Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry

TL;DR

The paper develops a comprehensive fusion higher-category framework for symmetries in quantum many-body systems, introducing algebraic higher symmetry as a generalization beyond group-like and higher-group symmetries. It identifies categorical symmetry with a bulk topological order in one higher dimension, enabling a holographic (boundary–bulk) perspective on symmetry, anomalies, and gauging. The authors formulate how to gauge algebraic higher symmetries, classify gapped liquid phases (SET and SPT orders), and relate different realizations through holo-equivalence and dualities, with explicit constructions via boundaries, centers, and condensable algebras. A thorough higher-category theory of topological orders in higher dimensions is presented, together with concrete $G$-gauge theory examples and a detailed treatment of emergent categorical symmetry. The work provides a unified language to analyze symmetry, duality, and phase structure in bosonic and fermionic systems across dimensions, with potential implications for understanding entanglement-driven constraints in quantum matter.

Abstract

We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also refer to non-invertible gravitational anomaly as categorical symmetry to stress its connection to symmetry. This provides a holographic and entanglement view of symmetries. For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low energy excitations, as long as they have the same categorical symmetry; (4) the classification of gapped liquid phases for bosonic/fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry). This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.

Figures (31)

• Figure 1: (a) an anomaly-free topological order $\mathsf{C}^{n+1}_\text{af} \in \mathds{TO}_{\mathrm{af}}^{n+1}$ in $\hbox{($n + 1$)}$-dimensional spacetime can be realized on lattice in the same dimension, which can also be viewed as a boundary of a trivial product state in one higher dimension. The excitations in $\mathsf{C}^{n+1}_\text{af}$ are described by fusion $n$-category $\cC^n_\text{af}$. (b) an anomalous topological order $\mathsf{C}^{n+1} \in \mathds{TO}^{n+1}$ in $\hbox{($n + 1$)}$-dimensional spacetime can be realized as a boundary of an anomaly-free topological order $\mathsf{M}^{n+2}$ in one higher dimension. The excitations in $\mathsf{C}^{n+1}$ are described by fusion $n$-category $\cC^n$. The excitations in $\mathsf{M}^{n+2}$ are described by fusion $\hbox{($n + 1$)}$-category $\cM^{n+1}$.
• Figure 2: (a) an anomaly-free domain wall $\mathsf{D}$ between two (potentially anomalous) topological orders $\mathsf{C}_1$ and $\mathsf{C}_2$. (b) an anomalous domain wall $\mathsf{D}$ between two (potentially anomalous) topological orders $\mathsf{C}_1$ and $\mathsf{C}_2$. $\mathsf{D}$ is a boundary of domain wall $\mathsf{C}_3$ which is a (potentially anomalous) topological order.
• Figure 3: (a) An algebraic higher symmetry in $n$d bosonic systems is fully characterized by its charge objects (the excitations in trivial symmetric state), which form a local fusion $n$-category $\cR$. The symmetry selects a set of local operators $\{ O_\cR\}$ which are said to have the algebraic higher symmetry $\cR$. (b) A categorical symmetry for bosonic systems in $n$-dimensional space is characterized by an anomaly-free topological order $\mathsf{M}$ in one higher dimension. The categorical symmetry $\mathsf{M}$ also select a set of local operators, which is given by all the boundary local interactions, $\{O_\mathsf{M}\}$, of the bulk topological order $\mathsf{M}$. An algebraic higher symmetry $\cR$ is holo-equivalent to a categorical symmetry given by $\mathsf{M}=\hbox{$\mathsf{bulk}$}(\cR)$. The holo-equivalence means that the algebraic higher symmetry $\cR$ and the categorical symmetry $\mathsf{M}=\hbox{$\mathsf{bulk}$}(\cR)$ select equivalent sets of local operators, there is one-to-one correspondence between $\{O_\cR\}$ and $\{O_\mathsf{M}\}$, such that the two corresponding local operators have the same operator algebra relations. In this sense, the systems with an algebraic higher symmetry $\cR$ also have the categorical symmetry $\mathsf{M}=\hbox{$\mathsf{bulk}$}(\cR)$.
• Figure 4: (a) Two algebraic higher symmetries $\cR$ and $\cR'$ are holo-equivalent if they have the same categorical symmetry $\hbox{$\mathsf{bulk}$}(\cR)\simeq \hbox{$\mathsf{bulk}$}(\cR')$. (b) Two sets of low energy excitations $\cC$ and $\cC'$ are holo-equivalent if they have the same categorical symmetry $\hbox{$\mathsf{bulk}$}(\cC)\simeq \hbox{$\mathsf{bulk}$}(\cC')$. Here holo-equivalent means the states with symmetry $\cR$ or $\cR'$ (or formed by $\cC$ or $\cC'$) have an one-to-one correspondence.
• Figure 5: Gauging the $\cR$-symmetry: stacking two local fusion $n$-category $\cR$ over their common bulk $Z_1(\cR)$ gives rise to an fusion $n$-category $\cR\underset{Z_1(\cR)}{\otimes}\cR^\mathrm{rev}$, describing the excitations in $n$d anomaly-free topological order $\mathsf{GT}_\cR^{n+1}$, which is the $\cR$-gauge theory. The boundary Hamiltonians of $\cR$-gauge theory share the same low energy properties with the Hamiltonians with algebraic higher symmetry $\Om\cR$, if $\cR=\Si\Om\cR$. We like to remark that $\cR$'s on the two boundaries differ by a parity transformation as indicated by the arrows and superscript $^\mathrm{rev}$.

Theorems & Definitions (75)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 11
• Definition 12
• Proposition 4
• Proposition 5
• Proposition 6