content_image/384414.jpg content_image/384416.jpg content_image/384419.jpg content_image/384422.jpg content_image/384423.jpg content_image/384424.jpg
## SUPPLEMENTARY MATERIAL 

## A: Derivation of Eq. (4) 

To obtain ρ$_{A}$ , we trace out the spin degrees of freedom in B . For the spins in B away from the partition boundary, the trace can be carried out in the Hilbert space of fermions and gauge field respectively. However, the spins on the boundary sites need more careful treatment. Suppose the links across the boundary are denoted as a$_{n}$b$_{n}$ , n = 1 , · · · 2 L , as shown in Fig. 1. (Note that we implicitly assumed the boundary length is even here. The odd boundary length case need some extra care but our results below remain valid.) The gauge field ˆ u$_{a}$$_{n}$$_{b}$$_{n}$ = iγ α $_{a$_{n}$}$γ α b$_{n}$ residing on a$_{n}$b$_{n}$ has spin degrees of freedom from both A and B . To trace out B we introduce new Z$_{2}$ variables ˆ w$_{A,n}$ = iγ α $_{a$_{2}$$_{n}$$_{−}$$_{1}$}$γ β a$_{2}$$_{n}$ and ˆ w$_{B,n}$ = iγ α $_{b$_{2}$$_{n}$$_{−}$$_{1}$}$γ β $_{b$_{2}$$_{n}$}$, which are defined on the dotted links in Fig. 1 and belong to either A or B exclusively. The eigenstates of ˆ w$_{A,n}$, ˆ w$_{B,n}$ are related to those of ˆ u$_{a}$$_{2}$$_{n}$$_{−}$$_{1}$$_{b}$$_{2}$$_{n}$$_{−}$$_{1}$, ˆ u$_{a}$$_{2}$$_{n}$$_{b}$$_{2}$$_{n}$ by a unitary transformation. Since | u 〉 is a direct product of gauge fields, we denote | u 〉 = | u$_{A}$, u$_{B}$, u$_{p}$ 〉 , where | u$_{A}$ 〉 , | u$_{B}$ 〉 , and | u$_{p}$ 〉 are gauge fields in A , B , and on the links across the partition boundary respectively. In term of eigenstates of ˆ w$_{A}$ and ˆ w$_{B}$ defined above, we obtain 

<img src='content_image/384414.jpg'>

if all boundary links have eigenvalues u$_{a}$$_{n}$$_{b}$$_{n}$ = 1. For more generic values of u$_{a}$$_{n}$$_{b}$$_{n}$ , the only change are the signs of the terms in the right hand side of Eq. (A1), which does not enter the reduced density matrix we are interested in. Here w$_{A}$$_{(}$$_{B}$$_{)}$ = { w$_{A}$$_{(}$$_{B}$$_{)}$$_{,n}$ } and w$_{A}$ = w$_{B}$ means w$_{A,n}$ = w$_{B,n}$ for all n = 1 , ..., L . In the new basis, the physical eigenstate is given by 

<img src='content_image/384416.jpg'>

from which one can obtain the reduced density matrix ρ$_{A}$ and compute Tr [ ρ n $_{A}$]. | g | ( | g |− 1) / x y z | g | ( | g |− 1) / 

We define X$_{g}$ = i $^{2}$∏ j ∈ $_{g}$γ $_{j}$γ $_{j}$γ j and Y$_{g}$ = i $^{2}$∏ j ∈ $_{g}$η$_{j}$ , where | g | denotes to the number of sites in g and the ordering of sites in the two products is implicitly taken to be the same such that X$_{g}$Y$_{g}$ = D$_{g}$ . X$_{g}$ and Y$_{g}$ are the gauge transformation operators that only act on the gauge fields and Majorana fermions, respectively. We further denote X$_{g}$ = ( − i ) | g$_{A}$ || g$_{B}$ $^{|}$X$_{g}$$_{B}$$_{≡}$$_{g}$$_{∩}$$_{B}$ X$_{g}$$_{A}$$_{≡}$$_{g}$$_{∩}$$_{A}$ and Y$_{g}$ = i | g$_{A}$ || g$_{B}$ $^{|}$Y$_{g}$$_{A}$ Y$_{g}$$_{B}$ , where X$_{g}$$_{A}$ = i | g$_{A}$ | ( | g$_{A}$ |− 1) / $^{2}$∏ i ∈ g$_{A}$ γ x i γ y $_{i}$γ z i and Y$_{g}$$_{A}$ = i | g$_{A}$ | ( | g$_{A}$ |− 1) / $^{2}$∏ i ∈ g$_{A}$ η$_{i}$ are the gauge transformation operator acting on A region, and similar for X$_{g}$$_{B}$ and Y$_{g}$$_{B}$ . The ground state (5) is written as 

<img src='content_image/384419.jpg'>

In the last line we have denoted w$_{A}$ = w$_{B}$ as w . The reduced density matrix ρ$_{A}$ is expressed as 

<img src='content_image/384422.jpg'>

To make the interproduct 〈 u$_{B}$, w $^{′}$| X † g ′ $_{B}$X$_{g}$$_{B}$ | u$_{B}$, w 〉 nonzero, the gauge transformation X$_{g}$$_{B}$ and X$_{g}$ ′ B must be either identical in B region, or different by a gauge transformation on all sites in the B region. We define X$_{B}$ as X$_{g}$ with g = B , and similarly for X$_{A}$ and Y$_{A,B}$ . Note that X † g$_{B}$ = ( − 1) | g$_{B}$ $^{|}$X$_{g}$$_{B}$ . So it is consistent to define B − g$_{B}$ such that X † B − $_{g$_{B}$}$X$_{g}$$_{B}$ = X$_{B}$ for any g$_{B}$ , where X$_{B}$ ≡ X$_{g}$$_{=}$$_{B}$ . Consequently, we have 

with 

<img src='content_image/384423.jpg'>

<img src='content_image/384424.jpg'>

