# Is resonating valence bond (RVB) states long-range entangled?

Quantum liquid is at the core of condensed matter theory study, examples include superfluid in Bose Hubbard model, quantum spin liquid around the RK point of a quantum dimer model, string-net condensation in a quantum many-string system, etc. Recently it was realized that quantum entanglement may be the key to differentiate these liquids. For example, the Bose superfluid is simply a direct product state with no entanglement, while string-net condensate is claimed to be long-range entangled. To me their differences would be that the former is a condensation of point objects, and the latter is a condensation of string-like objects. I would conclude from this observation that the condensation of extended objects leads to long-range entanglement. It seems to me that it is the extended objects that pass on the quantum entanglement in the system. So when I come to the RVB state in the quantum dimer model, I think of condensing the dimers -- stick-like objects. Sticks are more extended than points but less than strings. So I guess the dimer condensation (or the RVB state) would have an entanglement between the superfluid and the string-net condensation. I can see that the RVB state can not be written as a direct product state, so there is definitely some entanglement there. My question is: is RVB state long-range or short-range entangled?

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The quick answer is "yes". See this recent review by Sachdev: arxiv.org/abs/1203.4565. Considering X.G. Wen's recent activity on this site though, I'm quite looking forward to his inevitable answer. :) –  wsc Jun 2 '12 at 3:43
@wsc Thanks for pointing out Sachdev's nice review. Can I consider that it is because the dimer condensate RG flows to a string-net condensate in the low energy limit, that the long-range entanglement becomes evident? –  Everett You Jun 2 '12 at 3:55

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 $Z_2$ gauge theory.

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.

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Thanks for your wonderful answer. I have one more question: regarding the string representation you have mentioned, is it true that the string solid state would actually correspond to a valence bound solid (VBS) state, which can be considered as frozen from a RVB liquid? –  Everett You Jun 2 '12 at 4:18
Indeed, the string solid is the VB solid. One may wonder, why RVB state become a VB solid? Some time a RVB state is really a liquid (and it is long range entangled). But some times, a RVB state may secretly contain a long range order and actually correspond to a VB solid. We need to do some non-trivial calculation to determine if a RVB state is a liquid or a solid. –  Xiao-Gang Wen Jun 2 '12 at 4:22
Thanks for the quick answer. I would consider that VB solid can be described as well within the scope of a general RVB wave function. It is because the string tension gets stronger and stronger that froze the quantum fluctuation and drive a quantum liquid into solid. Is there any method to extract the effective string tension from a RVB wave function? –  Everett You Jun 2 '12 at 4:28