Hierarchical Parallelisation of Functional Renormalisation Group Calculations -- hp-fRG

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

Quasi-particle functional Renormalisation Group calculations in the two-dimensional t-t'-Hubbard model

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

Fermionic renormalization group flow into phases with broken discrete symmetry: charge-density wave mean-field model

We generalize the application of the functional renormalization group (fRG) method for the fermionic flow into the symmetry-broken phase to finite temperatures. We apply the scheme to the case of a broken discrete symmetry: the charge-density wave (CDW) mean-field model at half filling. We show how an arbitrarily small initial CDW order parameter starts to grow at the CDW instability and how it flows to the correct final value, suppressing the divergence of the effective interaction in the fRG flow. The effective interaction peaks at the instability and saturates at low energy scales or temperatures. The relation to the mean-field treatment, differences compared to the flow for a broken continuous symmetry, and the prospects of the new method are discussed.Comment: 10 pages, 7 figures. V2: minor corrections, journal references and DOI adde

Pseudogap opening in the two-dimensional Hubbard model: A functional renormalization group analysis

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.Comment: 15 pages, 15 figures, version as publishe

Pseudogap at hot spots in the two-dimensional Hubbard model at weak coupling

We analyze the interaction-induced renormalization of single-particle excitations in the two-dimensional Hubbard model at weak coupling using the Wick-ordered version of the functional renormalization group. The self energy is computed for real frequencies by integrating a flow equation with renormalized two-particle interactions. In the vicinity of hot spots, that is points where the Fermi surface intersects the umklapp surface, self energy effects beyond the usual quasi-particle renormalizations and damping occur near instabilities of the normal, metallic phase. Strongly enhanced renormalized interactions between particles at different hot spots generate a pronounced low-energy peak in the imaginary part of the self energy, leading to a pseudogap-like double-peak structure in the spectral function for single-particle excitations.Comment: 14 pages, 7 figure

Interaction flow method for many-fermion systems

We propose an interaction flow scheme that sums up the perturbation expansion of many-particle systems by successively increasing the interaction strength. It combines the unbiasedness of renormalization group methods with the simplicity of straight-forward perturbation theory. Applying the scheme to fermions in one dimension and to the two-dimensional Hubbard model we find that at one-loop level and low temperatures there is ample agreement with previous one-loop renormalization group approaches. We furthermore present results for the momentum-dependence of spin, charge and pairing interactions in the two-dimensional Hubbard model.Comment: 14 pages, 14 figure

Pair fluctuation induced pseudogap in the normal phase of the two-dimensional attractive Hubbard model at weak coupling

One-particle spectral properties in the normal phase of the two-dimensional attractive Hubbard model are investigated in the weak coupling regime using the non-selfconsistent T-matrix approximation. The corresponding equations are evaluated numerically directly on the real frequency axis. For temperatures sufficiently close to the superconducting transition temperature a pseudogap in the one-particle spectral function is observed, which can be assigned to the increasing importance of pair fluctuations.Comment: 22 pages, 13 figure

REPRESENTAÇÕES SOCIAIS DO CRACK NA MÍDIA

Crack cocaine has been a recurrent topic in newspaper articles and in recent political speeches. This researchaimed to identify how the local media represents the drug. Based on the Theory of Social Representations 76 articles publishedduring 2009 in the Correio Braziliense newspaper were analyzed. The content of the articles was analyzed with the softwareALCESTE, which generated six classes. Three social representations were highlighted: drug as a scourge of humanity,indistinct police actions against users and drug dealers and hospitalization as a solution of the problem. We observed that thesocial representations are consistent with a stigmatizing and repressive approach, which still consider drug users as criminalor sick persons, reinforcing the maintenance of their underground status and limiting the understanding of the phenomenon.O crack tem desafiado o sistema de saúde, configurando-se em um dos alvos das políticas de governo. O artigo tem como objetivo identificar as representações sociais veiculadas pela mídia sobre o usuário de crack. Com base na Teoria das Representações Sociais, foram analisadas as 76 reportagens do jornal Correio Braziliense publicadas em 2009 sobre o tema. Utilizou-se o software ALCESTE que gerou uma análise lexical das 93 unidades de contexto inicial, agrupando-as em seis classes. A partir das classes, foram destacadas três representações sociais: a droga como flagelo da humanidade, as ações indistintas da polícia contra usuários ou traficantes e a internação do usuário como solução do problema. Observou-se que as representações sociais se coadunam com a abordagem estigmatizante e repressiva do usuário de drogas, a qual continua a considerar o usuário como criminoso ou doente, reforçando a manutenção de sua clandestinidade e limitando a compreensão do fenômeno

Social representation of crack cocaine in Brasilia's media press

O crack tem sido tema recorrente em matérias jornalísticas e discursos políticos. O objetivo do estudo foi identificar como a mídia local representa a droga. Baseadas na Teoria das Representações Sociais foram analisadas 76 reportagens de 2009 do jornal Correio Braziliense, utilizando-se o software ALCESTE que gerou seis classes. Destacaram-se três representações sociais: droga como flagelo da humanidade, ações policiais indistintas contra usuários ou traficantes e internação do usuário como solução do problema. As representações sociais encontradas alinham-se com a abordagem estigmatizante e repressiva ao usuário de drogas que ainda o considera ora como criminoso, ora como doente e perpetuam sua clandestinidade e limitam a compreensão do fenômeno.Crack cocaine has been a recurrent topic in newspaper articles and in recent political speeches. This research aimed to identify how the local media represents the drug. Based on the Theory of Social Representations 76 articles published during 2009 in the Correio Braziliense newspaper were analyzed. The content of the articles was analyzed with the software ALCESTE, which generated six classes. Three social representations were highlighted: drug as a scourge of humanity, indistinct police actions against users and drug dealers and hospitalization as a solution of the problem. We observed that the social representations are consistent with a stigmatizing and repressive approach, which still consider drug users as criminal or sick persons, reinforcing the maintenance of their underground status and limiting the understanding of the phenomenon