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We formulate a stochastic gauge fixing method to study the gauge dependence of the Abelian projection. We consider a gauge which interpolates between the maximal Abelian gauge and no gauge fixing. We have found that Abelian dominance for the heavy quark potential holds even in a gauge which is far from maximally Abelian one. The heavy quark potentials from monopole and photon contribution are calculated at several values of the gauge parameter, and the former part shows always the confinement behavior.

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We report the preliminary results of the studies of hadron spectrum under the background of abelian and monopole gauge elds in quenched Wilson SU(3) QCD. Abelian gauge elds alone reproduce the same chiral limit as in the full case. Critical hopping parameter c and m are the same in both cases. We need more time to get a denite result in the case of monopole background. The photon contribution do not produce any mass gap in the chiral limit ( = c 0:17). The behavior is similar to those in the free photon case for c = 0:125.

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Abstract-An extremely scalable linear-algebraic algorithm was developed for quantum material simulation (electronic state calculation) with 10 8 atoms or 100-nm-scale materials. The mathematical foundation is generalized shifted linear equations ((zB − A)x = b), instead of conventional generalized eigenvalue equations. The method has a highly parallelizable mathematical structure. The benchmark shows an extreme strong scaling and a qualified time-to-solution on the full system of the K computer. The method was demonstrated in a real material research for ultra-flexible (organic) devices, key devices of next-generation Internet-of-Things (IoT) products. The present paper shows that an innovative scalable algorithm for a real research can appear by the co-design among application, algorithm and architecture.

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