Open AccessBook Chapter10.1007/978-3-319-53862-4_15

Parallel Adaptive Integration in High-Performance Functional Renormalization Group Computations

- 04 Oct 2016

- pp 170-184

TL;DR: The functional Renormalization group (fRG) has become a formidable tool to study correlated electron systems on lattices which, in turn, provided great insights to our understanding of complex many-body phenomena, such as high-temperature superconductivity or topological states of matter as mentioned in this paper.

Abstract: The conceptual framework provided by the functional Renormalization Group (fRG) has become a formidable tool to study correlated electron systems on lattices which, in turn, provided great insights to our understanding of complex many-body phenomena, such as high-temperature superconductivity or topological states of matter. In this work we present one of the latest realizations of fRG which makes use of an adaptive numerical quadrature scheme specifically tailored to the described fRG scheme. The final result is an increase in performance thanks to improved parallelism and scalability.

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1. What have the authors contributed in "Parallel adaptive integration in high-performance functional renormalization group computations" ?

The conceptual framework provided by the functional Renormalization Group ( fRG ) has become a formidable tool to study correlated electron systems on lattices which, in turn, provided great insights to their understanding of complex many-body phenomena, such as hightemperature superconductivity or topological states of matter.. In this work the authors present one of the latest realizations of fRG which makes use of an adaptive numerical quadrature scheme specifically tailored to the described fRG scheme.

2. What have the authors stated for future works in "Parallel adaptive integration in high-performance functional renormalization group computations" ?

In the future, the authors envision to expand the internal parallelism of PAID to distributed memory.

Fig. 2. The number of function evaluations needed for calculating all integrals of a fixed l value is plotted against Ω. We use a form-factor expansion that results in 45 independent integrals for each external momentum l, which is fixed to (1.57, 1.31) (left plot) and (2.88, 0.26) (right plot) respectively. The results of both implementations— the one using DCUHRE (red) and the one using PAID (blue)—are shown in the same plot in favor of a direct comparison. In order to use the same number of evaluations per subregion as in DCUHRE, we set the PAID parameter N to 4. Further we use MaxTask = 10.

Fig. 1. The value of the integrand is plotted against the two-dimensional momentum p for Ω = 1.0 (left plot) and Ω = 0.1 (right plot). In this example case the external momentum l is set to (3.14, 0.78) and both form-factor indices label the lowest order function, which is constant in momentum space.

Fig. 4. The computation time needed for calculating all integrals of a fixed l value using 48 threads is plotted against Ω. We use a form-factor expansion that results in 325 independent integrals for each l, which takes the same values as in Fig. 2. For reasons of comparison we use N = 4 in PAID as in the analysis shown in Fig. 2. Further, MaxTask is set to 7.

Fig. 3. These plots show the speedup at Ω = 10−3 against the number of threads for the implementation based on PAID. For thread numbers up to 24 each compute core executes only a single thread. At 48 threads each compute core processes two threads at a time using simultaneous multithreading. The l-values are chosen as in Fig. 2 and there are 325 integrals to calculate per l. For this analysis we use the PAID parameters that result in the best performance: N = 6 and MaxTask = 18.

Citations

Book•10.1007/978-3-319-53862-4

High-performance scientific computing : first JARA-HPC Symposium, JHPCS 2016, Aachen, Germany, October 4-5, 2016 : revised selected papers

Edoardo Di Napoli, +4 more

- 01 Jan 2017

•Journal Article•10.3389/FPHY.2018.00032

Truncated-unity functional renormalization group for multiband systems with spin-orbit coupling

G. A. H. Schober, +4 more

- 08 May 2018

- Frontiers of Physics in China

TL;DR: The functional renormalization group (fRG) is by now a well-established method for investigating correlated electron systems, but it is still undergoing significant technical and conceptual improvements.

...read moreread less

•Journal Article•10.1103/PHYSREVB.96.205155

Antiferromagnetism and competing charge instabilities of electrons in strained graphene from Coulomb interactions

David Sánchez de la Peña, +3 more

- 29 Nov 2017

- Physical Review B

TL;DR: In this article, a truncated-unity functional renormalization group (TU-fRG) approach was employed to model the wave-vector dependence of the interaction vertex.

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References

•Journal Article•10.1103/REVMODPHYS.84.299

Functional renormalization group approach to correlated fermion systems

Walter Metzner, +4 more

- 12 Mar 2012

- Reviews of Modern Physics

TL;DR: The functional renormalization group as discussed by the authors is a flexible and unbiased tool for dealing with scale-dependent behavior of correlated fermion systems, such as Luttinger liquid behavior and the Kondo effect.

...read moreread less

817

•Journal Article•10.1137/060659831

Is Gauss Quadrature Better than Clenshaw-Curtis?

Lloyd N. Trefethen

- 01 Feb 2008

- Siam Review

TL;DR: Comparisons of the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis are compared, and experiments show that the supposed factor-of-2 advantage of Gaussian quadratures is rarely realized.

...read moreread less

751

•Journal Article•10.1103/PHYSREVB.79.195125

Efficient Parametrization of the Vertex Function, $Ω$-Scheme, and the (t,t')-Hubbard Model at Van Hove Filling

Christoph Husemann, +1 more

- 26 May 2009

- Physical Review B

TL;DR: In this paper, a parametrization of the four-point vertex function in the one-loop one-particle irreducible renormalization group (RG) scheme for fermions is proposed.

...read moreread less

192

•Journal Article•10.1209/EPL/I1998-00462-X

Weakly correlated electrons on a square lattice: a renormalization group theory

D. Zanchi, +1 more

- 17 Dec 1998

- arXiv: Strongly Correlated Electrons

TL;DR: In this paper, the weakly interacting Hubbard model on the square lattice using a one-loop renormalization group approach is studied and the transition temperature T_c between the metallic and (nearly) ordered states is found.

...read moreread less

123

•Book

Numerical Integration on Advanced Computer Systems

Arnold R. Krommer, +1 more

- 01 Sep 1994

TL;DR: Application of numerical integration software, including architecture adaptive integration algorithms, and potential parallelism in integration algorithms are studied.

...read moreread less

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Parallel Adaptive Integration in High-Performance Functional Renormalization Group Computations

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Most frequently asked questions

1. What have the authors contributed in "Parallel adaptive integration in high-performance functional renormalization group computations" ?

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Citations

High-performance scientific computing : first JARA-HPC Symposium, JHPCS 2016, Aachen, Germany, October 4-5, 2016 : revised selected papers

Truncated-unity functional renormalization group for multiband systems with spin-orbit coupling

Antiferromagnetism and competing charge instabilities of electrons in strained graphene from Coulomb interactions

References

Functional renormalization group approach to correlated fermion systems

Is Gauss Quadrature Better than Clenshaw-Curtis?

Efficient Parametrization of the Vertex Function, $Ω$-Scheme, and the (t,t')-Hubbard Model at Van Hove Filling

Weakly correlated electrons on a square lattice: a renormalization group theory

Numerical Integration on Advanced Computer Systems

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