EDIpack: A parallel exact diagonalization package for quantum impurity problems

We present EDIpack, an exact diagonalization package to solve generic quantum impurity problems. The algorithm, based on a generalization of the look-up method introduced by Lin and Gubernatis, enables a massively parallel execution of the matrix-vector linear operations required by Lanczos and Arnoldi algorithms. We show that a suitable Fock basis organization is crucial to optimize the inter-processors communication in distributed memory setup and, thus, to reach sub-linear scaling in sufficiently large systems. We discuss the algorithm in details, indicating how to deal with multiple-orbitals and electron-phonon coupling. Finally, we detail the download, installation and functioning of this package.Comment: 33 pages, 6 figure

Study of Condensed Matter Systems with Monte Carlo Simulation on Heterogeneous Computing Systems

We study the Edwards-Anderson model on a simple cubic lattice with a finite constant external field. We employ an indicator composed of a ratio of susceptibilities at finite momenta, which was recently proposed to avoid the difficulties of a zero momentum quantity, for capturing the spin glass phase transition. Unfortunately, this new indicator is fairly noisy, so a large pool of samples at low temperature and small external field are needed to generate results with a sufficiently small statistical error for analysis. We thus implement the Monte Carlo method using graphics processing units to drastically speed up the simulation. We confirm previous findings that conventional indicators for the spin glass transition, including the Binder ratio and the correlation length do not show any indication of a transition for rather low temperatures. However, the ratio of spin glass susceptibilities does show crossing behavior, albeit a systematic analysis is beyond the reach of the present data. This reveals the difficulty with current numerical methods and computing capability in studying this problem. One of the fundamental challenges of theoretical condensed matter physics is the accurate solution of quantum impurity models. By taking expansion in the hybridization about an exactly solved local limit, one can formulate a quantum impurity solver. We implement the hybridization expansion quantum impurity solver on Intel Xeon Phi accelerators, and aim to apply this approach on the Dynamic Hubbard Models

Tensor Network States: Optimizations and Applications in Quantum Many-Body Physics and Machine Learning

Tensor network states are ubiquitous in the investigation of quantum many-body (QMB) physics. Their advantage over other state representations is evident from their reduction in the computational complexity required to obtain various quantities of interest, namely observables. Additionally, they provide a natural platform for investigating entanglement properties within a system. In this dissertation, we develop various novel algorithms and optimizations to tensor networks for the investigation of QMB systems, including classical and quantum circuits. Specifically, we study optimizations for the two-dimensional Ising model in a transverse field, we create an algorithm for the $k$-SAT problem, and we study the entanglement properties of random unitary circuits. In addition to these applications, we reinterpret renormalization group principles from QMB physics in the context of machine learning to develop a novel algorithm for the tasks of classification and regression, and then utilize machine learning architectures for the time evolution of operators in QMB systems

Bulk-boundary correspondence in non-equilibrium dynamics of one-dimensional topological insulators

Dynamical phase transitions (DPT) are receiving a rising interest. They are known to behave analogously to equilibrium phase transitions (EPT) to a large extend. However, it is easy to see that DPT can occur in finite systems, while EPT are only possible in the thermodynamic limit. So far it is not clear how far the analogy of DPT and EPT goes. It was suggested, that there is a relation between topological phase transitions (TPT) and DPT, but many open questions remain. Typically, to study DPT, the Loschmidt echo (LE) after a quench is investigated, where DPT are visible as singularities. For one-dimensional systems, each singularity is connected to a certain critical time scale, which is given by the dispersion in the chain. In topological free-fermion models with winding numbers 0 or 1, only the LE in periodic boundary conditions (PBC) has been investigated. In open boundary conditions (OBC), these models are characterized by symmetry protected edge modes in the topologically non-trivial phase. It is completely unclear how these modes affect DPT. We investigate systems with PBC governed by multiple time scales with a Z topological invariant. In OBC, we provide numerical evidence for the presence of bulk-boundary correspondence in DPT in quenches across a TPT.Dynamische Phasenübergänge (DP) erfreuen sich eines wachsenden Interesses. Es ist bekannt, dass sie sich analog zu Gleichgewichtsphasenübergängen (GP) verhalten. Andererseits prüft man leicht nach, dass DP im Gegensatz zu EP auch in endlichen Systemen auftreten können, während Letztere nur im thermodynamischen Limes stattfinden. Bisher ist also unklar, wie weit die zuvor erwähnte Analogie zwischen DP und GP geht. Außerdem wurde ein Zusammenhang zwischen DP und topologischen Phasenübergängen (TP) aufgezeigt, aber es sind noch viele Fragen offen. Üblicherweise werden DP untersucht, indem man das Loschmidtecho (LE) nach einem Quench betrachtet, in welchem DP als Singularitäten sichtbar werden. In eindimensinalen Systemen kann jeder Singularität eine bestimmte kritische Zeitskala zugeordnet werden, die durch die Dispersionsrelation des Systems gegeben ist. In topologischen freien Fermionenmodellen (FFM) mit einer Invariante in Z 2 wurde bisher nur das LE in periodischen Randbedingungen (PRB) erforscht. Diese topologisch nichttrivialen Modelle werden in offenen Randbedingungen (ORB) durch topologiegeschützte Randmoden charakterisiert. Der Einfluss dieser Rand- moden auf DP und die Zeitentwichlung des LE ist noch vollkommen unerforscht. Wir untersuchen Systeme in PRB, die aufgrund der ganzzahligen topologischen Invariante durch mehrere kritische Zeitskalen bestimmt werden. Für Systeme mit ORB erbringen wir numerische Nachweise der Existanz eines Äquivalents der holo- graphischen Volumenrandkorrenpondenz in DP nach einem Quench über einen TP

GPU Accelerated Approach to Numerical Linear Algebra and Matrix Analysis with CFD Applications

A GPU accelerated approach to numerical linear algebra and matrix analysis with CFD applications is presented. The works objectives are to (1) develop stable and efficient algorithms utilizing multiple NVIDIA GPUs with CUDA to accelerate common matrix computations, (2) optimize these algorithms through CPU/GPU memory allocation, GPU kernel development, CPU/GPU communication, data transfer and bandwidth control to (3) develop parallel CFD applications for Navier Stokes and Lattice Boltzmann analysis methods. Special consideration will be given to performing the linear algebra algorithms under certain matrix types (banded, dense, diagonal, sparse, symmetric and triangular). Benchmarks are performed for all analyses with baseline CPU times being determined to find speed-up factors and measure computational capability of the GPU accelerated algorithms. The GPU implemented algorithms used in this work along with the optimization techniques performed are measured against preexisting work and test matrices available in the NIST Matrix Market. CFD analysis looked to strengthen the assessment of this work by providing a direct engineering application to analysis that would benefit from matrix optimization techniques and accelerated algorithms. Overall, this work desired to develop optimization for selected linear algebra and matrix computations performed with modern GPU architectures and CUDA developer which were applied directly to mathematical and engineering applications through CFD analysis

Laboratory Directed Research and Development FY2010 Annual Report

A premier applied-science laboratory, Lawrence Livermore National Laboratory (LLNL) has at its core a primary national security mission - to ensure the safety, security, and reliability of the nation's nuclear weapons stockpile without nuclear testing, and to prevent and counter the spread and use of weapons of mass destruction: nuclear, chemical, and biological. The Laboratory uses the scientific and engineering expertise and facilities developed for its primary mission to pursue advanced technologies to meet other important national security needs - homeland defense, military operations, and missile defense, for example - that evolve in response to emerging threats. For broader national needs, LLNL executes programs in energy security, climate change and long-term energy needs, environmental assessment and management, bioscience and technology to improve human health, and for breakthroughs in fundamental science and technology. With this multidisciplinary expertise, the Laboratory serves as a science and technology resource to the U.S. government and as a partner with industry and academia. This annual report discusses the following topics: (1) Advanced Sensors and Instrumentation; (2) Biological Sciences; (3) Chemistry; (4) Earth and Space Sciences; (5) Energy Supply and Use; (6) Engineering and Manufacturing Processes; (7) Materials Science and Technology; Mathematics and Computing Science; (8) Nuclear Science and Engineering; and (9) Physics