Finite-volume scaling of the Wilson-Dirac operator spectrum

The microscopic spectral density of the Hermitian Wilson-Dirac operator is computed numerically in quenched lattice QCD. We demonstrate that the results given for fixed index of the Wilson-Dirac operator can be matched by the predictions from Wilson chiral perturbation theory. We test successfully the finite volume and the mass scaling predicted by Wilson chiral perturbation theory at fixed lattice spacing.

Figure 1

The eigenvalue density of $D_5$ in the sector with $\nu =0$ for $V={16}^4$ (top) and $V={20}^4$ (bottom) both with ${\beta }_{Iw}=2.635$ and $m_0=-0.216$. The $x$-axis has been rescaled by $\Sigma V$, where $\Sigma {16}^4=157.5$. In the top plot the smooth red curve displays a fit of the analytic prediction for the microscopic density of $D_5$ to the ${16}^4$ data. The fit values are $\widehat{m}=3.5$ and ${\widehat{a}}_8=0.35$ (${\widehat{a}}_6={\widehat{a}}_7=0$). In the plot below the smooth red line shows the prediction for the ${20}^4$ data ($\widehat{m}=8.5$ and ${\widehat{a}}_8=0.55$) obtained from finite-volume scaling of the ${16}^4$ values. We observe that the finite-volume scaling at fixed lattice spacing is well respected.

Figure 2

The eigenvalue density of $D_5$ in the sector with $\nu =1$ for $V={16}^4$ (top) and $V={20}^4$ (bottom) both with ${\beta }_{Iw}=2.635$ and $m_0=-0.216$. The $x$-axis has been rescaled by $\Sigma V$, where $\Sigma {16}^4=157.5$. The smooth red curves displays the analytic prediction for the microscopic density of $D_5$. The values, $\widehat{m}=3.5$ and ${\widehat{a}}_8=0.35$ for ${16}^4$ and $\widehat{m}=8.5$ and ${\widehat{a}}_8=0.55$ are obtained from the fit in the top panel of Fig. 1 and by finite-volume scaling from ${16}^4$ to ${20}^4$ (${\widehat{a}}_6={\widehat{a}}_7=0$).

Figure 3

The eigenvalue density of $D_5$ for ${\beta }_{Iw}=2.79$, $V={20}^4$ and $m_0=-0.184$ top: $\nu =0$ and bottom: $|\nu |=1$. For the $\nu =0$ data we find the best values $\widehat{m}=3.5$, ${\widehat{a}}_8=0.35$ and $\Sigma V=216$. With these values we then test the prediction against the $|\nu |=1$ data.

Figure 4

The eigenvalue density of $D_5$ for ${\beta }_{Iw}=2.79$, $V={20}^4$ and $m_0=-0.178$ top: $\nu =0$ and bottom: $|\nu |=1$. We use the value $\Sigma V=216$ (found for $m_0=-0.184$) and $\delta m=-0.006$ to determine the value $\widehat{m}=4.8$ for the ${\beta }_{Iw}=2.79$, $V={20}^4$, $m_0=-0.178$ data. Since the coupling is unchanged also ${\widehat{a}}_8=0.35$ is fixed by the previous fit. The parameter-free predictions for $\nu =0$ and $|\nu |=1$ are then plotted against the data. We observe that the mass scaling at fixed lattice spacing works well.

Figure 5

The eigenvalue density of $D_5$ for ${\beta }_{Iw}=2.79$, $V={20}^4$ and $|\nu |=2$ top: $m_0=-0.184$ and bottom: $m_0=-0.178$. The smooth curves are the parameter-free predictions from WCPT (all parameters are fixed from the fit to the $\nu =0$, $m_0=-0.184$ data).

Figure 6

The dependence of the eigenvalue density of $D_5$ for $\nu =0$ on top: the quark mass $\widehat{m}$ and on bottom: the lattice artifacts ${\widehat{a}}_8$. The central (red) line in both plots correspond to the (red) analytic curve which is plotted against the data in the top panel of Fig. 4.

Figure 7

The eigenvalue density of $D_5$ for $\nu =0$. Shown are the lattice data and best fit, previously displayed in the top panel of Fig. 3, along with the $a=0$ result at the same value of $\widehat{m}$. Clearly it is essential to include the order $a^2$ effects in WCPT in order to match the data. Note that scale on the $y$-axis is different from that in the other plots.

Figure 8

The eigenvalue density of $D_5$ for $\nu =0$ top: the dependence on ${\widehat{a}}_6$ and bottom: the dependence on ${\widehat{a}}_7$. The more oscillating (red) line in both plots has $a_6=a_7=0$ and corresponds to the (red) analytic curve which is plotted against the data in the top panel of Fig. 4. Both ${\widehat{a}}_6^2<0$ or ${\widehat{a}}_7^2<0$ tends to smear out the oscillations. Since the magnitude of the oscillations in the lattice data presented in Fig. 4 is of the same magnitude as those in the analytic prediction with ${\widehat{a}}_6={\widehat{a}}_7=0$ we can put an upper bound on these $|{\widehat{a}}_6^2|$, $|{\widehat{a}}_7^2|<0.01$.

Figure 9

The distribution of the first (black) and second (blue) eigenvalue top: in the ${16}^4$ ${\beta }_{Iw}=2.635$ data set and bottom: in the ${20}^4$ ${\beta }_{Iw}=2.79$ data set. The (red) histograms show the first and second eigenvalue distribution in a WRMT simulation with $n=100$ and 100000 matrices.

Figure 10

The distribution of the real modes of $D_W$ before subtraction of the bare mass and scaling of the axis.

Figure 11

top: The distribution of the real modes. bottom: the distribution ${\rho }_{\chi }$ defined in Eq. (20). All parameters are fixed from the fits to ${\rho }_5$. The eigenvalue data are first rescaled by $\Sigma V$ and then shifted by $m_0\Sigma V-\widehat{m}$ where $m_0$ is the lattice mass and $\widehat{m}$ is the value obtained previously for this $m_0$.