Breakdown of large-N quenched reduction in SU(N) lattice gauge theories

TL;DR

This work probes the large-$N$ reduction idea by testing whether the four-dimensional SU($N$) lattice gauge theory is equivalent to its quenched Eguchi-Kawai model. Through analytic arguments and extensive nonperturbative lattice simulations for $20 \le N \le 200$, the authors uncover dynamical correlations between Euclidean components of the gauge field, manifested as momentum locking, that spoil the required factorization and center-symmetry conditions for reduction. They demonstrate significant discrepancies between QEK and infinite-volume SU($N$) results, including a strong mismatch in the transition couplings and nonzero $M_{\mu,\nu}$ signaling locked vacua, even at large $N$. Consequently, large-$N$ reduction in the QEK framework does not survive in the continuum limit, leaving only a few single-site alternatives such as deformed EK models or adjoint-fermion variants as viable paths to volume reduction.

Abstract

We study the validity of the large-N equivalence between four-dimensional SU(N) lattice gauge theory and its momentum quenched version--the Quenched Eguchi-Kawai (QEK) model. We find that the assumptions needed for the proofs of equivalence do not automatically follow from the quenching prescription. We use weak-coupling arguments to show that large-N equivalence is in fact likely to break down in the QEK model, and that this is due to dynamically generated correlations between different Euclidean components of the gauge fields. We then use Monte-Carlo simulations at intermediate couplings with 20 <= N <= 200 to provide strong evidence for the presence of these correlations and for the consequent breakdown of reduction. This evidence includes a large discrepancy between the transition coupling of the "bulk" transition in lattice gauge theories and the coupling at which the QEK model goes through a strongly first-order transition. To accurately measure this discrepancy we adapt the recently introduced Wang-Landau algorithm to gauge theories.

Figures (16)

• Figure 1: Illustration of the extra source terms, eq. (\ref{['eq:loop0']}), in the Dyson-Schwinger equations for a closed Wilson loop in the QEK model. The dashed line represents $W_{\rm open}$ while the solid line represents $W'_{\rm open}$.
• Figure 2: A two-dimensional example of the embedding suggested in Refs. GKBars of the color indices in the Brilluoin zone for $N=16$.
• Figure 3: The dependence of $f=F_{\rm EK}(p)/((d-2)N(N-1)/2)$ on the combined order parameter $M$ for random permutations of the clock momenta. Results are for $d=4$ and $N=10,20,40,200$. The fully locked points with $M=6$ are included by hand. There are also partially unlocked states which, for each $N$, interpolate between the mass of unlocked states and the locked ones. These do not appear in the random sampling except for $N=10$.
• Figure 4: The free energy $F_{\rm EK}(p)/(d-2)$, divided by $N$ (left panel) or $N^2$ (right panel), along the two paths between locked vacua described in the text. Results are for $d=4$ and $N=10$, $50$, $200$ and $1000$. The horizontal axis gives the fraction of the total transpositions required, with the starting point being plotted at position $1/N$ (left panel) and $2/[N(N-1)+1]$ (right panel). The left panel is for path 1 (vacua related by center-symmetry), the right panel for path 2 (vacua related by a reflection).
• Figure 5: Hysteresis plots of the plaquette variable $u_p$ versus $b$ for $SU(50)$ ([blue] crosses), $SU(80)$ ([magenta] open squares), and $SU(100)$ ([light blue] filled squares). Results are for $\rho_{\rm clock}$ and self-averaging. The curves are the predictions for $SU(\infty)$ in the strong-coupling expansion to leading order (solid [red] curve) and of the weak-coupling expansion to three loop order (dashed [blue] curve) (taken from, for example, Ref. TEK). The lower-panel shows a close-up of the strong-to-weak transition region.