Entanglement Spectrum Resolved by Loop Symmetries

TL;DR

The paper develops a rigorous, topology-driven framework to resolve the entanglement spectrum of quantum states with generalized Rep$(G)$ loop symmetries, accommodating non-invertible, higher-form, and non-Abelian structures across arbitrary dimensions and (non-)orientable manifolds. By leveraging the Seifert–van Kampen theorem and fundamental groupoids, it derives an explicit block structure for reduced density matrices, separating topological and geometric degrees of freedom. Upon incorporating gauge invariance, the framework reproduces the topological entanglement entropy and proves the Li–Haldane correspondence for the Kitaev quantum double model, connecting entanglement blocks to RCFT data via the modular $S$-matrix of the Drinfeld double $D(G)$. The approach yields concrete block decompositions for a variety of low-dimensional manifolds (torus, Klein bottle) and higher-dimensional tori, and extends naturally to 3D via Heegaard splittings. Overall, the work provides a principled, algorithmic pathway to understand how generalized, non-invertible symmetries shape entanglement in topological phases and gauge theories, with potential extensions to higher-form and higher-categorical symmetry structures.

Abstract

A rigorous analysis is presented for the entanglement spectrum of quantum many-body states possessing a higher-form group-representation symmetry generated by topological Wilson loops, which is generally non-invertible. A general framework based on elementary algebraic topology and category theory is developed to determine the block structure of reduced density matrices for arbitrary bipartite manifolds on which the states are defined. Within this framework, we scrutinize the impact of topology on the entanglement structure for low-dimensional manifolds, including especially the torus, the Klein bottle, and lens spaces. By further incorporating gauge invariance, we refine our framework to determine the entanglement structure for topological gauge theories in arbitrary dimensions. In particular, in two dimensions, it is shown for the Kitaev quantum double model that not only the topological entanglement entropy can be reproduced, but also the Li-Haldane conjecture concerning the full entanglement spectrum holds exactly.

Figures (13)

• Figure 1: (a) Procedure for defining a lattice and its bipartition. (a1) Fix a manifold $M$, for example a genus-2 surface. (a2) Fix a sufficiently fine-grained discretization of $M$. (a3) Fix a bipartition of $M$ into $X$ and $Y$ with no common edges. (b) Definition of Hilbert space and the symmetry action on it. (b1) Fix a finite group $G$. For each edge, a local Hilbert space $\mathbb{C}^{|G|}$ spanned by $|g\rangle$ (blue dot) and an orientation (arrows) is assigned. (b2) Building block of the matrix product operator (MPO) of ${\rm Rep}(G)$ loop symmetry. Here $D^\alpha_{ij}(g)$ is the representation matrix of irrep $\alpha$. (b3) Action of MPO on the state. The orientation of an MPO is fixed. If its orientation aligns with (opposite to) that of the edge, we adopt the building block in (b2) (with $D^\alpha_{ij}(g)$ replaced by $D^\alpha_{ij}(g^{-1})$). (c) Constraint on the many-body state from the loop symmetries. Holonomies along contractible loops (red solid loop) are enforced to be trivial, while those along non-contractible loops (red dashed loop) may not be trivial.
• Figure 2: Torus has essentially two topologically different ways of bipartition. (a) Disk + the other. Since $\partial$ is simply connected, only one base point is necessary. The holonomies along red and blue curves must commute. (b) Two tubes. Here $\partial$ has two connected components. The non-contractible loop along the horizontal axis is inevitably segmented.
• Figure 3: Nonzero elements in $W$ in two settings under fixing $G=D_6$. The blue square denotes the blocks generated by gauge transformation on $V \backslash A$, corresponding to the geometric part. (a) Nontrivial bipartition of torus. (a1) A table showing where nonzero elements can be placed for nontrivial bipartition of torus. The rows are labeled by $(a_X,b_X)$, and the columns are labeled $(a_Y,b_Y)$. The part where the blue square is placed is where nonzero elements are assigned when the basis is arranged in the lexicographical order $(1,1),(1,r),\dots,(1,sr^2),(r,1),\dots, (sr^2,sr^2)$, and the parts where it is not placed must be zero. (a2) Result of rearranging the basis vectors into blocks. The direct sum decomposition of the topological part is $\mathbb{C}^{6\times 6}\oplus 4 \mathbb{C}^{3\times 3}\oplus 9 \mathbb{C}^{2\times 2}$. (b) Symmetric bipartition of genus-2 surface. (b1) A table showing where nonzero elements can be placed for this bipartition. The rows are labeled by $(a_X,b_X)$, and the columns are labeled $(a_Y,b_Y)$. The basis is arranged in the lexicographical order again. It appears to have a far more complex structure than the examples seen before, and it is not so obvious that it can be reduced to a block structure simply by rearranging the basis. (b2) The result of rearranging the basis vectors into blocks. The direct sum decomposition of the topological part is $\mathbb{C}^{18\times 18}\oplus 2 \mathbb{C}^{9\times 9}$.
• Figure 4: (a) Constructing the Klein bottle from a square. The relation of holonomies is a bit different from the case of torus. (b) Bipartition of the Klein bottle to two Möbius bands. Since the boundary of the Möbius band is $S^1$, we only need one base point. The holonomy along the horizontal axis (blue line) can be understood as a product of two vertical holonomies (red and green curves) respectively associated to $X$ and $Y$.
• Figure 5: Heegaard splitting of lens space is achieved by gluing together two solid tori. One of these tori is generally twisted by the action of the modular group $\mathrm{SL}(2;\mathbb{Z})$. By precisely aligning and joining the meshes drawn on the two tori in the figure, a lens space is obtained.