Large-$N$ analysis of the critical behavior of the topological Ginzburg-Landau theory of self-dual Josephson junction arrays

Using large-N technique at fixed dimension ($d=3$), I examine the multicritical behavior of a $U(N/2)\times U(N/2)$ Ginzburg-Landau theory of two multicomponent complex fields interacting through gauge fields described by Maxwell terms and a mixed Chern-Simons term. This model is relevant to the dynamics of Cooper pairs and vortices in a self-dual Josephson junction array system near its superconductor-insulator quantum transition. I present calculations of the various critical exponents including $1/N$ corrections to the $N=\infty$ saddle point. I investigate in the scaling region the behavior of the renormalized zero-momentum four-point quartic couplings $u$ and $w$ in the action, and I calculate the $1/N$ correction to the $\beta$-functions and their fixed-point values. It is shown that the decoupled fixed point is destabilized in the presence of the mixed Chern-Simons term at the next-to-leading order. Finally, I examine the universal character of the conductivity at the critical point up to the next-to-leading order in $1/N$ expansion.

Figure 1

(a) ${\mathrm{\Gamma }}_{11}^{(4)}$ and (b) ${\mathrm{\Gamma }}_{12}^{(4)}$ to one loop. The solid lines represent the scalar field propagators with 1 corresponding to $\mathrm{\Psi }$ and 2 to $\mathrm{\Phi }$. The wiggly lines represent the gauge propagators with 1 corresponding to $a_{\mu }$ and 2 to $b_{\mu }$.

Figure 2

One-loop renormalization group flows in the $(\widehat{u},\widehat{w})$ plane at fixed dimension $d=3$.

Figure 3

Feynman rules for the $1/N$ expansion.

Figure 4

One-loop diagrams that determine the effective action of the gauge fields at large $N$.

Figure 5

Self-energy diagrams that enter the two-point scalar fields correlation function at order $1/N$.

Figure 6

Diagrammatic scalar field contributions to the four-point vertex function ${\mathrm{\Gamma }}_{1,1}^{(4)}$.

Figure 7

Diagrammatic gauge field contributions to the four-point vertex function ${\mathrm{\Gamma }}_{1,1}^{(4)}$.

Figure 8

Diagrammatic scalar field contributions to the four-point vertex function ${\mathrm{\Gamma }}_{1,2}^{(4)}$.

Figure 9

Diagrammatic gauge field contributions to the four-point vertex function ${\mathrm{\Gamma }}_{1,2}^{(4)}$.

Figure 10

Diagrammatic gauge field contributions to the four-point vertex function ${\mathrm{\Gamma }}_{1,2}^{(4)}$.

Figure 11

The flow of the interaction coupling constants for $N=\infty$.

Figure 12

Contributions with tadpole insertion to the vacuum polarization tensor to order $1/N$.

Figure 13

Contributions to the vacuum polarization tensor to order $1/N$.