Minimally Truncated SU(3) Lattice Gauge Theory and String Tension

TL;DR

This work develops a minimally truncated (qutrit) SU(3) lattice gauge theory in $(2+1)$D on trivalent lattices, deriving the Kogut–Susskind Hamiltonian for square plaquette chains and honeycomb geometries and extending the construction toward $SU(N_c)$. By exact diagonalization on small lattices, it analyzes the gauge-field spectrum, reveals quantum-chaotic level statistics, and extracts the SU(3) string tension from flux-boundary configurations, alongside static $Q\bar{Q}$ and $QQQ$ potentials and their finite-temperature screening. The study also elucidates the SU(3) $6j$-symbol structure governing the truncated dynamics and investigates the large-$N_c$ limit to identify surviving flux configurations, highlighting the balance between truncation and gauge dynamics. Collectively, these results demonstrate how a low-dimensional, minimally truncated model captures essential nonperturbative features such as confinement and flux-tube melting in a computationally tractable setting, with implications for quantum simulation and finite-temperature gauge dynamics.

Abstract

We study SU(3) gauge theory on small lattices in the minimal (qutrit) electric field truncation retaining only the ${\bf 1}, {\bf 3}, {\bf \overline{3}}$ representations for the link variables. Explicit expressions are given for the Kogut-Susskind Hamiltonian for the square plaquette chain and the two-dimensional honeycomb lattice. Our formalism can be easily extended to the minimally truncated general SU($N_c$) gauge theory. The addition of (static) quarks is discussed. We present results for the energy spectrum of the gauge field on these lattices by exact diagonalization of the Hamiltonian and analyze its statistical properties. We also compute the SU(3) string tension and discuss how it is modified by vacuum fluctuations. Finally, we calculate the potential energies of a static quark-antiquark pair and three static quarks and study their screening at finite temperature.

Figures (26)

• Figure 1: Labeling of links and plaquettes for the hexagonal lattice. $P$ is the plaquette on which the plaquette operator acts. The arrows indicate the orientation of the links.
• Figure 2: Structure of an active plaquette $P$ in the linear plaquette chain.
• Figure 3: Insertion of a static quark or antiquark at vertex $V_3$ by shifting the insertion point infinitesimally away from the lattice site and creating an auxiliary link $L_Q$ of infinitesimal length and a new trivalent vertex $V_Q(L_Q,L_4;1)$ (quark) or $V_Q(L_Q,L_4;-1)$ (antiquark). The links of the active plaquette $P$ are shown as solid arrows, and those of the adjacent plaquette $P_L$ as dashed arrows.
• Figure 4: Level spectrum $\rho(E)\Delta E$ for the $N=11$ plaquette chain with asymmetric (singlet/triplet) boundary conditions at $g^2=1$ with the bin size $\Delta E=0.5$. The energies are shown without the diagonal term in the magnetic energy. Compared with the spectrum for an SU(2) chain (see Fig. 1 of Ebner:2023ixq) the level density for the SU(3) chain is more strongly skewed to the upside.
• Figure 5: Energy gap distribution $N(\delta)$ for the middle $70\%$ spectrum (see Fig. \ref{['fig:spectrum']}) of the $N=11$ plaquette chain with asymmetric (singlet/triplet) boundary conditions at $g^2=1$. The bin size is $6\times10^{-6}$. The red curve is a fit of the Wigner surmise form $N(\delta) = a\delta\,e^{-b\delta^2}$ with $a=5.2894\times10^8$ and $b=2.7554\times10^8$. The distribution is well described by a Wigner-Dyson distribution for the GOE ensemble.