${\mathrm{QED}}_3$-Inspired Three-Dimensional Conformal Lattice Gauge Theory without Fine-Tuning

We construct a conformal lattice theory with only gauge degrees of freedom based on the induced nonlocal gauge action in ${\mathrm{QED}}_3$ coupled to large number of flavors $N$ of massless two-component Dirac fermions. This lattice system displays signatures of criticality in gauge observables, without any fine-tuning of couplings and can be studied without Monte Carlo critical slowdown. By coupling exactly massless fermion sources to the lattice gauge model, we demonstrate that nontrivial anomalous dimensions are induced in fermion bilinears depending on the dimensionless electric charge of the fermion. We present a proof-of-principle lattice computation of the Wilson-coefficients of various fermion bilinear three-point functions. Finally, by mapping the charge $q$ of fermion in the model to a flavor $N$ in massless ${\mathrm{QED}}_3$, we point to a universality in low-lying Dirac spectrum and an evidence of self-duality of $N=2$ ${\mathrm{QED}}_3$.

Figure 1

Mass anomalous dimension as computed at different charges $q$. Left: the dependence of smallest Dirac eigenvalue ${\tilde{\mathrm{\Lambda }}}_1(q)$, normalized by free theory value, on $L$. The curves are the fits to extract the leading $L^{-{\gamma }_S}$ dependence. Right: the finite size scaling of the scalar two-point function $\tilde{G}(|x_{12}|)$ at separations $|x_{12}|=L/4$. The lines are the expected asymptotic dependence $\tilde{G}(|x_{12}|=L/4)\sim L^{2{\gamma }_S}$ at different $q$, with ${\gamma }_S$ determined from ${\tilde{\mathrm{\Lambda }}}_n$.

Figure 2

Left: a configuration of collinearly placed operators. Right: the effective OPE coefficients ${\tilde{C}}_{ijk}(z_2,z_3;q)$ of three different collinear three-point functions (distinguished by colors and slightly displaced) are shown as a function of $z_3$ at three different fixed $z_2=6$ (open triangles), 8 (open circles), 10 (filled circles).

Figure 3

Bottom panel: mass anomalous dimension ${\gamma }_S$ is shown as a function of charge $q$. The filled circles are numerical determinations in the lattice model and the black band is the resulting spline interpolation of the data. The expected region corresponding to $N=2$, 4, 6, 8 flavor ${\mathrm{QED}}_3$ are shown by the rectangular boxes, so as to match the values of ${\gamma }_S$. The dashed line is the unitarity bound on ${\gamma }_S$. Top panel: the $C_V$ in the lattice model and $C_V^{\mathrm{top}}=q^2/(4{\pi }^4)$ are shown as a function of $q$. The two intersect in the region of $q$ corresponding to $N=2$ ${\mathrm{QED}}_3$, as inferred from the bottom panel.

Figure 4

Distribution of scaled eigenvalues $z_i=({\mathrm{\Lambda }}_i/⟨{\mathrm{\Lambda }}_i⟩)$ for the three lowest eigenvalues (left to right) from the conformal lattice model at $q=2.5$ (top) and $q=2.0$ (bottom) are compared with those from $N=2$ and $N=8$ ${\mathrm{QED}}_3$. For the lattice model, results from $L=24$, 28, 32 are shown, where as for ${\mathrm{QED}}_3$, results from two large box sizes $\ell$ (measured in units of coupling $g^2$) are shown.