We extend and apply a recently introduced quasi-particle functional renormalisation group scheme to the two-dimensional Hubbard model with next-nearest-neighbour hopping and away from half filling. We confirm the generation of superconducting correlations in some regions of the phase diagram, but also find that the inclusion of self-energy feedback by means of a decreasing quasi-particle weight can suppress superconducting tendencies more than anti-ferromagnetic correlations by which they are generated. As a supplement, we provide sample results for the self-energy in second-order perturbation theory and address some conceptual matters.

A recently proposed extension of the interaction flow method is applied to exemplary cases of selected physical and methodical parameters for the two-dimensional Hubbard model away from half-filling and perfect nesting. In this scheme, the self-energy is calculated on the real-frequency axis and its feedback on the flow of interactions is included in a simple manner via a momentum-dependent quasi-particle weight. Results for two different types of self-energy feedback are compared to the case without feedback and to existing results stemming from calculations for imaginary frequencies. Various physical and non-physical aspects which influence the outcome qualitatively and quantitatively are addressed. Some tentative directions for future developments are suggested.

A recently proposed extension of the interaction flow method is applied to exemplary cases of selected physical and methodical parameters for the two-dimensional Hubbard model away from half-filling and perfect nesting. In this scheme, the self-energy is calculated on the real-frequency axis and its feedback on the flow of interactions is included in a simple manner via a momentum-dependent quasi-particle weight. Results for two different types of self-energy feedback are compared to the case without feedback and to existing results stemming from calculations for imaginary frequencies. Various physical and non-physical aspects which influence the outcome qualitatively and quantitatively are addressed. Some tentative directions for future developments are suggested.

The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations. Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state-of-the-art quantum many-body methods. Upon cooling into its insulating antiferromagnetic ground state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate-temperature metallic regime, and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations. We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantumMonte Carlo methods) serving as a benchmark. By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes. Based on this analysis, we explain how quasiparticles can coexist with increasingly long-range antiferromagnetic correlations and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime. We also critically discuss whether imaginary-time methods are able to capture the non-Fermi-liquid singularities of this fully nested system.

Background Energy system models (ESM) are widely used in research and industry to analyze todays and future energy systems and potential pathways for the European energy transition. Current studies address future policy design, analysis of technology pathways and of future energy systems. To address these questions and support the transformation of today’s energy systems, ESM have to increase in complexity to provide valuable quantitative insights for policy makers and industry. Especially when dealing with uncertainty and in integrating large shares of renewable energies, ESM require a detailed implementation of the underlying electricity system. The increased complexity of the models makes the application of ESM more and more difficult, as the models are limited by the available computational power of today’s decentralized workstations. Severe simplifications of the models are common strategies to solve problems in a reasonable amount of time – naturally significantly influencing the validity of results and reliability of the models in general. Solutions for Energy-System Modelling Within BEAM-ME a consortium of researchers from different research fields (system analysis, mathematics, operations research and informatics) develop new strategies to increase the computational performance of energy system models and to transform energy system models for usage on high performance computing clusters. Within the project, an ESM will be applied on two of Germany’s fastest supercomputers. To further demonstrate the general application of named techniques on ESM, a model experiment is implemented as part of the project. Within this experiment up to six energy system models will jointly develop, implement and benchmark speed-up methods. Finally, continually collecting all experiences from the project and the experiment, identified efficient strategies will be documented and general standards for increasing computational performance and for applying ESM to high performance computing will be documented in a best-practice guide.

Zentrales Ziel des BEAM-ME-Projekts war die signifikante Reduktion der Losungszeiten fur Energiesys-temmodelle, die als lineares Optimierungsproblem in GAMS (General Algebraic Modeling System) formuliert sind, sowie die Losung bisher unlosbarer derartiger Probleme. Um dieses Ziel zu erreichen wurden verschiedene Methoden evaluiert, und deren Wirkung im Rahmen eines Modellexperiments fur verschieden formulierte und fokussierte Modelle verglichen. Die zwei zentralen Saulen der Modellbeschleunigung sind einerseits die Anwendung von modellbasierten Methoden (Kapitel 2) und andererseits die Nutzung von Losungsalgorithmus-basierten Methoden (Kapitel 3). Die modellbasierten Methoden umfassen im Wesentlichen verschiedene Ansatze der raumlichen und zeitlichen Aggregierung, sowie der heuristischen und mathematisch exakten Problemzerlegung (Dekomposition). Sie stellen den Hauptteil der bislang genutzten Beschleunigungsstrategien fur Energiesystem-Optimierungsmodelle dar und bieten den Vorteil, dass sie in der Regel durch Anpassungen von Quellcodes oder Vorprozessieren von Eingangsdaten von Modellanwendern selbst entwickelt und implementiert werden konnen. Im Rahmen von BEAM-ME wurden sie uberwiegend in das Energiesystemmodell REMix, punktuell aber auch in andere Modelle implementiert, sowie bewertet. Neben der Reduktion der Losungszeit war bei der Untersuchung auch die Auswirkung auf die Genauigkeit der Ergebnisse von wesentlicher Bedeutung (Abbildung 1). Als zentrales Ergebnis der Evaluation modellbasierter Methoden konnte gezeigt werden, dass mit vertretbaren Einschrankungen bezuglich der Genauigkeit, Heuristiken die Rechenzeit groser Modelle bis zu einem Faktor 10 senken konnen. Gleichzeitig wurde dabei deutlich, dass fur weitere Modellbeschleunigungen parallelisiertes Rechnen unabdingbar ist. Diese Lo-sungsalgorithmus-basierten Methoden wurden ebenfalls im Projekt untersucht. Kern der Untersuchung von Losungsalgorithmus-basierten Methoden war die Entwicklung des Losers PIPS-IPM++, welcher das parallelisierte Losen von linearen Optimierungsproblemen auf Hochleistungs-rechnerarchitekturen ermoglicht. Er basiert auf dem Open-Source Loser PIPS-IPM und wurde im Rahmen von BEAM-ME umfangreich weiterentwickelt um eine Nutzung fur typische Probleme der Energiesystemoptimierung zu ermoglichen. Diese Problemkategorie zeichnet sich durch viele so genannte „Linking Constraints“ (verknupfende Restriktionen) aus, welche zur Modellierung von Netzen, Speichern und CO2-Emissionsbeschrankungen erforderlich sind. Dafur wurde der Loser unter anderem so angepasst, dass er neben verknupfenden Variablen („Linking Variables“) auch eine grose Anzahl an verknupfenden Restriktionen („Linking Constraints“) parallel verarbeiten kann. Fur eine bessere Integration wurde im Rahmen des Projektes eine Schnittstelle zwischen GAMS und PIPS-IPM++ entwickelt. Grundlage hierfur ist die ebenfalls neu geschaffene Moglichkeit der Annotation von Blockstrukturen in GAMS-Modellen, die eine Zerlegung sehr groser Probleme in viele kleine Blocke erlaubt, was wiederum fur eine parallele Anwendung der Losungsalgorithmen unabdingbar ist. Mit der Entwicklung von PIPS-IPM++ und der Anpassung der Energiesystemmodelle wurde die Nutzung von Hochstleistungsrechnern (High Performance Computer, HPC) fur die Energiesystemanalyse erschlossen. Die im Rahmen von BEAM-ME erreichten Fortschritte erlauben eine Reduktion der Losungszeit linearer Optimierungsprobleme der Energiesystemanalyse bis zu einem Faktor 26. Durch effiziente Problemzerlegung und Nutzung von PIPS-IPM++ sind fernerhin vorher unlosbare Probleme losbar geworden. Die erreichten Fortschritte auf technischer Seite lassen sich exemplarisch anhand eines typischen wissenschaftlichen Rechencluster aus gekoppelten Servern darstellen. Auf einem solchen System konnte mittels PIPS-IPM++ eine Reduktion der Laufzeit um 76% und eine Reduktion des benotigten Arbeitsspeicherbedarfes je Rechenknoten um 96% erreicht werden. Auf dedizierten hochparallelen Grosrechnern wie dem Supercomputer Juwels am Julich Supercomputing Centre (JSC) reduzierte sich die Laufzeit in einigen Fallen um 96%. Diese Fortschritte erlauben eine signifikante Erweiterung der Analysetiefe von Energiesystemmodellen (Kapitel 4). Fur REMix wurde die zugangliche raumliche Auflosung von etwa 50 auf uber 1000 Modellknoten erhoht, die Optimierung von Transformationspfaden statt einzelner Stichjahre realisiert, die Betrachtung weiterer Technologien und Sektoren ermoglicht, und die Moglichkeit gros angelegter Parameterraumanalysen geschaffen. Das BEAM-ME-Projekt sties auf breites Interesse in der Wissenschaftsgemeinde, was sich insbesondere durch groses Interesse an einer Teilnahme am Modellexperiment, vielfaltige Anfragen, rege Teilnahme am Abschlussworkshop sowie zahlreichen Veroffentlichungen manifestierte.

Using the recently introduced multiloop extension of the functional renormalization group, we compute the frequency- and momentum-dependent self-energy of the two-dimensional Hubbard model at half filling and weak coupling. We show that, in the truncated-unity approach for the vertex, it is essential to adopt the Schwinger-Dyson form of the self-energy flow equation in order to capture the pseudogap opening. We provide an analytic understanding of the key role played by the flow scheme in correctly accounting for the impact of the antiferromagnetic fluctuations. For the resulting pseudogap, we present a detailed numerical analysis of its evolution with temperature, interaction strength, and loop order.

We present a highly parallelisable scheme for treating functional Renormalisation Group equations which incorporates a quasi-particle-based feedback on the flow and provides direct access to real-frequency self-energy data. This allows to map out the boundaries of Fermi-liquid regimes and to study the effect of quasi-particle degradation near Fermi liquid instabilities. As a first application, selected results for the two-dimensional half-filled perfectly nested Hubbard model are shown.

The functional renormalisation group (fRG) has evolved into a versatile tool in condensed matter theory for studying important aspects of correlated electron systems. Practical applications of the method often involve a high numerical effort, motivating the question in how far High Performance Computing (HPC) can leverage the approach. In this work we report on a multi-level parallelisation of the underlying computational machinery and show that this can speed up the code by several orders of magnitude. This in turn can extend the applicability of the method to otherwise inaccessible cases. We exploit three levels of parallelisation: Distributed computing by means of Message Passing (MPI), shared-memory computing using OpenMP, and vectorisation by means of SIMD units (single-instruction-multiple-data). Results are provided for two distinct High Performance Computing (HPC) platforms, namely the IBM-based BlueGene/Q system JUQUEEN and an Intel Sandy-Bridge-based development cluster. We discuss how certain issues and obstacles were overcome in the course of adapting the code. Most importantly, we conclude that this vast improvement can actually be accomplished by introducing only moderate changes to the code, such that this strategy may serve as a guideline for other researcher to likewise improve the efficiency of their codes.

Running and combining HPC applications often leads to complex scientific workflows, even more when code is to be executed in several different computing platforms. We present a flexible and platform independent framework for workflow definition and execution based on a redesigned version of the benchmarking environment JUBE. By means of a generalised configuration method this new version of JUBE can now be applied to more complex production, development and testing scenarios. It provides user-defined parameter substitution at all workflow stages, automated job submission, extensive directory and result handling and customisable analysis steps. In this report we demonstrate how it can be used to implement a given workflow representation and how it relates to and differs from other generic workflow management systems.

We study the anisotropic two-dimensional Hubbard model at and near half filling within a functional renormalization group method, focusing on the structure of momentum-dependent correlations which grow strongly upon approaching a critical temperature from above. We find that a finite nearest-neighbor interchain hopping is not sufficient to introduce a substantial momentum dependence of single-particle properties along the Fermi surface. However, when a sufficiently large second-nearest neighbor inter-chain hopping is introduced, the system is frustrated and we observe the appearance of so-called "hot spots", specific points on the Fermi surface around which scattering becomes particularly strong. We compare our results with other studies on quasi-one-dimensional systems.

We extend the functional renormalization group technique in a modi cation of the one-particle irreducible scheme to study discrete symmetry breaking at nite temperature. As an instructive example, we employ the technique to access both the symmetric and the symmetry-broken phase of a charge-density wave mean- eld model. We study the half- lled case, and thus the breaking of a discrete symmetry, at nite temperature. A small external symmetry-breaking eld allows us to access the symmetry-broken state without encountering any divergence in the o w. We show diagrammatically that our method is equivalent to an exact resummation treatment. We numerically study the dependence of the o w on the external eld and on temperature.

Electron-electron interactions can induce Fermi surface deformations which break the point-group symmetry of the lattice structure of the system. In the vicinity of such a the Fermi surface is easily deformed by anisotropic perturbations, and exhibits enhanced collective fluctuations. We show that critical Fermi surface fluctuations near a d-wave Pomeranchuk instability in two dimensions lead to large anisotropic decay rates for single-particle excitations, which destroy Fermi liquid behavior over the whole surface except at the Brillouin zone diagonal.