We compute several free energy bounds of increasing accuracy for the triangular-lattice Ising model to assess the utility of this method. \\ This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.   

Stabilized quasi-Newton optimization of noisy potential energy surfaces

Optimizations of atomic positions belong to the most frequently performed tasks in electronic structure calculations. Many simulations like global minimum searches or the identification of chemical reaction pathways can require the computation of hundreds or thousands of minimizations or saddle points. To automatize these tasks, optimization algorithms must not only be efficient but also very reliable. Unfortunately, computational noise in forces and energies is inherent to electronic structure codes. This computational noise poses a severe problem to the stability of efficient optimization methods like the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm. In this talk a recently published technique that allows to obtain significant curvature information of noisy potential energy surfaces is presented. This technique was used to construct both, a stabilized quasi-Newton minimization method and a stabilized quasi-Newton saddle finding approach. With the help of benchmarks both the minimizer and the saddle finding approach were demonstrated to be superior to comparable existing methods.    

Phonon softening and mechanical failure of graphene under tensile strain

Phonon softening of graphene under tensile strain is investigated based on ab initio density functional theory calculations. From calculated phonon band structures, we show that the Kohn anomaly point shifts from a high symmetry $K$ point to a lower symmetry one as a consequence of the Dirac point shift in the electronic band structure. We demonstrate that, over a wide range of tensile strain directions, the strain-induced enhancement of phonon softening can give rise to phonon instabilities resulting in a mechanical failure of graphene at lower strains. It is shown that there are two types of instabilities associated with phonons near $K$ and $\Gamma$ points, respectively, which induce symmetry-breaking structural distortions, and both of them lead to mechanical failure prior to the elastic failure commonly expected when the structural symmetry is retained.   

Adjoint based data assimilation for phase field model using second order information of a posterior distribution

Phase field (PF) method, which phenomenologically describes dynamics of microstructure evolutions during solidification and phase transformation, has progressed in the fields of hydromechanics and materials engineering. How to determine, based on observation data, an initial state and model parameters involved in a PF model is one of important issues since previous estimation methods require too much computational cost. We propose data assimilation (DA), which enables us to estimate the parameters and states by integrating the PF model and observation data on the basis of the Bayesian statistics. The adjoint method implemented on DA not only finds an optimum solution by maximizing a posterior distribution but also evaluates the uncertainty in the estimations by utilizing the second order information of the posterior distribution. We carried out an estimation test using synthetic data generated by the two-dimensional Kobayashi's PF model. The proposed method is confirmed to reproduce the true initial state and model parameters we assume in advance, and simultaneously estimate their uncertainties due to quality and quantity of the data. This result indicates that the proposed method is capable of suggesting the experimental design to achieve the required accuracy.   

Quantum Monte Carlo study of 4d vs 5d atomic and molecular systems.

We investigate the electronic properties of Mo and W atomic and molecular systems by quantum Monte Carlo (QMC) methods. One area of interest in these systems are the systematic changes in the fixed-node errors from 4d to 5d elements and corresponding changes in the correlation effects. We find that similarly to first and second-row systems the fixed-node biases grow with increasing degree of charge localization for similarly complex wave functions and bonding patterns. The second area of interest is the impact of relativistic effects on the electronic structure, in particular, to which extent they affect the bonding properties. We use scalar-relativistic energy-consistent pseudopotentials with averaged spin-orbit effects and we contrast these calculations with the explicit inclusion of the spin-orbit in the two-component framework.   

Information geometry with correlated data: Bayesian explorations of cosmological predictions for the microwave background radiation

We developed a new, generalized fitting algorithm for miltiparameter models which incorporates varying and correlated errors. This was combined with geometrical methods of sampling to explore model prediction space, notably to plot geodesics and determine the size and edges of the model manifold. We illustrate this using the microwave background spectra for all possible universes, as described by the standard $\Lambda$-cold dark matter ($\Lambda$-CDM) cosmological model. In this case, the predicted data are fluctuations and highly correlated with varying errors, resulting in a manifold with a varying metric (as the natural metric to use is given by the Fisher information matrix). Furthermore, the model manifold shares the hyperribbon structure seen in other models, with the edges forming a strongly distorted image of a hypercube. Practical applications of such an analysis include optimizing experimental instrumentation designed to test more detailed cosmological theories.   

Recasting the 3D Wigner-Liouville equation with spectral components of the force

The phasespace approach to many-body quantum mechanics, by means of the Wigner-function is interesting through its connection to classical mechanics. Time-evolution of any statistical distribution of states under influence of a (time-dependent) Hamiltonian is obtained through use of the Wigner-Liouville equation. The standard form of this equation contains two 3D integrals, over the entire phase space. As a result, this form emphasizes the non-locality of the interaction of the potential, but lacks simplicity and ease of understanding. Furthermore, the integrals make numerical solution of the Wigner-Liouville equation challenging. We present an alternative form to the Wigner-Liouville equation based on the force rather than the potential, in alignment with the classical Boltzmann equation. Decomposition of the force in its spectral components yields a simpler form of the Wigner-Liouville equation. This new form has only one 3D integral over the spectral force components, and is local in position, simplifying both interpretation and numerical implementation. Because of its use of the force, it straightforwardly reduces to the Boltzmann equation under classical conditions.   

Exciton condensation in one dimension

We show the existence of a stable bilayer exciton condensate in one dimension, which demonstrates both true long-range order and non-negligible pairing amplitude. The condensate is stabilized by a finite inter-wire tunneling between two parallel quasi-1D wires, which we propose as the system in which to realize the condensate. Combining numerical DMRG, mean-field approaches, and bosonization to go beyond perturbation theory, we analyze experiments which will demonstrate the off-diagonal long-range order and verify the associated non-negligible pairing amplitude of the exciton condensate.   

Rotationally invariant ensembles of integrable matrices

We construct ensembles of \textit{random integrable matrices} with any prescribed number of nontrivial integrals and formulate \textit{integrable matrix theory} (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-$M$ family of integrable matrices consists of exactly $N-M$ independent commuting $N\times N$ matrices linear in a real parameter. We first develop a rotationally invariant parameterization of such matrices, previously only constructed in a preferred basis. For example, an arbitrary choice of a vector and two commuting Hermitian matrices defines a type-1 family and vice-versa. Higher types similarly involve a random vector and two matrices. The basis-independent formulation allows us to derive the joint probability density for integrable matrices, in a manner similar to the construction of Gaussian ensembles in the RMT.   

Matrix Elements for Hylleraas CI

The limitation to at most a single interelectron distance in individual configurations of a Hylleraas-type multiconfiguration wave function restricts significantly the types of integrals occurring in matrix elements for energy calculations, but even then if the formulation is not handled efficiently the angular parts of these integrals escalate to create expressions of great complexity. This presentation reviews ways in which the angular-momentum calculus can be employed to systematize and simplify the matrix element formulas, particularly those for the kinetic-energy matrix elements.