Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions

TL;DR

The paper develops a curvature-based framework to analyze entanglement entropy in gapped phases across dimensions, separating local (boundary-patch) and topological contributions. In 3D, it shows there is a single topological entanglement entropy governed by boundary connectivity, while higher dimensions host additional TEEs linked to higher Betti numbers, realized in discrete p-form gauge theories and toric-code–like models. It generalizes Kitaev–Preskill/Levin–Wen-type constructions to extract TEEs in D≥3 and clarifies when constant terms arise from geometry versus topology. The results provide a unified view of how topology and boundary geometry shape entanglement, with implications for identifying and classifying topological orders in higher dimensions.

Abstract

We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an `entropy density' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size, and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the `zeroth Betti number'). In D > 3 dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in D >=3.

Figures (10)

• Figure 1: The local part of the entropy of region $A$ is the sum of contributions of small patches on the boundary.
• Figure 2: Illustration of the Z$_2$ symmetry for the curvature expansion discussed in the text.
• Figure 3: Fig.(a) and (b) show two valid $ABC$ constructions (Eq. \ref{['kitpres']}) in three dimensions that can be used to extract the topological entanglement entropy. In Fig.(a) the cross-section of a torus has been divided into three tori $A, B$ and $C$ while in Fig.(b) a torus that has been divided into three cylinders $A, B$ and $C$. The Fig.(c) shows an invalid construction as explained in the text. In all three figures, we define region $D$ to be the rest of the system.
• Figure 4: Defining the orientation of a hypersurface from an orientation of space, illustrated in three dimensions. A pair of axes on the surface $\hat{a},\hat{b}$ is defined to be right-handed if the triad $\hat{a},\hat{b},\hat{n}$ is right-handed. Formally speaking, $\gamma$ is defined by contracting the D-dimensional epsilon tensor with the normal $\hat{n}$ and then transforming to curvilinear coordinates, $\gamma^{\alpha_1\alpha_2\dots\alpha_{D-1}}=n^{i_d}\epsilon_{i_1i_2\dots i_d}\frac{\partial x^{i_1}}{\partial u^{\beta_1}}\dots\frac{\partial x^{i_{D-1}}}{\partial u^{\beta_{D-1}}}g^{\alpha_1\beta_1}g^{\alpha_2\beta_2}\dots g^{\alpha_{D-1}\beta_{D-1}}$.
• Figure 5: Extracting entanglement entropy when a three dimensional topological ordered state coexists with a layered two dimensional topological order. Fig.(a): The $ABC$ construction for this geometry yields $\gamma = L_z \gamma_{2D}$. Fig.(b): The $ABC$ construction for this geometry yields $\gamma = \gamma_{3d}$.