entanglement entropy of disconnected regions - Staff.city.ac.uk - City ...

Some simple limit cases
• When r 1 , r 2 ≪ r 3 , r 4 the 4-point function factorizes as
lim
n→1
d
dn 〈T (r 1) ˜T (r 2 )T (r 3 ) ˜T (r 4 )〉 → lim
n→1
= lim
n→1
〈T (r 3 ) ˜T (r 4 )〉
} {{ }
=1
+ lim
n→1
〈T (r 1 ) ˜T (r 2 )〉
} {{ }
=1
d
dn 〈T (r 1) ˜T (r 2 )〉
lim
n→1
} {{ }
=−S A
d
lim
n→1 dn 〈T (r 3) ˜T (r 4 )〉 = −S A − S B ,
} {{ }
=−S B
d
dn 〈T (r 1) ˜T (r 2 )〉〈T (r 3 ) ˜T (r 4 )〉
• Regions A and B are so far from each other (the size of region C tends to ∞ whilst the
sizes of regions A and B are fixed and finite) that the bipartite entropy becomes simply
the sum of the entanglement entropy between each of the regions and the rest of the
system.