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

TL;DR

The paper addresses large-$N$ volume independence in SU($N$) gauge theory using the twisted Eguchi-Kawai model with two-flavor adjoint Wilson fermions and a minimal twist $k=1$, showing that heavy adjoint fermions stabilize the $(\mathbb{Z}_N)^4$ center symmetry even when $k/N<1/9$ (N=121). It extends the framework to a partially reduced model on $\mathbb{R}^3 \times S^1$ and demonstrates adiabatic continuity under periodic adjoint fermions, with no deconfinement as the $S^1$ circumference is reduced, unlike the anti-periodic case. The results also compare symmetric and modified twists for volume independence, finding that the modified twist preserves volume independence more robustly. Together, these findings validate the use of minimal-twist adjoint TEK models as efficient, large-$N$ surrogates for 4D gauge theories and provide evidence for smooth confinement behavior across different compactifications and twist choices, with implications for computing hadronic observables in this reduced setting.

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 $κ=0.01$-$0.03$ ($\llκ_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 $(κ,b)$ 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 $κ=0.03$-$0.16$ 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.

Figures (13)

• Figure 1: Monte Carlo results for the magnitude of the Polyakov loop $P_\mu$ in the adjoint (T)EK model with $N=121$, $N_f=2$ and $b=0.36$ for $k=0,1$. (Left) the results for $k=0$, namely the adjoint EK model without twist. (Right) the results for $k=1$, namely the adjoint TEK model with the minimal twist.
• Figure 2: Monte Carlo results of Wilson loop for different sizes of loops. The logarithm of the expectation values of the Wilson loops are roughly proportional to their areas, which suggest the area law and the existence of nonzero string tension.
• Figure 3: Comparison of the schematic phase diagrams in the plane of gauge coupling $b$ and hopping parameter $\kappa$ for the adjoint EK model without twist (Left) and the adjoint TEK model with $k=1$ (Right).
• Figure 4: Another volume-independence order parameter $W_{123}$ in the partially reduced TEK model with the symmetric twist, $N=16,36$, $N_f=2$, $k=1$ and $L_4 = 2$ as a function of $b$. (Left) Heavy adjoint fermions $\kappa=0.03$: $W_{123}$ has nonzero values. (Right) Light adjoint fermions $\kappa=0.12$: $W_{123}$ gets smaller with larger $N$.
• Figure 5: Another volume-independence order parameter $W_{123}$ in the partially reduced TEK model with the modified twist, $N=16\,(l=1),\,36\,(l=3)$, $N_f=2$, and $L_4 = 2$ as a function of $b$. (Left) Heavy adjoint fermions $\kappa=0.03$: $W_{123}$ remains near zero over the entire range. (Right) Light adjoint fermions $\kappa=0.12$: $W_{123}$ remains near zero over the entire range.