Entanglement entropy of gapped phases and topological order in three dimensions

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 $D\ge 3$.

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

Panels (a) and (b) show two valid $ABC$ constructions [Eq. (15)] in three dimensions that can be used to extract the TEE. In (a) the cross section of a torus has been divided into three tori $A$, $B$, and $C$, while in (b) a torus that has been divided into three cylinders $A$, $B$, and $C$. Panel (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 $\widehat{a},\widehat{b}$ is defined to be right-handed if the triad $\widehat{a},\widehat{b},\widehat{n}$ is right-handed. Formally speaking, $\gamma$ is defined by contracting the $D$-dimensional epsilon tensor with the normal $\widehat{n}$ and then transforming to curvilinear coordinates, ${\gamma }^{{\alpha }_1{\alpha }_2,\cdots ,{\alpha }_{D-1}}=n^{i_d}{\epsilon }_{i_1{i}_2,\cdots ,i_d}\frac{\partial x^{i_1}}{\partial u^{{\beta }_1}}\cdots \frac{\partial x^{i_{D-1}}}{\partial u^{{\beta }_{D-1}}}{g}^{{\alpha }_1{\beta }_1}{g}^{{\alpha }_2{\beta }_2}\cdots g^{{\alpha }_{D-1}{\beta }_{D-1}}$.

Figure 5

Extracting entanglement entropy when a three-dimensional topological ordered state coexists with a layered two-dimensional topological order. (a) The $ABC$ construction for this geometry yields $\gamma =L_z{\gamma }_{2D}$. (b) The $ABC$ construction for this geometry yields $\gamma ={\gamma }_{3d}$.

Figure 6

Illustration for proving that the entanglement entropy is linear in the first Betti number. Panel (a) shows a genus $k+1$ torus ($k=3$), and (b) and (c) give two ways of dividing it up into $A$, $B$, $C$ regions. In (b) $A$ is a $k$ torus, $B$ is a ball, and $C$ is also a ball that has been stretched out. In (c) $A$ is a $k$-torus, $B$ a 1 torus, and $C$ a ball filling up the hole of the 1 torus. Applying strong subadditivity to both configurations gives a recursion formula for $S(k+1)$ in terms of $S\left(k\right)$. Strong subadditivity gives inequalities, but the left- and right-hand sides of the inequality in (b) and (c) are reverses of one another, leading to an exact relation.

Figure 7

Linear dependence of the Betti number on $b_0$. (a) A spherical region with $k+1$ hollow cavities in it ($b_0$$=k+2$ including the outer surface). (b),(c) Two sets of regions for applying strong subadditivity in order to prove that the entropy for $k+1$ holes minus the entropy for $k$ holes is a constant. (b) The region is divided up into $C$, containing $k-1$ of the cavities, a slab $B$, and a region $A$ containing the remaining cavity. (c) $C$ again contains $k-1$ of the holes, $B$ is the rest of the region, and $A$ is a solid sphere that fills one of the cavities.

Figure 8

Illustration of gluing and drilling. (a) Gluing on a general handle: The torus region is $R_k$, and the region $H$ is the handle. The handle meets the boundary of region $R$ in two disks. Attaching the handle gives $R_{k+1}=R_k\cup H$. (b) Drilling a hole: The initial region here, $R_k$, is a solid sphere, and $H$ is a knot that is hollowed out from the inside of the ball to obtain $R_{k+1}=R_k-H$. The boundary of this region is topologically a genus one torus, although it is not possible to continuously deform it into one. Hence drilling increases the genus by one just as adding a handle does.

Figure 9

Figure for the proof that links passing through the handle do not affect the entropy. Panel (a) shows region $R_k$ and the handle (in gray) that is added to it to get $R_{k+1}$. As in Fig. 6, the goal is to relate the entropy of a $k+1$-holed torus to a $k$-holed one, where $k=3$. The dotted circle indicates the portion of the figure that is enlarged in the later panels of this figure. The portion of $R_k$ that passes through the hole is not shown in the subsequent frames for clarity. Parts (b) and (c) show the two decompositions that are used to prove the lower bound on $S_{\mathrm{topo}}\left(R_{k+1}\right)-S_{\mathrm{topo}}\left(R_k\right)$. In (b), there are three regions: Region $B$ (checkered) is $R_k$. The handle is flattened so that it is ribbonlike, and then striplike regions $A$ and $C$ are demarcated along its edges. These together form $A\cup B\cup C:=R_{\mathrm{double}}$ which has the topology of $R_k$ with two handles attached. In (c) we decompose $R_{k+1}$ into $A$, the center of the ribbon, $B$ (checkered), the border of the ribbon closed up with parts of $R_k$ to form a loop, and $C$, the rest of $R_k$. In this figure, $B\cup C=R_{\mathrm{double}}$. Part (D) shows the decomposition used to prove the upper bound, which corresponds to the division in Fig. 6b.

Figure 10

Showing that the entropy of a knotted handle is the same as if it were an unknotted. (a) Shows the knotted handle $H$ (the gray region) that is attached to region $R_k$ (the white region). Passing one strand through the other so the topology changes to panel (e) causes the handle to become unknotted: The loops in (e) can be untwisted, giving a simple handle. Any knot can become unknotable if the right strands are passed through each other. The intermediate panels show that this process does not change the entropy. Panel (b) is a magnification of panel (a). In panel (c) the topology is changed so that one strand passes through an eyelet in the other. This configuration is obtained by adding a handle along the dotted lines in (b); hence, the entropy changes by $S_{\mathrm{torus}}-S_{\mathrm{sphere}}$ (the handle is unknotted). Panel (d) is obtained by removing a handle from (c); the entropy therefore decreases back to its original value. Thus, the change from (a) to (e) does not affect the topological entropy.