Entanglement Asymmetry for Higher and Noninvertible Symmetries

TL;DR

The paper generalizes entanglement asymmetry beyond traditional symmetries to encompass generalized, higher-form, and noninvertible symmetries described by fusion categories. It introduces a universal symmetrizer based on symmetric separability idempotents (SSI) and, when available, Hopf/weak-Hopf structures, to quantify symmetry breaking via relative entropy and Rényi measures on circle and interval subsystems. By contrasting fusion, tube, and strip algebras, the authors show that tube algebras provide a finer detector of symmetry breaking for noninvertible cases, with clear implications for excited-state CFTs, SSB in 2d gapped phases, and SPT classifications. A large suite of explicit examples (Ising, Fibonacci, Haagerup, TY categories) demonstrates how boundary conditions, subsystem size, and algebra choice shape asymmetry outcomes, including nonmonotonic Rényi behavior and vacua inequivalence. The work connects topological data, boundary conditions, and entanglement measures, offering a robust framework to diagnose and distinguish symmetry realizations in quantum field theories and lattice models.

Abstract

Entanglement asymmetry is an observable in quantum systems, constructed using quantum-information methods, suited to detecting symmetry breaking in states -- possibly out of equilibrium -- relative to a subsystem. In this paper we define the asymmetry for generalized finite symmetries, including higher-form and noninvertible ones. To this end, we introduce a "symmetrizer" of (reduced) density matrices with respect to the $C^*$-algebra of symmetry operators acting on the subsystem Hilbert space. We study in detail applications to (1+1)-dimensional theories: First, we analyze spontaneous symmetry breaking of noninvertible symmetries, confirming that distinct vacua can exhibit different physical properties. Second, we compute the asymmetry of certain excited states in conformal field theories (including the Ising CFT), when the subsystem is either the full circle or an interval therein. The relevant symmetry algebras to consider are the fusion, tube, and strip algebras. Finally, we comment on the case that the symmetry algebra is a (weak) Hopf algebra.

Figures (18)

• Figure 1: The Hilbert space splits as $\mathcal{H} = \bigoplus_r n_r V_r$ where $V_r$ is the $r$-th irreducible representation of $\mathcal{A}$, while $n_r$ is its degeneracy. A similar block decomposition applies to density matrices. The symmetrizer acts by setting to zero the off-diagonal blocks, and keeping sub-blocks proportional to the identity on the diagonal.
• Figure 2: Left: action of the fusion algebra \ref{['fusion algebra']} of the symmetry category $\mathscr{C}$ on the untwisted Hilbert space on $S^1$. The product of two elements of the algebra is realized by stacking the corresponding lines on the cylinder and fusing them. Right: the corresponding action of the fusion algebra on local operators.
• Figure 3: Left: action of the tube algebra $\mathsf{Tube}(\mathscr{C})$ on the total Hilbert space on $S^1$ (left) and on (un)twisted point operators (right). The product of two elements of the algebra is realized by stacking the corresponding elements on the cylinder and then resolving the configuration.
• Figure 4: Left: the map $\psi_{\mathcal{B}_L, \mathcal{B}_R}\colon \mathcal{H} \to \mathcal{H}_A \otimes \mathcal{H}_B$ constructed with the path-integral on a strip, where $\mathcal{B}_{L,R}$ are boundary conditions at two removed half-discs of radius $\varepsilon$. Right: the map $\Psi_{\mathcal{B}_L, \mathcal{B}_R}\colon \mathop{\mathrm{End}}\nolimits(\mathcal{H}) \to \mathop{\mathrm{End}}\nolimits(\mathcal{H}_{A})$ constructed via the path-integral on a strip with a finite-size cut, with the same boundary conditions imposed.
• Figure 5: Left: the clustering states, or vacua, $|\mathcal{B}_m\rangle$ on the circle are created by the simple boundaries. Right: Correlation functions in those states are computed by disc partition functions.