Emergent photons and mechanisms of confinement

TL;DR

The paper investigates 4D $\mathbb{Z}_N$ lattice gauge theories with a $\mathbb{Z}_N$ 1-form symmetry and demonstrates that, for all $N\ge 3$, the system exhibits three phases: confined, a photon phase with emergent $U(1)$ 1-form symmetry, and a phase with spontaneous $\mathbb{Z}_N$ 1-form symmetry breaking. Using Monte Carlo simulations of $\mathbb{Z}_7$, $\mathbb{Z}_4$, and $\mathbb{Z}_3$ models, it links the photon phase to the proliferation of center vortices with suppressed monopole-junctions and shows the confined phase requires monopole proliferation; it also identifies a photon phase in deformed $Z_3$/$Z_4$ models and confirms Coulomb-like long-distance behavior via plaquette-plaquette correlators. The results challenge the conventional center-vortex-only picture of confinement in 4D gauge theories and illustrate how higher-form symmetries organize nonperturbative dynamics, indicating a robust massless photon phase in 4D and refining the Landau paradigm for $1$-form symmetries. Implications span intuition for confinement mechanisms in YM theories and the role of higher-form symmetries in nonperturbative QFT dynamics.

Abstract

We numerically study $\mathbb{Z}_N$ lattice gauge theories in 4D as prototypical models of systems with $\mathbb{Z}_N$ 1-$\textit{form symmetry}$. For $N \geq 3$, we provide evidence that such systems exhibit not only the expected phases with spontaneously broken/restored symmetry but also a third photon phase. When present, the 1-form symmetry provides a precise notion of confinement, and it is commonly believed that confinement ensues due to the proliferation of extended, string-like objects known as $\textit{center vortices}$, which carry a $\mathbb{Z}_N$ flux. However, this picture is challenged by the three-phase scenario investigated here. We show that both the confined and the photon phases are associated with the proliferation of center vortices and that the key difference between them lies in whether or not vortex-junctions - the $\textit{monopoles}$ - proliferate.

Figures (8)

• Figure 1: a) The definition of the plaquette Wilson loop $W_p=U_{\ell_1}U_{\ell_2}U_{\ell_3}U_{\ell_4}$. b) The illustration of the $\mathbb{Z}_N$ 1-form symmetry transformation in 2D. c) The center vortex as an excitation of the plaquette fluxes.
• Figure 2: Phase structure of $\mathbb{Z}_7$ Lattice Gauge Theory. Plots show the average Polyakov loop and the corresponding distributions of the smeared Polyakov loop $\bar{P}$. The histograms of $\bar{P}$ indicate confinement (left), emergent U(1) symmetry (middle), and spontaneous $\mathbb{Z}_7$ 1-form symmetry breaking (right). Insets: Typical 3D Monte Carlo configurations showing center vortices (colored by charge ±1, ±2, ±3, arrows indicating the sign) and monopoles/anti-monopoles (green/yellow boxes).
• Figure 3: Histogram of the smeared Polyakov loop as a function of $\beta$ and $\tilde{\beta}$ (see \ref{['eq:S-Z4']}) of the $\mathbb{Z}_4$ lattice gauge theory.
• Figure 4: Susceptibility $\chi_S$ versus $\beta$ for the $\mathbb{Z}_3$ model with monopole suppression ($\mu$=1) for lattices $L=8,10,12$ and $14$. Two prominent peaks indicate phase transitions. Bottom panels show zooms of the low-$\beta$ (confined $\to$ photon) and high-$\beta$ (photon $\to$ SSB) transition regions. Inset: smeared Polyakov loop histograms near the high-$\beta$ transition for $L=6, 10$ and $14$.
• Figure 5: The correlator ratio $C(n)/C(n+1)$ in the photon phases of $\mathbb{Z}_3, \mathbb{Z}_4$, and $\mathbb{Z}_7$ on an $L=16$ lattice compared to the long-distance expectation (see \ref{['eq:asympt']}). Insets: Corresponding smeared Polyakov loops.