Spectral properties of the Wilson-Dirac operator and random matrix theory

Random matrix theory has been successfully applied to lattice quantum chromodynamics. In particular, a great deal of progress has been made on the understanding, numerically as well as analytically, of the spectral properties of the Wilson-Dirac operator. In this paper, we study the infrared spectrum of the Wilson-Dirac operator via random matrix theory including the three leading order $a^2$ correction terms that appear in the corresponding chiral Lagrangian. A derivation of the joint probability density of the eigenvalues is presented. This result is used to calculate the density of the complex eigenvalues, the density of the real eigenvalues, and the distribution of the chiralities over the real eigenvalues. A detailed discussion of these quantities shows how each low-energy constant affects the spectrum. Especially we consider the limit of small and large (which is almost the mean field limit) lattice spacing. Comparisons with Monte Carlo simulations of the random matrix theory show a perfect agreement with the analytical predictions. Furthermore we present some quantities which can be easily used for comparison of lattice data and the analytical results.

Figure 1

Schematic plots of the effects of $W_6$ (left) and of $W_7$ (right). The low-energy constant $W_6$ broadens the spectrum parallel to the real axis according to a Gaussian with width $4{\widehat{a}}_6=4\sqrt{-V{W}_6{\tilde{a}}^2}$, but does not change the continuum spectrum in a significant way. When $W_7$ is switched on and $W_6=0$ the purely imaginary eigenvalues invade the real axis through the origin and only the real (green crosses along the real axis) are broadened by a Gaussian with width $4{\widehat{a}}_7=4\sqrt{-V{W}_7{\tilde{a}}^2}$.

Figure 2

The density of additional real modes is shown for various parameters ${\widehat{a}}_{6/7/8}$. The analytical results (solid curves) agree with the Monte Carlo simulations of the random matrix theory (histogram [MC] with bin size 0.5 and with different ensemble and matrix sizes such that statistics are about 1%–5%) for $\nu =1$. We plot only the positive real axis since ${\rho }_{\mathrm{r}}$ is symmetric. Notice that the height of the two curves for ${\widehat{a}}_7={\widehat{a}}_8=0.1$ (right) are 2 orders smaller than the height of the other curves (left) and because of bad statistics we have not performed simulations for this case. Notice the soft repulsion of the additional real modes from the origin at large ${\widehat{a}}_7=\sqrt{-V{W}_7}\tilde{a}$ as discussed in the introductory section. The parameter ${\widehat{a}}_6=\sqrt{-V{W}_6}\tilde{a}$ smoothens the distribution.

Figure 3

Log-log plots of $N_{\mathrm{add}}$ as a function of ${\widehat{a}}_8=\sqrt{V{W}_8{\tilde{a}}^2}$ for $\nu =0$ (left) and $\nu =2$ (right). The analytical results (solid curves) are compared to Monte Carlo simulations of RMT (symbols; ensemble and matrix size varies such that the statical error is about 1%–5%). Notice that $W_6$ has no effect on $N_{\mathrm{add}}$. The saturation around zero is due to a nonzero value of ${\widehat{a}}_7=\sqrt{-V{W}_7{\tilde{a}}^2}$. For ${\widehat{a}}_7=0$ (lowest curves) the average number of additional real modes behaves like ${\widehat{a}}_8^{2\nu +2}$; see Ref. 23.

Figure 4

At large lattice spacing the density of additional real modes develops square root singularities at the boundaries. The analytical results at $\widehat{a}\to \infty$ (solid curves) are compared to Monte Carlo simulations at nonzero, but large lattice spacing (histogram [MC], with bin size 50, ${\widehat{a}}_6=\sqrt{-V{W}_6{\tilde{a}}^2}=0.01$ and $n=2000$ for an ensemble of 1000 matrices). Due to the finite matrix size and the finite lattice spacing, ${\rho }_{\mathrm{r}}$ has a tail which drops off much faster than the size of the support. The low-energy constant ${\widehat{a}}_8=\sqrt{V{W}_8{\tilde{a}}^2}$ is chosen equal to 10. Therefore the boundary is at $\widehat{x}=800$ which is confirmed by the Monte Carlo simulations. The dependence on $W_6$ and $\nu$ is completely lost.

Figure 5

Comparison of the analytical result (solid curves) and Monte Carlo simulations of the random matrix theory (histogram [MC] with bin size equal to 0.4 and varying ensemble size and matrix size such that the statistical error is about 1%–5%) for the density of the complex eigenvalues projected onto the imaginary axis. The index of the Wilson-Dirac operator is $\nu =1$ for all curves. Notice that ${\widehat{a}}_6=\sqrt{-V{W}_6}\tilde{a}$ does not affect this density. The comparison of ${\widehat{a}}_7={\widehat{a}}_8=0.1$ with the continuum result (black, thick curve) shows that ${\rho }_{\mathrm{cp}}$ is still a good quantity to extract the chiral condensate $\Sigma$ at small lattice spacing.

Figure 6

The analytical result (solid curves) for ${\rho }_{\chi }$ is compared to Monte Carlo simulations of RMT (histogram [MC] with bin size 0.6 and varying ensemble and matrix size such that the statistical error is about 1%–5%) for $\nu =1$. We plotted only the positive real axis since the distribution is symmetric around the origin. At small ${\widehat{a}}_8=\sqrt{V{W}_8{\tilde{a}}^2}$ the distributions for $({\widehat{a}}_6,{\widehat{a}}_7)=(\sqrt{-V{W}_6{\tilde{a}}^2},\sqrt{-V{W}_7{\tilde{a}}^2})=(1,0.1)$, (0.1, 1) are almost the same Gaussian as the analytical result predicts. At large ${\widehat{a}}_8$ the maximum reflects the predicted square root singularity which starts to build up. We have not included the case ${\widehat{a}}_{6/7/8}=0.1$ since it exceeds the other curves by a factor of 10 to 100.

Figure 7

We compare the analytical result of ${\rho }_{\chi }$ (solid curves) with Monte Carlo simulations of RMT (histogram [MC] with bin size 0.6 and with varying ensemble and matrix size such that the statistical error is about 1%–5%) for $\nu =2$. Again we only plotted the positive real-axis because ${\rho }_{\chi }$ is symmetric in the quenched theory. Two curves with $W_{6/7}=0$ and $W_8=0.1$, 0.5 (highest, purple and thick, black curve) are added to emphasize that the two peaks (${\rho }_{\chi }$ has to be reflected at the origin) can be strongly suppressed by nonzero $W_{6/7}$ although they are only of the same order as $W_8$. Recall that the two peaks are relics of a $2\times 2$ GUE which is formed by $W_8$.