Remarks on strange-quark simulations with Wilson fermions

In the simulation of QCD with $2+1$ flavors of Wilson fermions, the positivity of the fermion determinant is generally assumed. We present evidence that this assumption is in general not justified and discuss the consequences of this finding.

Figure 1

Examples of the identification of eigenvalues of $\widehat{Q}$ crossing as the strange-quark mass is increased by $\mathrm{\Delta }m$. We plot measurements on close-by configurations from the S400r001 run. The value of $\mathrm{\Delta }m$ at which zero is crossed is a measure for the eigenvalue $-\lambda$ of $D(m_{\mathrm{s}})$. Note that crossings occur when the other eigenvalues have at least doubled their value, again indicated by the increased gap in the other eigenvalues.

Figure 2

Monte Carlo time history of the number of negative real eigenvalues of the Dirac operator. Configurations are spaced by 4 units of molecular dynamics time; only for J303, this spacing amounts to 8 units. We observe that changes in the number are rare.

Figure 3

On the left, function for a single variable which replaces the $-\log (x)$ of the fermion action, for the light quark due to the twisted-mass reweighting and for the strange due to the RHMC. We show the parameters of the H105r001 and H105r005 runs. Note that for the latter the potential barrier is roughly 1 unit higher. On the right, we give the measured distribution of the lowest eigenvalue of $\widehat{Q}$, not including the reweighting factors and without giving uncertainties, which are difficult to compute in the case of the H105r005 runs due to the large autocorrelations discussed in the text. The scales of the x axis in the two plots is different for better readability.

Figure 4

The violation of the HMC Hamiltonian in the top row is confronted with the number of negative eigenvalues of the Dirac operator in the bottom. We show the data for the runs H105r001 and H105r005. There is clear evidence that the period of poor acceptance, i.e., small $\exp (-dH)$, is correlated with the appearance of a negative real eigenvalue in the latter run. The different choice of molecular dynamics integrator might be responsible for the system staying in this state for an extended time.