Entropic order parameters in weakly coupled gauge theories

TL;DR

The paper develops entropic order parameters for generalized symmetries in weakly coupled gauge theories by analyzing smeared non‑local operators in an enlarged non‑gauge invariant Hilbert space. It provides a unified framework using complementarity diagrams and conditional expectations to bound the relative entropies that define the order and disorder parameters, and it validates the entropic certainty relation $S_{\textrm{order}}+S_{\textrm{disorder}}=\log|Z|$ in both Abelian and non‑Abelian settings. For Maxwell theory with a finite center $Z_N$ and for non‑Abelian gauge theories, the Wilson and ’t Hooft loops are shown to be labeled by weight and co‑weight lattices, yielding explicit weak‑coupling scaling: smeared Wilson loops saturate to the maximal topological value while smeared ’t Hooft loops are exponentially suppressed, with the exact coefficients controlled by optimal fluxes and the center of the gauge group. The results illuminate how generalized symmetries evolve under RG flow and establish quantitative bounds that connect geometric ring configurations, loop operators, and the center structure of the gauge group, with potential extensions to matter‑charged theories.

Abstract

The entropic order parameters measure in a universal geometric way the statistics of non-local operators responsible for generalized symmetries. In this article, we compute entropic order parameters in weakly coupled gauge theories. To perform this computation, the natural route of evaluating expectation values of physical (smeared) non-local operators is prevented by known difficulties in constructing suitable smeared Wilson loops. We circumvent this problem by studying the smeared non-local class operators in the enlarged non-gauge invariant Hilbert space. This provides a generic approach for smeared operators in gauge theories and explicit formulas at weak coupling. In this approach, the Wilson and 't Hooft loops are labeled by the full weight and co-weight lattices respectively. We study generic Lie groups and discuss couplings with matter fields. Smeared magnetic operators, as opposed to the usual infinitely thin ones, have expectation values that approach one at weak coupling. The corresponding entropic order parameter saturates to its maximum topological value, except for an exponentially small correction, which we compute. On the other hand, smeared 't Hooft loops and their entropic disorder parameter are exponentially small. We verify that both behaviors match the certainty relation for the relative entropies. In particular, we find upper and lower bounds (that differ by a factor of 2) for the exact coefficient of the linear perimeter law for thin loops at weak coupling. This coefficient is unphysical/non-universal for line operators. We end with some comments regarding the RG flows of entropic parameters through perturbative beta functions.

Figures (7)

• Figure 1: We show an operator $a$ which is non-local in the topologically non-trivial region $R$. However, the same operator $a$ can be locally generated in the topologically trivial region $B$ containing $R$.
• Figure 2: The support of $J(x)$ and $\tilde{J}(x)$ for linked loops.
• Figure 3: A ring formed by the revolution around the $z$ axis of a disk $D$ of radius $r$, such that the inner radius of the ring is $l$.
• Figure 4: Optimal flux squared as a function of the ring cross ratio.
• Figure 5: Numerical upper and lower bounds for $S_{A_{W_q}}(\omega|\omega\circ E_{Z_N})$ with $N=3$ for 3 different charges: electron charge $q=\sqrt{4 \pi \alpha}$, $1$, and the self dual point $q=\sqrt{2 \pi}$.