does not hold, modifications were proposed for this issue. They are quenched Eguchi-Kawai model (QEK model) [@Bhanot:1982sh; @Parisi:1982gp; @Gross:1982at] and twisted Eguchi-Kawal model (TEK model) [@GonzalezArroyo:1982ub]. Historically, most of the works previously done were based on the TEK model because this model is theoretically interesting and numerically more practical (and this model describes the NCYM as mentioned before).

In the TEK model, twisted boundary conditions are imposed and then the $\mathbb{Z}_N^D$ symmetry is ensured in the weak coupling limit. It is not obvious whether the symmetry is broken or not in the intermediate coupling region. There is no guarantee for not violating the symmetry. Numerical simulations in the 1980s, however, suggested that the $\mathbb{Z}_N^D$ symmetry is not broken throughout the whole coupling region. Then we have believed that the TEK model actually describes the large-$N$ limit of the gauge theory.

Recently some indication about the $\mathbb{Z}_N^D$ symmetry breaking was surprisingly reported in several context around the TEK model [@IO03; @Teper:2006sp][@Guralnik:2002ru; @Bietenholz:2006cz]. The most relevant discussion for the present article was done by Teper and Vairinhos in [@Teper:2006sp][^1] . They showed that the $\mathbb{Z}_N^D$ symmetry is really broken in the intermediate coupling region by the Monte-Carlo simulation for the $D=4$ TEK 