There's something known as a "resonating valence bond" (RVB) state, which plays a role in at least some attempts to understand physics of high-$T_c$ superconductors. This, roughly, involves a state that's in a superposition (hence the "resonating" part of the name, if I understand correctly) of different ways to pair electrons into strongly-bonded spin singlets. My question is: what is a more precise definition of this sort of state? What's the underlying physics, when does it arise, and why is it interesting?
Points an answer might address: is there a simple toy model for which this is the ground state, that sheds light on what sort of system it could arise in? Is there an interesting continuum limit, in which we can characterize this state in a more field-theoretic language? Are there particular kinds of instabilities such a state tends to be subject to?
I think I know where I would start digging if I wanted to really understand this for myself, but mostly I'm asking it to probe the community and see what kind of expertise might be lurking here, since there haven't been so many condensed matter questions.
Best Answer
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