More precisely, an anomalous higher-form symmetry $S$ cannot preserve any short-range-entangled state; that is, there is no state $|\mathrm{\Psi }⟩$ obtained from a product state by a finite-depth quantum circuit that satisfies $S|\mathrm{\Psi }⟩\propto |\mathrm{\Psi }⟩$.

In the literature, this unitary operator is sometimes referred to as a shallow-depth circuit, meaning that the circuit depth is small compared to the system size under consideration.

In condensed matter physics, a particle is called deconfined if it can be separated from its antiparticle at an arbitrarily large distances without incurring an energy cost that grows with the distance. In contrast, a confined particle has an energy cost proportional to the separation between excitations, since the corresponding string operator $U(s)$ violates Hamiltonian terms all along its length, not just near the end points.

For $N=2$, the statistics can be simplified as $Z_4^I(a,b)=[U(a{)}_C,[U(a{)}_B,[U(a{)}_A,U(b{)}_{A+B+C+D}]]]$.

A different boundary theory (“anomalous projective semion states”) of this SPT phase was studied in Ref. [95].

In the case of loop excitations, the membrane operators $U_{0ij}$ and $U_{0ji}$ represent the same operator.

For $N=2$, a simpler expression is $Z_5(g)≔[U(g{)}_D,[U(g{)}_C,[U(g{)}_B,U(g{)}_A^2]]]$.

For simplicity, we typically assume that $M$ is a sphere, ensuring that the ground state has no topological degeneracy. This assumption is valid since generalized statistics are local properties and are insensitive to the global topology.

One can extend Eq. (57) to a more generic setup $$\begin{gathered} ⟨a|[U(s_n),[\cdots ,[U(s_2),U(s_1)]]]|a⟩ \\ =⟨b|[U(s_n),[\cdots ,[U(s_2),U(s_1)]]]|b⟩, \end{gathered}$$(D28)for all states $|a⟩,|b⟩$ that are identical at the mutual support of unitaries $\mathrm{supp}(s_1)\cap \dots \cap \mathrm{supp}(s_n)$. Note that this is equivalent to Eq. (57) when $X\backslash \{\mathrm{supp}(s_1)\cap \dots \cap \mathrm{supp}(s_n)\}$ is connected, allowing us to find an additional operator that connects states $|a⟩$ and $|b⟩$ without overlapping with the mutual support of unitaries. Therefore, when $X$ is a triangulation of a closed manifold, such as a sphere $S^d$, it is sufficient to consider Eq. (57). This paper will focus on this case.

This property follows from the commutator decomposition $[AB,C]=B^{-1}[A,C]B[B,C]$, which expresses the commutator of a product as a product of commutators up to conjugation. Therefore, if all higher commutators involving each $U(s)$ are trivial, then the higher commutators of any product $V={\mathrm{\Pi }}_{s\in \mathcal{S}}U(s{)}^{n_s}$ with $n_s\in \mathbb{Z}$ are also trivial, provided that the intersections of their supports vanish.

The minimal triangulation of $S^d$ is formed by $d+1$ $d$-simplices, which are identified as the boundary of a single ($d+1$)-simplex. Its vertices and edges form a complete graph $K_{d+2}$.

To be precise, $T$ behaves more similar to the homology of Eilenberg-MacLane space: $T=H_{d+2}(B^{d-p}G,\mathbb{Z})$ (but they are isomorphic) [108].

To be precise, Eq. (109) states only that the state $|jk⟩$ has the form of $|jk⟩=|\psi {⟩}_{jk}\otimes |0{⟩}_{\overline{j,k}}$, where $|\psi {⟩}_{jk}$ is a possibly entangled state localized at two vertices $j$, $k$. Since a pair of zero-dimensional excitations is disconnected, we need extra work to show that $|\psi {⟩}_{jk}$ is factorized as $|\psi {⟩}_{jk}=|j⟩\otimes |k⟩$. To see this, we consider a state of four Abelian anyons $|jklm⟩$, which is expressed as either $|jklm⟩=|\psi {⟩}_{jk}\otimes |\psi {⟩}_{lm}\otimes |0{⟩}_{\overline{jklm}}$ or $|jklm⟩=|\psi {⟩}_{jm}\otimes |\psi {⟩}_{kl}\otimes |0{⟩}_{\overline{jklm}}$, depending on whether we use $U_{jk}{U}_{lm}$ or $U_{jm}{U}_{kl}$ to create them. We then have $|\psi {⟩}_{jk}\otimes |\psi {⟩}_{lm}=|\psi {⟩}_{jm}\otimes |\psi {⟩}_{kl}$, which implies that $|\psi {⟩}_{jk}$ does not have entanglement between $j$ and $k$, hence, $|\psi {⟩}_{jk}=|j⟩\otimes |k⟩$. See Ref. [115] for this discussion on general Abelian anyons.

To be precise, when we define the phases such as $\theta (U_{j;jkl}^{(0)},a),\theta ({\tilde{U}}_{jkl},a)$, we are fixing the choice of the intermediate states, e.g., ${\tilde{U}}_{jkl}|a{⟩}_A$ in terms of the patch of MPS. This amounts to fixing the MPS at the intermediate edges expressed as the green lines in Fig. 6.

A key fact is that ${\gamma }_{14}/a_{14}(\text{mod}2)$ remains invariant under ${\prod }_v{X}_v$, which shifts each $a_i\to a_i+1$.

Here, the vertex 1 is regarded as a zero-dimensional face.