Entropic order parameters for the phases of QFT

TL;DR

The paper develops entropic order parameters for generalized symmetries in QFT by analyzing Haag-Kastler nets, showing non-local operators organize into dual Abelian groups tied to region topology. It defines relative-entropy-based order/disorder measures via conditional expectations and proves a certainties relation linking complementary region statistics, enabling quantitative distinctions among global, gauge, conformal, and massive phases. The framework reproduces known phenomena (Wilson/’t Hooft area vs perimeter laws, confinement/Higgs duality) and provides controlled bounds and RG-flow intuition through illustrative toy models. Overall, it offers a unifying algebraic-information-theoretic perspective on symmetry breaking and phase structure across QFTs, including higher-form generalizations.

Abstract

We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT's. We do it through an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show that the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the 't Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.

Figures (15)

• Figure 1: A region formed by two disjoint balls $R_1$ and $R_2$ (grey region) containing the intertwiner formed by a charge-anti-charge operator. In the complement $S$ of the two balls, which has a non-contractible $d-2$ dimensional surface, lives the twist operator.
• Figure 2: Left: Duality and additivity cannot be valid simultaneously for a ring-like region (solid torus) $R$. The operator $a$ is not additive in $R$. The interlocked operator $b$ is again not additive in the complement $R'$. $a$ and $b$ do not commute with each other. For commuting algebras attached to $R$ and $R'$, either $a$ or $b$ have to belong to the respective algebra (but not both of them at the same time), and additivity is lost. Right: Violation of the intersection property. The figure shows a section of two spherical cap regions $A$ and $B$ intersecting in the ring $R$ (here $d= 4$). A non-additive operator in $R$ is additive in both the topologically trivial regions $A$ and $B$. It then necessarily belongs to the intersection of the algebras of $A$ and $B$. This implies that additivity for $R$ cannot be maintained at the same time as the intersection property.
• Figure 3: Upper panel: reconnecting a loop operator using local operators in the shaded region. Lower panel: this cannot be done for non Abelian twists.
• Figure 4: The operators of the form $a\bar{a}$ are locally generated inside a ring $R$, marked with the dashed line.
• Figure 5: The construction of Bachas that shows the convexity of the quark-antiquark potential Bachas:1985xs.