Gross-Witten-Wadia phase transition in induced QCD on the graph

In this paper, we examine a modification of the Kazakov-Migdal (KM) model with gauge group $U(N_c)$, where the adjoint scalar fields in the conventional KM model are replaced by $N_f$ fundamental scalar fields (FKM model). After tuning the coupling constants and eliminating the fundamental scalar fields, the partition function of this model is expressed as an integral of a graph zeta function weighted by unitary matrices. The FKM model on cycle graphs at large $N_c$ exhibits the Gross-Witten-Wadia (GWW) phase transition only when $N_f>N_c$. In the large $N_c$ limit, we evaluate the free energy of the model on a general graph in two distinct parameter regimes and demonstrate that the FKM model generally consists of multiple phases. The effective action of the FKM model reduces to the standard Wilson action by taking an appropriate scaling limit when the graph consists of plaquettes (fundamental cycles) of the same size, as in the square lattice case. We show that, for the FKM model on such a graph, the third-order GWW phase transition universally occurs in this scaling limit.

Figure 1

A cycle graph $C_n$ with $n$ vertices and edges.

Figure 2

This show that an example of the eigenvalue distributions in the different phases. We set $\gamma =10$, then the critical point is ${\alpha }^{*}=\frac{1}{19}=0.0526316\cdots$. (a) For $\alpha <{\alpha }^{*}$, the eigenvalue distribution is attached at $\theta =\pm \pi$ with the periodicity. (b) For $\alpha >{\alpha }^{*}$, the eigenvalue distribution behaves in the same manner as Wigner’s semi-circle.

Figure 3

A triangle-square (TS) graph with five vertices and six edges.

Figure 4

For the triangle-square asymmetric graph, this plot shows the behavior of $\frac{d}{d\lambda }(F_{\mathrm{TS}}^{-}-F_{\mathrm{TS}}^{+})$ for $\gamma =20$, 200, 2000 and the $\gamma \to \infty$ limit in the region of the coupling $\lambda \sim \mathcal{O}(1)$, with setting $q=(\gamma \lambda {)}^{-1/3}$. Since these curves do not cross nor touch the $\lambda$-axis, $F_{\mathrm{TS}}^{-}$ and $F_{\mathrm{TS}}^{+}$ cannot be connected smoothly in this region.

Figure 5

A double triangle (DT) graph with four vertices and five edges.

Figure 6

For the double triangle (DT) graph, this plot shows that the behavior of $\frac{d}{d\lambda }(F_{\mathrm{DT}}^{-}-F_{\mathrm{DT}}^{+})$ for $\gamma =20$, 200, 2000 and the $\gamma \to \infty$ limit with setting $q=(\gamma \lambda {)}^{-1/3}$. We see that the third-order phase transition never occurs for the finite $\gamma$, since the corresponding curves cross the $\lambda$-axis. The curve for $\gamma \to \infty$ contacts with the $\lambda$-axis at $\lambda =2$, which is the third-order phase transition point of the GWW model.