Symmetry Breaking In Twisted Eguchi-Kawai Models

TL;DR

This study numerically investigates four-dimensional SU($N$) twisted Eguchi-Kawai (TEK) models with a symmetric twist to test the validity of large-$N$ planar reduction. It finds that for $N\geq100$ the $Z_N^4$ center symmetry spontaneously breaks over a broad range of couplings, while for $N\leq81$ the symmetry remains unbroken and no physical confining/deconfining window appears in the reduced model. The breaking proceeds through a sequence of $Z_N^k$ phases, with stable extrema near $U_\mu\in Z_N$ preventing tunnelling to the twist-eater vacuum, consistent with fluxon-based interpretations. These results cast doubt on the utility of conventional TEK reductions for capturing large-$N$ planar physics and motivate exploring alternative reduced-model constructions or higher-$N$ analyses, including anisotropic or partially reduced variants.

Abstract

We present numerical evidence for the spontaneous breaking of the centre symmetry of four-dimensional twisted Eguchi-Kawai models with SU(N) gauge group and symmetric twist, for sufficiently large N. We find that for N greater or equal than 100 this occurs for a wide range of bare couplings. Moreover for N less or equal than 144, where we have been able to perform detailed calculations, there is no window of couplings where the physically interesting confined and deconfined phases appear in the reduced model. We provide a possible interpretation for this in terms of generalised 'fluxon' configurations. We discuss the implications of our findings for the validity and utility of space-time reduced models as N goes to infinity.

Figures (11)

• Figure 1: Average value of the real part of the plaquette, $\left\langle \mathrm{Re}~u_p\right\rangle$, in the SU(64) TEK model versus the inverse bare 't Hooft coupling, $b$. The squares ($\square$) represent the $N\rightarrow\infty$ extrapolation of the average plaquette in Wilson's lattice gauge theory. The solid line (---) and the dashed line (- - -) represent the strong-coupling and the 3-loop weak-coupling expansions of Wilson's lattice gauge theory, respectively.
• Figure 2: Average value of the real and imaginary parts of traced link variables, $\frac{1}{N}\mathrm{Tr}~U_\mu$, in the SU(64) TEK model versus the inverse bare 't Hooft coupling, $b$ (from a cold start simulation).
• Figure 3: Average value of the real part of the plaquette, $\left\langle\mathrm{Re}~u_p\right\rangle$ in the SU(100) TEK model versus the inverse bare 't Hooft coupling, $b$. The squares ($\square$) represent the $N\rightarrow\infty$ extrapolation of the average plaquette in Wilson's lattice gauge theory.
• Figure 4: Average value of the real and imaginary parts of traced link variables, $\frac{1}{N}\mathrm{Tr}~U_\mu$, in the SU(100) TEK model versus the inverse bare 't Hooft coupling, $b$ (from a cold start simulation); the dot-dashed line (-- $\cdot$ --) represents the average real plaquette.
• Figure 5: Average value of the real and imaginary parts of traced link variables, $\frac{1}{N}\mathrm{Tr}~U_\mu$, in the SU(100) TEK model versus the inverse bare 't Hooft coupling, $b$ (from a hot start simulation); the dot-dashed line (-- $\cdot$ --) represents the average real plaquette.