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We consider a dual representation of an effective three-dimensional Polyakov loop model for the SU(3) theory at nonzero real chemical potential. This representation is free of the sign problem and can be used for numeric Monte-Carlo simulations. These simulations allow us to locate the line of second order phase transitions, that separates the region of first order phase transition from the crossover one. The behavior of local observables in different phases of the model is studied numerically and compared with predictions of the mean-field analysis. Our dual formulation allows us to study also Polyakov loop correlation functions. From these results, we extract the screening masses and compare them with large-N predictions.
The broad class of U(N) and SU(N) Polyakov loop models on the lattice are solved exactly in the combined large N, Nf limit, where N is a number of colors and Nf is a number of quark flavors, and in any dimension. In this ’t Hooft-Veneziano limit the ratio N/Nf is kept fixed. We calculate both the free energy and various correlation functions. The critical behavior of the models is described in details at finite temperatures and non-zero baryon chemical potential. Furthermore, we prove that the calculation of the N-point (baryon) correlation function reduces to the geometric median problem in the confinement phase. In the deconfinement phase we establish an existence of the complex masses and an oscillating decay of correlations in a certain region of parameters.
Quenched QCD at zero baryonic chemical potential undergoes a first-order deconfinement phase transition at a critical temperature Tc, which is related to the spontaneous breaking of the global center symmetry. Including heavy, dynamical quarks breaks the center symmetry explicitly and weakens the first-order phase transition. For decreasing quark masses the first-order phase transition turns into a smooth crossover at a Z2-critical point. The critical quark mass corresponding to this point has been examined with Nf=2 Wilson fermions for several Nτ in a recent study within our group. For comparison, we also locate the critical point with Nf=2 staggered fermions on Nτ=8 lattices. For this purpose we perform Monte Carlo simulations for several quark mass values and various aspect ratios in order to extrapolate to the thermodynamic limit. The critical mass is obtained by fitting to a finite size scaling formula of the kurtosis of the Polyakov loop. Our results indicate large discretization effects, requiring simulations on lattices with Nτ>8.
In the strong coupling and heavy quark mass regime, lattice QCD dimensionally reduces to effective theories of Polyakov loops depending on the parameters of the original Wilson action β,κ and Nτ. We apply coarse graining techniques to such theories in 1d and 2d, corresponding to lattice QCD at finite temperature and non-zero chemical potential in 1+1d and 2+1d, respectively. In 1d the method is applied to the effective theories up to O(κ4). Using the transfer matrix, the recursion relations are solved analytically. The thermodynamic limit is taken for some observables. Afterwards, continuum extrapolation is performed numerically and results are discussed. In 2d the coarse graining method is applied in the pure gauge and static quark limit. Running couplings are obtained and the fixed points of the transformations are discussed. Finally, the critical coupling of the deconfinement transition is determined in both limits. Agreement to about 12% with Monte Carlo results of 2+1d Yang-Mills theory from the literature is observed.
We empirically investigate algorithms for solving Connected Components in the external memory model. In particular, we study whether the randomized O(Sort(E)) algorithm by Karger, Klein, and Tarjan can be implemented to compete with practically promising and simpler algorithms having only slightly worse theoretical cost, namely Borůvka’s algorithm and the algorithm by Sibeyn and collaborators. For all algorithms, we develop and test a number of tuning options. Our experiments are executed on a large set of different graph classes including random graphs, grids, geometric graphs, and hyperbolic graphs. Among our findings are: The Sibeyn algorithm is a very strong contender due to its simplicity and due to an added degree of freedom in its internal workings when used in the Connected Components setting. With the right tunings, the Karger-Klein-Tarjan algorithm can be implemented to be competitive in many cases. Higher graph density seems to benefit Karger-Klein-Tarjan relative to Sibeyn. Borůvka’s algorithm is not competitive with the two others.
The order of the chiral phase transition of lattice QCD with unimproved staggered fermions is known to depend on the number of quark flavours, their masses and the lattice spacing. Previous studies in the literature for Nf∈{3,4} show first-order transitions, which weaken with decreasing lattice spacing. Here we investigate what happens when lattices are made coarser to establish contact to the strong coupling region. For Nf∈{4,8} we find a drastic weakening of the transition when going from Nτ=4 to Nτ=2, which is consistent with a second-order chiral transition reported in the literature for Nf=4 in the strong coupling limit. This implies a non-monotonic behaviour of the critical quark or pseudo-scalar meson mass, which separates first-order transitions from crossover behaviour, as a function of lattice spacing.