${\mathrm{QED}}_3$ on a space-time lattice: Compact versus noncompact formulation

We study quantum electrodynamics in a ($2+1$)-dimensional space-time with two flavors of dynamical fermions by numerical simulations on the lattice. We discretize the theory using both the compact and the noncompact formulations and analyze the behavior of the chiral condensate and of the monopole density in the finite lattice regime as well as in the continuum limit. By comparing the results obtained with the two approaches, we draw some conclusions about the possible equivalence of the two lattice formulations in the continuum limit.

Figure 1

Phase diagram in the ($x$, $T$) plane [14]; here $x$ represents the doping and $T$ the temperature.

Figure 2

Correlation between $⟨\overline{\chi }\chi ⟩$ and ${\rho }_m$ for the compact (circles) and the noncompact (boxes) theories for $N_f=2$ and $8^3$ lattice according to Ref. 33.

Figure 3

As in Fig. 2, according to our results.

Figure 4

${\beta }^2⟨\overline{\chi }\chi ⟩$ versus $\beta m$ in the compact formulation.

Figure 5

${\beta }^2⟨\overline{\chi }\chi ⟩$ versus $\beta m$ in the noncompact formulation.

Figure 6

$⟨\overline{\chi }\chi ⟩$ versus ${\rho }_m$ in both the compact and the noncompact formulations on a ${12}^3$ lattice.

Figure 7

$⟨\overline{\chi }\chi ⟩$ versus ${\rho }_m$ in both the compact and the noncompact formulations on a ${32}^3$ lattice.

Figure 8

${\beta }^3{\rho }_m$ versus $\beta m$ in the compact formulation.

Figure 9

${\beta }^3{\rho }_m$ versus $\beta m$ in the noncompact formulation.