Broken quantum symmetry and confinement phases in planar physics

TL;DR

This work addresses how Hopf algebraic symmetries govern topological excitations in planar quantum systems and how spontaneous symmetry breaking reorganizes these excitations into confined and non-confined sectors. By introducing a hierarchical sequence of Hopf algebras $A \to T \to U$, the authors show that a condensate in a representation of $A$ selects a maximal subalgebra $T$ and then a residual symmetry $U = \operatorname{Im}(\Gamma)$ that governs the low-energy non-confined spectrum; confinement is diagnosed via braiding with the condensed vacuum, and strings attached to confined states are described by the Hopf kernel $\mathrm{Ker}(\Gamma)$. The paper provides explicit breaking patterns for electric, magnetic, and flux condensates, deriving the corresponding $U$ and kernel structures, and demonstrates that hadronic composites and domain walls emerge systematically from the $T$-tensor product rules. This framework links topological order, anyon braid statistics, and confinement in 2D field theories, with potential applications to discrete gauge theories, fractional quantum Hall states, and related CFT descriptions.

Abstract

Many two-dimensional physical systems have symmetries which are mathematically described by quantum groups (quasi-triangular Hopf algebras). In this letter we introduce the concept of a spontaneously broken Hopf symmetry and show that it provides an effective tool for analysing a wide variety of phases exhibiting many distinct confinement phenomena.