Emergent Geometry from Entanglement Structure
353
literature [16] (see also [12, 17–22]). The entanglement contour function for a given
block estimates the contribution of each site to the entanglement entropy obtained
for that bipartition. For a given block A, mathematically it can be expressed as
S A =
i∈A
s A (i),
s A (i) ≥ 0 .
(6)
Interestingly, from Eq. (2) one can observe that the entanglement adjacency matrix
provides a natural entanglement contour which can be expressed as
s A (i) ≡
j ∈A c
J ij .
(7)
The contour function defined above satisfies all the standard constraints listed in
Ref. [16]. Here, we stress the fact that unlike the actual formulation of the entanglement contour introduced in Ref. [16], our approach aims to provide an overall
description of bipartite entanglement by considering contributions of all bipartitions
and not just the ones consisting of simply connected intervals. Moreover, the
formalism includes any general quantum systems, including interacting cases.
Contour Plot for Free-Fermionic Hamiltonian For free-fermionic model, described
below
H free−ferm = −
1
2
ij
t ij
c
†
i c j + hc
,
(8)
where c i (c
†
i )’s is the fermionic annihilation (creation) operators at site i, and t ij is
the hopping matrix, a proposal for the contour is given in Ref. [16],
s A (i) =
|A|
p=1
Φ
(A)
p,i
2 H (ν p ),
(9)
where Φ
(A)
p,i is the eigenvector with eigenvalue ν p , of the correlation matrix block [8,
28] restricted to A and H (x) = −
x log x + (1 − x) log(1 − x)
. Using the above
equation, in Fig. 3a, we compute the entanglement contour function for the ground
state of the dimerized Hamiltonian, which can be obtained from the free-fermionic
model described in Eq. (8), for t ij = (1+δ(−1) i ), |i−j | = 1 and compare that to the
contour function obtained using the elements of the entanglement adjacency matrix,
J ij , as described in Eq. (7). From the figure, we note that the contour functions
obtained using these two different methods are very similar to each other.
Contour Plot for XXZ Hamiltonian Subsequently, we move one step further and
apply the formalism to an interacting model, namely, the one-dimensional XXZ
model, expressed as
353
literature [16] (see also [12, 17–22]). The entanglement contour function for a given
block estimates the contribution of each site to the entanglement entropy obtained
for that bipartition. For a given block A, mathematically it can be expressed as
S A =
i∈A
s A (i),
s A (i) ≥ 0 .
(6)
Interestingly, from Eq. (2) one can observe that the entanglement adjacency matrix
provides a natural entanglement contour which can be expressed as
s A (i) ≡
j ∈A c
J ij .
(7)
The contour function defined above satisfies all the standard constraints listed in
Ref. [16]. Here, we stress the fact that unlike the actual formulation of the entanglement contour introduced in Ref. [16], our approach aims to provide an overall
description of bipartite entanglement by considering contributions of all bipartitions
and not just the ones consisting of simply connected intervals. Moreover, the
formalism includes any general quantum systems, including interacting cases.
Contour Plot for Free-Fermionic Hamiltonian For free-fermionic model, described
below
H free−ferm = −
1
2
ij
t ij
c
†
i c j + hc
,
(8)
where c i (c
†
i )’s is the fermionic annihilation (creation) operators at site i, and t ij is
the hopping matrix, a proposal for the contour is given in Ref. [16],
s A (i) =
|A|
p=1
Φ
(A)
p,i
2 H (ν p ),
(9)
where Φ
(A)
p,i is the eigenvector with eigenvalue ν p , of the correlation matrix block [8,
28] restricted to A and H (x) = −
x log x + (1 − x) log(1 − x)
. Using the above
equation, in Fig. 3a, we compute the entanglement contour function for the ground
state of the dimerized Hamiltonian, which can be obtained from the free-fermionic
model described in Eq. (8), for t ij = (1+δ(−1) i ), |i−j | = 1 and compare that to the
contour function obtained using the elements of the entanglement adjacency matrix,
J ij , as described in Eq. (7). From the figure, we note that the contour functions
obtained using these two different methods are very similar to each other.
Contour Plot for XXZ Hamiltonian Subsequently, we move one step further and
apply the formalism to an interacting model, namely, the one-dimensional XXZ
model, expressed as