$\mathbb{Z}_N$ stability and continuity in twisted Eguchi-Kawai model with two-flavor adjoint fermions

Abstract: We investigate the twisted Eguchi-Kawai (TEK) reduced model of four-dimensional $SU(N)$ gauge theory in the presence of two-flavor adjoint fermions (adjoint TEK model). Using Monte Carlo simulations with $N=121$, twist parameter $k=1$, hopping parameter $\kappa=0.01$-$0.03$ ($\ll\kappa_c $) and inverse 't Hooft coupling $b=0.30$-$0.45$, we show that heavy adjoint fermions stabilize the $(\mathbb{Z}_N)^4$ center-symmetric vacuum even for the minimal twist satisfying $k/\sqrt{N} < 1/9$, where the $(\mathbb{Z}_N)^4$ symmetry is spontaneously broken in the absence of adjoint fermions. This result also suggests that the adjoint TEK model with the minimal twist is equivalent to $SU(N)$ gauge theory over a broader $(\kappa,b)$ parameter region than the adjoint EK model without twist. We further extend our analysis to a partially reduced model to realize a geometry akin to $\mathbb{R}^3 \times S^1$ and study the theory under $S^1$ compactification with periodic adjoint fermions. Numerical simulations with $N=16$-$49$, $b=0.30$-$0.46$ and $\kappa=0.03$-$0.16$ supports the adiabatic continuity conjecture: with periodic adjoint fermions, the theory remains in a center-symmetric (confined) phase as the $S^1$ circle size is reduced, in contrast to the deconfining transition observed in the pure TEK model or in the TEK model with antiperiodic adjoint fermions. We present the Polyakov loop measurements and consistency checks supporting these findings.