Polyakov loops and the Hosotani mechanism on the lattice

TL;DR

This work investigates nonperturbative realization of the Hosotani mechanism in SU(3) gauge theory on a $R^3\times S^1$-like lattice setup using adjoint and fundamental fermions. By analyzing Polyakov loop eigenvalue densities and comparing with one-loop effective potential predictions, the authors identify lattice phases corresponding to SU(3) confinement, SU(3) deconfinement, SU(2)×U(1), and U(1)×U(1), thereby linking AB-phase minima to observable lattice data. They also explore fundamental fermions with varying boundary conditions to reveal Z3 symmetry breaking and Roberge-Weiss-type structures, establishing a consistent nonperturbative picture of Hosotani dynamics and its phase structure. The results pave the way for future nonperturbative studies of gauge-Higgs unification, including spectrum analyses and higher-dimensional implementations.

Abstract

We explore the phase structure and symmetry breaking in four-dimensional SU(3) gauge theory with one spatial compact dimension on the lattice ($16^3 \times 4$ lattice) in the presence of fermions in the adjoint representation with periodic boundary conditions. We estimate numerically the density plots of the Polyakov loop eigenvalues phases, which reflect the location of minima of the effective potential in the Hosotani mechanism. We find strong indication that the four phases found on the lattice correspond to SU(3)-confined, SU(3)-deconfined, SU(2) x U(1), and U(1) x U(1) phases predicted by the one-loop perturbative calculation. The case with fermions in the fundamental representation with general boundary conditions, equivalent to the case of imaginary chemical potentials, is also found to support the $Z_3$ symmetry breaking in the effective potential analysis.

Figures (19)

• Figure 1: A sketch of the possible values of Polyakov loop in each phase. $(A_1,A_2,A_3)$ and $(B_1,B_2,B_3)$ form ${\rm Z}_3$ triplets. $\Theta$ and $\Phi$ are interpolating configurations useful in the description of the mixed fermion content case.
• Figure 2: Three contributions to the effective potential $V_{\rm eff}^{\rm g+gh}(\theta_H)$, $V_{\rm eff}^{\rm ad}(\theta_H)$ and $V_{\rm eff}^{\rm fd}(\theta_H)$ are plotted for the case with $d=4$, $m_{\rm fd} = m_{\rm ad} =0$ and $\alpha_{\rm fd} = \alpha_{\rm ad} =0$. $R$ is normalized to unity.
• Figure 3: Effective potential for the case of $N_{\rm ad} = 2$ adjoint fermion with periodic boundary condition ($\alpha_{\rm ad}=0$) for the values of $m_{\rm ad} R$ in $d=4$. They are corresponding to the $A$ phase, the $A$-$B$ transition point, the $B$ phase, the $B$-$C$ transition point and the $C$ phase, respectively. Lower values of $V_{{\rm eff}}$ are indicated by lighter colors.
• Figure 4: The effective potential with four massless fundamental fermions in the $(\theta_1/\pi,\theta_2/\pi)$ plane. Boundary condition of the fermions are changed from $\alpha_{\rm fd}=0$ to $\alpha_{\rm fd}=5\pi/3$. We plot the three phase transitions $A_2$-$A_3$, $A_2$-$A_1$ and $A_1$-$A_3$, and three phases $A_1$, $A_2$ and $A_3$. Lower values of $V_{{\rm eff}}$ are indicated by lighter colors.
• Figure 5: $V_{{\rm eff}}$ for $(N_{\rm fd},N_{\rm ad}) = (4,0)$ versus $\alpha_{\rm fd}$. Solid, dashed and dot-dashed lines correspond to values at $A_1$, $A_2$ and $A_3$, respectively. Thick [thin] lines are for $m_{\rm fd}=0$ [$m_{\rm fd} R=0.4$].