Entanglement of several blocks in fermionic chains

In this paper we propose an expression for the entanglement entropy of several intervals in a stationary state of a free, translational invariant Hamiltonian in a fermionic chain. We check numerically the accuracy of our proposal and conjecture a formula for the asymptotic behavior of principal submatrices of a Toeplitz matrix.

Figure 1

Contour of integration, cuts, and poles for the computation of $S_{\alpha }\left(X\right)$. The cuts for the function $f_{\alpha }$ extend to $\pm \infty$.

Figure 2

By the diagonal bars in (a), we represent the defining property of a Toeplitz matrix: the entries along any subdiagonal parallel to the principal one are equal. Panel (b) represents a block matrix in which each block is Toeplitz but the full matrix is not.

Figure 3

Two blocks mutual information when $\alpha =1$ (left panel) and $\alpha =2$ (right panel) as a function of $y$ for state 3. With $\diamond$ we represent the numerical value for two blocks made out of 50 sites each, varying their separation from 1 up to 500 sites. The $•$ corresponds to two blocks of lengths 1000 and 500 sites, separating each other between 1 and 1000 sites. The continuous line depicts the function (20) with the $B_{\alpha }$ evaluated from the Fisher-Hartwig conjecture.

Figure 4

Mutual information, for state 3, of three $(•)$ and four $(\diamond )$ blocks as a function of the product of the possible cross ratios $y_{ij}$. For $p=3$, we have chosen two intervals of 1000 sites and one of 500 sites which is separated from one of the former by 1500 sites, while the remaining distance is modified from 1 up to $99000$ sites. For $p=4$, we take intervals of lengths 2000, 500, 1000, 2000. The distance between the first couple is 1500 sites, the break between the smallest blocks is also fixed, 5000 sites, whereas the other one is increased between 1 and $99000$ sites.

Figure 5

Graphical representation of conjecture (25) for $p=2$. In the left-hand side we represent the determinant of the shadowed submatrix that, in general, is not Toeplitz. In the right-hand side, however, the determinants of the shadowed submatrices are of the Toeplitz type (or product of these).

Figure 6

Analog of mutual information for determinants, (26), against the cross ratio of $(u_1,v_1;u_2,v_2)$. The $\diamond$ represents the numerical results for two subsets of size $|u_1-v_1|=|u_2-v_2|=50$ while the gap between them, $|v_1-u_2|$, varies between 1 and 200. The $•$ corresponds to two subsets of length 500 separated by a distance between 1 and 4500. The continuous line is the conjectured analytical expression of (27).