Effect of stout smearing on the phase diagram from multiparameter reweighting in lattice QCD

The phase diagram and the location of the critical endpoint (CEP) of lattice QCD with unimproved staggered fermions on a $N_t=4$ lattice was determined fifteen years ago with the multiparameter reweighting method by studying Fisher zeros. We first reproduce the old result with an exact algorithm (not known at the time) and with statistics larger by an order of magnitude. As an extension of the old analysis we introduce stout smearing in the fermion action in order to reduce the finite lattice spacing effects. First we show that increasing the smearing parameter $\rho$ the crossover at $\mu =0$ gets weaker, i.e., the leading Fisher zero gets farther away from the real axis. Furthermore as the chemical potential is increased the overlap problem gets severe sooner than in the unimproved case, therefore shrinking the range of applicability of the method. Nevertheless certain qualitative features remain, even after introducing the smearing. Namely, at small chemical potentials the Fisher zeros first get farther away from the real axis and later at around $a{\mu }_q=0.1–0.15$ they start to get closer, i.e., the crossover first gets weaker and later stronger as a function of $\mu$. However, because of the more severe overlap problem the CEP is out of reach with the smeared action.

Figure 1

The real and imaginary part of the Fisher zero closest to ${\beta }_0$ as a function of $\mu$ obtained with standard staggered rooting and with geometric matching on a ${12}^3\times 4$ lattice at $\rho =0$.

Figure 2

The real and imaginary part of the infinite volume extrapolation of the leading Fisher zero via standard rooting and geometric matching at $\rho =0$. The red point with error bars denotes our estimate of the location of the critical endpoint for $N_t=4$ unimproved staggered fermions.

Figure 3

The $\mu$ dependence of the imaginary part of the Fisher zeros extrapolated to infinite volume at several values of the smearing parameter $\rho$. The left figure shows only the values close to $\mu =0$.

Figure 4

The distribution of the weights $W_i$ of Eq. (9) for $N_s=12$ for different values of $\rho$ and $\mu$ obtained with geometric matching. The distribution of the positive and negative weights are shown separately on logarithmic scale. At zero smearing there is no outlier configurations even at large $\mu$, however at large $\mu$ and finite smearing configurations appears with extremely large weights. The few outlier configuration appearing indicate a bad sampling of the Monte Carlo integration.

Figure 5

Distributions of the positive weights $W_i$ of Eq. (9) on a ${12}^3\times 4$ lattice at $a\mu =0.2$ for different values of the smearing parameter.