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?
Quantum Mechanics – Are Resonating Valence Bond (RVB) States Long-Range Entangled?
condensed-matterquantum mechanicsquantum-entanglementresearch-leveltopological-order
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RVB states were first coined in 1938 by Pauling in the context of organic materials and they were later extended to metals. Anderson revived the interest in this concept in 1973 when he claimed that they explained the Mott insulators. (Mott, not Matt and not Motl, which is a shame because I was born in 1973.) He wrote a new important paper in 1987 in which he described the copper oxide as an RVB state.
If one has a lattice of atoms etc. and there is qubit at each site - e.g. the spin of an electron - then the RVB state in the Hilbert space of many qubits is simply $$|\psi\rangle = \sum |(a_1,b_1) (a_2,b_2) \dots (a_N,b_N)\rangle$$ which is a tensor product of "directed dimer" states of qubits which are simple singlets $$|M,N\rangle = \frac{1}{\sqrt{2}} \left( |\uparrow_M\downarrow_N\rangle - |\downarrow_M\uparrow_N\rangle \right) $$ The sum defining the RVB state goes over all arrangements or methods how to divide the lattice into (vertical or horizontal or whatever dimensions you have) pairs of adjacent lattice sites. For each lattice site, one puts the corresponding two qubits (usually spins) into the singlet state.
The singlet state above is $M,N$-antisymmetric, so one has to be careful about the signs. So all the tensor factors $(a_i,b_i)$ above are oriented and the orientation always goes in such a way that $a_i$ is a white site on a chessboard while $b_i$ is a black site on a chessboard, in the usual chessboard method to divide the lattice into two subsets.
Because all singlet states used in the RVB states are made out of nearest neighbors, it looks like a liquid, which is why the resulting material in this state is called the RVB liquid. (Imagine molecules in a liquid - they also like to interact with some neighbors only. If one doesn't rely on distant molecules to neutralize the spin, it is "liquid-like".)
The idea - related to the name - is that the information about the terms defining $|\psi\rangle$ is the information about which adjacent lattice sites are connected - these are the valence bonds ("valence" because the nearest neighbors are interacting through their valence electrons or degrees of freedom). However, a general term of this type could be claimed to evolve into another similar state where different links (valence bonds) are included to the creation of the singlets. If one tries to allow the valence bonds to jump anywhere - and switch from vertical to horizontal directions - one obtains a "resonating" system. This symmetrization (symmetric superposition) of all possibilities is a usual way to obtain a quantum eigenstate of the lowest energy, assuming that the different terms are able to change into each other by a transition amplitude.
The funny feature of this liquid state is that it is invariant under all the translations - those allowed by the lattice - and the rotations - allowed by the lattice, if any. This is very different from a particular chosen method how to divide the lattice sites (qubits) to pairs. By summing over all methods to divide into pairs, we reach a certain degree of "democracy" that gives the state very different and special properties - in comparison with some particle "vertical crystals" or other ways how you could orient the singlets.
Or you may look at it from the other side. It is somewhat nontrivial to construct singlet states of the material, and the RVB state is the most democratic one. It is often useful to look at mathematical guesses that look special and the RVB state was no exception.
You seem to be interested in the high-$T_c$ superconductors. I believe that the critical paper in this direction was this 1987 paper
http://prb.aps.org/abstract/PRB/v35/i16/p8865_1
by Kivelson, Rokhsar, and Sethna. They asked a simple question - what are the excitations above the RVB state. A fascinating feature was that the excitations inherit only 1 of 2 key properties of the electron: there are spin-1/2 fermionic excitations - like the electron - but the shock is that they're electrically neutral; and there are charged excitations - like the electron - but they're spin-0 bosons (similar to solitons in polyacetylene).
It's a cool property that by choosing a pretty natural state, you may obtain totally unfamiliar excitations - of course, it's a common theme in condensed matter physics. I suppose that if they can talk about the mass of the excitations, and they're non-negative, they also have a Hamiltonian for which the state occurs, and they show that it is stable along the way. But you must read the full paper.
I haven't mentioned the high-$T_c$ punch line yet. Of course, the bosonic charged excitations may produce a Bose gas and this Bose gas could exist at high temperatures.
But of course, one must be careful not to get carried away. The RVB state is not the only one that one can construct out of the spins. The experimental attempts to produce a full-fledged RVB liquids remained inconclusive, to put it lightly, and some previously believed applications of the RVB liquid are no longer believed to be true. For example, it was believed that the RVB state is a description of the disordering of anti-ferromagnets, but especially from a 1991 paper by Read and my ex-colleague Sachdev, it became much more likely that the spin-Peierls description is more likely.
An interesting theoretical by-product of the RVB considerations were things related to cQED - strong coupling compact quantum electrodynamics - with a $\pi$-flux RVB state in the continuum limit. This bizarre theory also has the neutral spin-1/2 excitations; an infinite bare coupling; and has been studied nicely for $SU(N)$ and $Sp(2k)$ gauge groups. It must be assumed that the spin Peierls ordering doesn't develop in the system.
Best wishes Lubos
See What is a resonating valence bond (RVB) state? for a related discussion.
2+1D Z2 RVB state is described by a Z2 gauge theory at low energies. A vison is simply the Z2 gauge vortex/flux. It is a defect in the phase change/twist of the ground wave function. It is hard to present it in your first picture which is classical and does not show the phases and superpositions of the wave function.
Z2 RVB state has a long range entanglement, but do not have gapless edge excitations. Not all topologically ordered states have gapless edge excitations. Topologically ordered states are define by its non-trivial topologically robust degenerate ground states on spaces with non-trivial topology, and the modular transformation properties of those degenerate ground states. Topological order is not defined by gapless edge excitations, since not every topologically ordered states have gapless edge excitations. See http://en.wikipedia.org/wiki/Topological_order and http://en.wikipedia.org/wiki/Topological_degeneracy
Best Answer
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.