content_image/417051.jpg content_image/417053.jpg content_image/417052.jpg content_image/417056.jpg content_image/417057.jpg content_image/417058.jpg content_image/417062.jpg content_image/417063.jpg content_image/417064.jpg content_image/417069.jpg
Using this result ρ$_{A}$ can be simplified to 

<img src='content_image/417051.jpg'>

Notice that Y$_{B}$ = i | B | ( | B |− 1) / $^{2}$∏ j ∈ B η$_{j}$ is actually the fermion number parity operator in B region, we see that the operator 1+ x$_{B}$ ( w ) Y$_{B}$ 2 is actually the projector of the fermion number parity Y$_{B}$ = x$_{B}$ ( w ). For convenience, we define 

and similar for A region. The reduced density matrix 

is the fermion density matrix with restriction to the fixed fermion number parity x$_{B}$ ( w ) in the B region. By using similar interproduct formula (A5) for A region, ρ 2 A can be computed as 

<img src='content_image/417053.jpg'>

<img src='content_image/417052.jpg'>

<img src='content_image/417056.jpg'>

The same calculation can be repeated to obtain 

<img src='content_image/417057.jpg'>

Finally we take the trace of ρ n A by using the same Eq. (A5) to obtain 

<img src='content_image/417058.jpg'>

For Eq. (A6), we see that flipping the sign of one link w$_{i}$ , for some integer 1 ≤ i ≤ L , changes the sign of both x$_{A}$ ( w ) and x$_{B}$ ( w ). If we define ∏ ij ∈ A ( B $_{)}$u$_{ij}$ = p$_{A}$$_{(}$$_{B}$$_{)}$ , then x$_{A}$$_{(}$$_{B}$$_{)}$ ( w ) = p$_{A}$$_{(}$$_{B}$$_{)}$ ∏ L n $_{=1}$w$_{n}$ , and the summation over all 2 L configurations of w = { w$_{i}$, i = 1 , 2 , ..., L } leads to 

In the main text we have discussed that the ground state | u 〉 | φ ( u ) 〉 must satisfy the constraint D | u 〉 | φ ( u ) 〉 = | u 〉 | φ ( u ) 〉 . Since D = X$_{A}$X$_{B}$Y$_{A}$Y$_{B}$ and X$_{A}$X$_{B}$ | u 〉 = p$_{A}$p$_{B}$ | u 〉 , we obtain Y$_{A}$Y$_{B}$ | φ ( u ) 〉 = p$_{A}$p$_{B}$ | φ ( u ) 〉 . In other words, the total fermion parity is fixed, so that P p$_{A}$ $_{A,F}$P − p$_{B}$ A,F | φ ( u ) 〉 = P − p$_{A}$ A,F P p$_{B}$ A,F | φ ( u ) 〉 = 0. Thus we have 

so that 

where ρ$_{A,F}$ = ρ + A,F + ρ − A,F = Tr$_{B}$ [ | φ ( u ) 〉 〈 φ ( u ) | ] is the free fermion density matrix without fermion number parity constraint in the B region. 

To understand this result more intuitively, we note that 

<img src='content_image/417062.jpg'>

<img src='content_image/417063.jpg'>

<img src='content_image/417064.jpg'>

<img src='content_image/417069.jpg'>

