The Nambu-Jona Lasinio mechanism and the electroweak symmetry breaking in the Standard Model

This is a short report of the entire work developed during the study of the possible realizations of the "Top Mode" Standard Model. Here it is examined the breaking of internal symmetries from another point of view showing that is possible to reproduce the gauged electroweak panorama of the traditional Standard Model in a exhaustive and selfconsistent way. The result is reached applying the main futures of the Nambu-Jona Lasinio (NJL) mechanism to an electroweak invariant Lagrangian. In this context the use of functional formalism for composite operators naturally leads to a different dynamical approach. Meanwhile the Higgs mechanism acts on the Lagrangian form, a NJL like model looks directly at the physics of the system showing the real dynamical content hidden in the Green functions of the theory.Comment: 18 Pages, no figures, LaTex, corrected typos, references remove

Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems

In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear. This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach.Comment: 12 Pages, 5 figures. Accepted for publication on Computer Physics Communication

Multiparametric Quantum Algebras and the Cosmological Constant

With a view towards applications for de Sitter, we construct the multi-parametric $q$-deformation of the so(5,\IC) algebra using the Faddeev-Reshetikhin-Takhtadzhyan (FRT) formalism.Comment: v4: cosmetic changes from the published versio

An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems

In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation. We propose a alternative approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers.Comment: 23 Pages, 6 figures. First revision of an invited submission to special issue of Concurrency and Computation: Practice and Experienc

An Example of Symmetry Exploitation for Energy-related Eigencomputations

One of the most used approaches in simulating materials is the tight-binding approximation. When using this method in a material simulation, it is necessary to compute the eigenvalues and eigenvectors of the Hamiltonian describing the system. In general, the system possesses few explicit symmetries. Due to them, the problem has many degenerate eigenvalues. The ambiguity in choosing a orthonormal basis of the invariant subspaces, associated with degenerate eigenvalues, will result in eigenvectors which are not invariant under the action of the symmetry operators in matrix form. A meaningful computation of the eigenvectors needs to take those symmetries into account. A natural choice is a set of eigenvectors, which simultaneously diagonalizes the Hamiltonian and the symmetry matrices. This is possible because all the matrices commute with each other. The simultaneous eigenvectors and the corresponding eigenvalues will be in a parametrized form in terms of the lattice momentum components. This functional dependence of the eigenvalues is the dispersion relation and describes the band structure of a material. Therefore it is important to find this functional dependence in any numerical computation related to material properties.Comment: To appear in the proceedings of the 7th International Conference on Computational Methods in Science and Engineering (ICCMSE '09

Efficient parallel implementation of the ChASE library on distributed CPU-GPU architectures

The Chebyshev Accelerated Subspace iteration Eigensolver (ChASE) is an iterative eigensolver developed at the JSC by the SimLab ab initio. The solver target principally sequences of dense eigenvalue problems as they arise in Density functional Theory, but can also work on the single eigenproblem. ChASE leverages on the preponderant use of BLAS 3 subroutines to achieve close-to-peak performance. Currently, the library can be executed in parallel on many- and multi-core platforms. The latest development of this project dealt with the extension of the CUDA build to encompass multiple GPUs on distinct CPUs. As such this hybrid parallelization will use MPI as well as CUDA interfaces effectively exploiting heterogeneous multi-GPU platforms. The extended library was tested on large and dense eigenproblems extracted from excitonic Hamiltonian. The ultimate goal is to integrate this new parallel implementation of ChASE with the VASP-BSE code

Towards an Efficient Use of the BLAS Library for Multilinear Tensor Contractions

Mathematical operators whose transformation rules constitute the building blocks of a multi-linear algebra are widely used in physics and engineering applications where they are very often represented as tensors. In the last century, thanks to the advances in tensor calculus, it was possible to uncover new research fields and make remarkable progress in the existing ones, from electromagnetism to the dynamics of fluids and from the mechanics of rigid bodies to quantum mechanics of many atoms. By now, the formal mathematical and geometrical properties of tensors are well defined and understood; conversely, in the context of scientific and high-performance computing, many tensor- related problems are still open. In this paper, we address the problem of efficiently computing contractions among two tensors of arbitrary dimension by using kernels from the highly optimized BLAS library. In particular, we establish precise conditions to determine if and when GEMM, the kernel for matrix products, can be used. Such conditions take into consideration both the nature of the operation and the storage scheme of the tensors, and induce a classification of the contractions into three groups. For each group, we provide a recipe to guide the users towards the most effective use of BLAS.Comment: 27 Pages, 7 figures and additional tikz generated diagrams. Submitted to Applied Mathematics and Computatio

Dissecting the FEAST algorithm for generalized eigenproblems

We analyze the FEAST method for computing selected eigenvalues and eigenvectors of large sparse matrix pencils. After establishing the close connection between FEAST and the well-known Rayleigh-Ritz method, we identify several critical issues that influence convergence and accuracy of the solver: the choice of the starting vector space, the stopping criterion, how the inner linear systems impact the quality of the solution, and the use of FEAST for computing eigenpairs from multiple intervals. We complement the study with numerical examples, and hint at possible improvements to overcome the existing problems.Comment: 11 Pages, 5 Figures. Submitted to Journal of Computational and Applied Mathematic

Dark Matter In Minimal Trinification

We study an example of Grand Unified Theory (GUT), known as trinification, which was first introduced in 1984 by S.Glashow. This model has the GUT gauge group as $[SU(3)]^3$ with a discrete $\mathbb{Z}_3$ to ensure the couplings are unified at the GUT scale. In this letter we consider this trinification model in its minimal formulation and investigate its robustness in the context of cosmology. In particular we show that for a large set of the parameter space the model doesn't seem to provide a Dark Matter candidate compatible with cosmological data.Comment: To appear in the LXXXVI session of the "Les Houches" summer school. 9 pages, 2 graph