Scale invariance of parity-invariant three-dimensional QED

We present numerical evidences using overlap fermions for a scale-invariant behavior of parity-invariant three-dimensional QED with two flavors of massless two-component fermions. Using finite-size scaling of the low-lying eigenvalues of the massless anti-Hermitian overlap Dirac operator, we rule out the presence of a bilinear condensate and estimate the mass anomalous dimension. The eigenvectors associated with these low-lying eigenvalues suggest critical behavior in the sense of a metal-insulator transition. We show that there is no mass gap in the scalar and vector correlators in the infinite-volume theory. The vector correlator does not acquire an anomalous dimension. The anomalous dimension associated with the long-distance behavior of the scalar correlator is consistent with the mass anomalous dimension.

Figure 1

The dimensionless improved eigenvalues ${\lambda }_i\ell$ at various $\ell$ are shown as a function of $1/L^2$ in order to show the remaining dominant $1/L^2$ lattice artifact after the improvement using $Z_m$. The left and the right panels are for the smallest ${\lambda }_1$ and a larger ${\lambda }_6$ respectively. The data points are from the simulations using $L=12$, 14, 16, 20 and 24 lattices. The continuum limits are taken using $1/L^2$ extrapolations, which are shown as the straight lines in the plots.

Figure 2

The top panel shows representative eigenvalues in the continuum along with the ones in finite lattice spacing, as a function of $\ell$. In the bottom panel, the continuum limits using overlap (filled symbols) and Wilson-Dirac fermions (open symbols) are compared.

Figure 3

The ${\chi }^2/\mathrm{DOF}$ for the fits using the ansatz in Eq. (40) to the eigenvalue data are shown as a function of the exponent ${\gamma }_m$. These are shown by the colored solid curves for the smallest six eigenvalues. The corresponding 68% confidence intervals for the exponent are shown by the error bars.

Figure 4

$\log (\lambda \ell )$ is shown as a function of $-\log (\ell )$ for the smallest six eigenvalues after taking their continuum limits. A power law $\lambda \sim {\ell }^{-{\gamma }_m-1}$ would be a straight line with a slope ${\gamma }_m$ in this plot. No distinct power law is seen in the volumes that we simulated. The $\ell$ dependence is well described by a $[1/1]$ Padé approximant in Eq. (40) which we use to estimate the eventual power-law behavior that would set in at even larger $\ell$ than we used. The best fits using ${\gamma }_m=1$ are shown by the solid curves.

Figure 5

(Top panel) The scaling $I_2\sim {\ell }^{-3+\eta }$ is shown using the $L=24$ data. (Bottom panel) The continuum limits of the exponent $-3+\eta$ using both $I_2$ (unimproved) and $I_2/Z_m$ (improved) are shown. We estimate $-3+\eta =-2.62(1)$ in the continuum limit. This is close to the value $-2.68(1)$ estimated using Wilson fermions.

Figure 6

The exponent $-3+\eta$ for the finite-size scaling of the IPR for various eigenvectors corresponding to the 20 low-lying eigenvalues ${\lambda }_i$.

Figure 7

Lattice-spacing effect on number variance. (Left) Number variance ${\mathrm{\Sigma }}_2(n)$ as a function of $n$ is shown for $\ell =250$ at various lattice sizes $L$. As lattice spacing is reduced, the number variance at larger $n$ increases and approaches the solid black line which has a slope of $\frac{\eta }{6}=0.063$. (Right) ${\mathrm{\Sigma }}_2(n)$ for the overlap ($L=20$) and the Wilson-Dirac fermions ($L=28$) at $\ell =250$ are compared.

Figure 8

The number variance is shown as a function of $n$ for various values of $\ell$ on $L=20$ lattice. The black line has a slope of $\frac{\eta }{6}=0.063$. The slope of the number variance increases with $\ell$ and approaches $\eta /6$ which is a trend opposite of that expected when a bilinear condensate is present.

Figure 9

The figure shows the dimensionless effective mass $M(t)\ell$ as a function of $\frac{t}{\ell }$ for different $\ell$, represented by different colored symbols. The top and the bottom panels are for the scalar and vector respectively, as determined on a $L=24$ lattice. The value of $M\ell$ along the plateau seen in both the panels gives an upper bound on the mass gap (times $\ell$) present in the scalar and vector correlators. Since the position of the plateau seems to be independent of $\ell$, $M\sim 1/\ell$.

Figure 10

Small lattice spacing effects in the correlators. The zero-spatial-momentum correlators $G(t)$ are plotted as a function of physical separation $t$. The top panels are for the vector and the bottom ones are for the scalar. The different colored symbols denote the different $L^3$ lattices used to determine the correlators at fixed $\ell$. Lattice effects are small in both finer ($\ell =16$ in the left panels) and coarser ($\ell =160$ in the right panels) lattices.

Figure 11

The figures show the vector and scalar correlators $G$ as a function of physical separation $t$, by putting together the data from various values of $\ell$, on a $L=24$ lattice. The different colored symbols correspond to various representative values of $\ell$. (Left) The vector shows a power-law behavior as a function of $t$. The small deviations are due to the effect of periodicity at finite $\ell$. (Right) The scalar does not show a simple power law, nevertheless massless, as seen by the concave-up nature of the correlator in the log-log plot.

Figure 12

The tangents, which have a slope $d\log (G)/d\log (t)$, are shown along with the correlator data, the same as the ones shown in Fig. 11. In the left panel, the red solid line corresponds to a $t^{-2}$ power law. In the right panel, the different colored lines are the tangents determined at various $t=t_o(\ell )\equiv \ell /L$ using the finite-$\ell$ scalar correlator data on a $L=24$ lattice, which are represented by symbols of the same color.

Figure 13

The scale-dependent exponent $k(t)$ for the scalar. The figure shows the behavior of $k(t_o)$ defined as $d\log (G(t))/d\log (t)$ determined at various $t=t_o(\ell )\equiv \ell /L$ using the various finite-$\ell$ scalar correlator data. The different colored symbols correspond to different $L$. At smaller $t_o$, the value of $k$ seems to flow to 2. At larger $t_o$, it keeps decreasing and the value it approaches as $t_o\to \infty$ is the power-law exponent corresponding to the infrared fixed point. By using a [$1/1$] Padé in $\tanh (1/t_o)$, shown by the red solid line, we estimate the infinite-$t$ limit of $k$ to be 0.4(2). This corresponds to a mass anomalous dimension ${\gamma }_m=0.8(1)$.

Figure 14

The monopole density ${\rho }_m$ at various $\ell$ as the continuum limit is approached by taking $L\to \infty$. The left panel shows the result for the unsmeared, thin gauge links. The solid lines show how the density approaches zero using a quadratic $1/L$ extrapolation. In the right panel, the result when optimal HYP smearing is used is shown. With this improvement, the monopole density is consistent with zero at all simulation points.

Figure 15

Run-time history of the lowest eigenvalue using two different stopping criteria for the inner CG, ${\epsilon }_{\mathrm{in}}={10}^{-6}$ and ${10}^{-8}$. The corresponding stopping criterion for the outer CG is $100{\epsilon }_{\mathrm{in}}$.