This is a very good question.
First let me clarify a point. So far long range entanglement is only defined for gapped quantum systems.
The gapless systems seems always "long range entangled". So the notion is useless.
Do RVB states have long range entanglements? I think the string idea that you mentioned is a very good idea: A string liquid leads to long range entanglements.
Are RVB states string liquid states? Actually, the answer is yes.
We may take a VB configuration as a reference, than the difference between any other VB configuration and the reference VB configuration can be described by a closed string! (See Sutherland, Phys. Rev. B 37 3786 1988; Kohmoto, Phys. Rev. B 37 3812, 1988).
So a RVB state is actually a string liquid! If the dimmers only connect
between A sub-lattice and B sub-lattice, then the strings are orientable
and the corresponding string liquid gives rise to an emergent U(1)
gauge theory. Otherwise, the strings are not orientable
and the corresponding string liquid gives rise to an emergent $Z_2$
In fact, the situation is a more complicated than the above discussion.
The string liquid from RVB is not an equal weight superposition of all loop
configurations. Different string configurations may have different weights.
So the string liquid from RVB may not be a liquid. It could be a string solid
+ a little fluctuations. In this case, there is no emergent gauge theory
and the corresponding RVB state is not long range entangled.
If a RVB state does correspond to a loop liquid, then the corresponding RVB state is long range entangled.
This post imported from StackExchange Physics at 2014-04-05 04:30 (UCT), posted by SE-user Xiao-Gang Wen