Bosonization and functional renormalization group approach in the framework of ${\mathrm{QED}}_2$

Known results on two-dimensional quantum electrodynamics (${\mathrm{QED}}_2$) have been used to study the dependence of functional renormalization group equations on renormalization schemes and approximations applied for its bosonized version. It is demonstrated that the singularity of flow equations can be avoided in the optimized and power-law schemes for the bosonized model and the drawback of renormalization on bosonization is shown: it is indicated that renormalization of ${\mathrm{QED}}_2$ possibly requires interaction terms corresponding to higher frequency modes of its bosonized version.

Figure 1

Phase diagram of the MSG model for ${\beta }^2=4\pi$. RG trajectories are obtained by the integration of Eq. (19). Since SI does not occur in the RG flow, the critical ratio of the MSG model can be determined, $[\tilde{u}/{\tilde{M}}^2{]}_c=0.0625$. The arrows indicate the direction of the flow.

Figure 2

Phase diagram of the MSG model for ${\beta }^2=4\pi$. RG trajectories are obtained by the integration of Eq. (20) with $a=100$. The dashed lines correspond to RG trajectories where SI occurs in the RG flow, thus the critical ratio of the MSG model cannot be obtained.

Figure 3

Phase diagram of the MSG model for ${\beta }^2=4\pi$. RG trajectories are obtained by the integration of Eq. (21). The dashed lines correspond to RG trajectories where SI occurs in the RG flow, thus the critical ratio of the single-frequency MSG model cannot be obtained.

Figure 4

This figure shows how the IR value of the ratio $\tilde{u}/{\tilde{M}}^2$ obtained by the integration of RG Eqs. (19, 20) depends on the parameter $a$ of the regulator function. The same initial condition has been used for the numerical integration, ($\tilde{u}(\Lambda )={10}^{-5}$, ${\tilde{M}}^2(\Lambda )={10}^{-9}$). SI has occurred for $a\ne 1$, thus the RG flow stops at some finite scale where the ratio has been read off and plotted. It is possible to avoid SI but only for $a=1$. For $a\to \infty$ (i.e., $x\to 1$) the RG Eq. (20) becomes identical to that that was obtained by the power-law type regulator with $b=1$ (21); consequently in this case the IR values of the ratio coincide.

Figure 5

Phase diagram of the MSG model for ${\beta }^2=4\pi$. RG trajectories are obtained by the direct numerical solution of Eq. (23) (solid lines) incorporating the effect of higher harmonics.

Figure 6

RG trajectories are obtained by the integration of Eq. (16). Dashed lines correspond to RG trajectories where SI occurs, thus the critical ratio $r_{b=1}^c$ cannot be determined unambiguously. Vertical line shows the dimensionful mass scale, which remains unchanged under RG transformations.

Figure 7

RG trajectories are obtained by the numerical integration of Eqs. (14, 15) for the power-law regulator with $b=2$. Dashed lines correspond to RG trajectories in the broken symmetric phase of the single-frequency MSG model. Vertical line shows the dimensionful mass scale, which remains unchanged under RG transformations. The inset shows the scaling of the wave-function renormalization in the two phases.