Entanglement Sum Rule from Higher-Form Symmetries

TL;DR

The paper addresses how entanglement can decompose additively in coupled quantum lattice models when finite abelian higher-form symmetries are present. It introduces a minimal coupling unitary U that ties two decoupled sectors, and shows that, under a Mayer–Vietoris criterion for a bipartition, U factorizes across the cut on the symmetry-invariant subspace, yielding an additive entanglement entropy for symmetric eigenstates. The approach recovers known sum rules in fermion–ℤ2 gauge theories and provides a constructive path to generate new examples by gauging higher-form symmetries, demonstrated explicitly for a (3+1)D transverse-field toric code. By tying entanglement structure to symmetry and topology via homology, the work reveals a deep connection between entanglement, higher-form symmetries, and gauging procedures, with potential extensions to continuous symmetries and broader topologies.

Abstract

We prove an entanglement sum rule for $(d-1)$-dimensional quantum lattice models with finite abelian higher-form symmetries, obtained by minimally coupling a sector on $p$-simplices carrying a $p$-form $G$ symmetry to a sector on $(p+1)$-simplices carrying the dual $(d-p-2)$-form $\widehat G$ symmetry (with $\widehat G$ being the Pontryagin dual of $G$). The coupling is introduced by conjugation with a symmetry-preserving operator $\mathcal{U}$ that dresses symmetry-invariant operators with appropriate Wilson operators. Our main result concerns symmetric eigenstates of the coupled model that arise by acting with $\mathcal{U}$ on direct-product symmetric eigenstates of the decoupled model: provided a topological criterion formulated via the Mayer--Vietoris sequence holds for the chosen bipartition, $\mathcal{U}$ factorizes across the cut when acting on the symmetric state, and the entanglement entropy equals the sum of the entropies of the two sectors. This framework explains and generalizes known examples in fermion-$\mathbb{Z}_2$ gauge theory, identifies when topology obstructs the sum rule, and provides a procedure to construct new examples by gauging higher-form symmetries.

Figures (4)

• Figure 1: An example of the 1-form $\mathbb{Z}_2$ symmetry where $d=3$. The space $X$ is a two-dimensional complex. The closed black dashed lines on the dual complex represent $\phi\in Z^1(X;\mathbb{Z}_2)\cong Z_1(\bar{X};\mathbb{Z}_2)$: For a link $\sigma\in\Delta^1(X)$, $\phi(\sigma)=1$ if $\sigma$ crosses a dashed line, otherwise $\phi(\sigma)=0$. $\phi$ represents a 1-form symmetry transformation $U(\phi)=\prod_{\sigma\in\Delta^1(X)}U_{\sigma}(\phi(\sigma))$, where $U_{\sigma}(0)=U^2_{\sigma}(1)=1$. The red lines represent $k\in C_1(X;\mathbb{Z}_2)$: $k_{\sigma}=1$ if the link $\sigma$ is colored red, otherwise $k_{\sigma}=0$. $k$ represents an irreducible representation of local unitary operators, the corresponding projector is $P(k)=\prod_{\sigma\in\Delta^1(X)} P_{\sigma}(k_{\sigma})$. $P_{\sigma}(1)$ is the projection to the sign representation where $U_{\sigma}(0)=1, U_{\sigma}(1)=-1$; $P_{\sigma}(0)$ is the projection to the trivial representation where $U_{\sigma}(0)=U_{\sigma}(1)=1$. Then we know by acting $U(\phi)$ on the subspace that $P(k)$ projects onto, we get $e^{i\braket{k,\phi}}$ where $\braket{k,\phi}$ equals the number of intersections of the black dashed lines and red lines times $\pi$. Thus, $P(k)$ projects onto a subspace that is invariant under all 1-form symmetry transformations if and only if the red lines have an even number of intersections with all closed loops on the dual complex. Lemma \ref{['lemma1']} yields that this is equivalent to $k$ being a boundary, or equivalently, the red lines being the domain walls between two sets of plaquettes. This is indeed the case in the figure.
• Figure 2: An example of the ambiguity of minimal coupling in fermion-$\mathbb{Z}_2$ gauge theory.
• Figure 3: An illustration of the bipartition in the case where $d=3$ and $p=1$. The 1-simplices are the links. The 2-simplices are the triangles colored yellow.
• Figure 4: An illustration of $f_1$ and $f_2$ in a special case where $p=1$. $k=k_A+k_{A^c}$ is depicted on the left. In this case $t_A([k_A-h(\partial k_A)])=t_B([k_{A^c}-h(\partial k_{A^c})])=0$, since $k_A-h(\partial k_A)$ and $k_{A^c}-h(\partial k_{A^c})$ are already boundaries in $A$ and $B$ respectively.

Theorems & Definitions (11)

• Lemma 1
• proof
• Corollary 1
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Remark 2
• Theorem 3