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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.

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This is an excellent question and @Lubos provides an excellent answer. –  user346 Jan 15 '11 at 21:55
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up vote 7 down vote accepted

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

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You've foiled my plot to lure condensed matter physicists to participate more in this site by answering the question before they get a chance. :-) But thanks for the nice answer; I wasn't aware of the spin-Peierls story. (I had to check that spin-Peierls means roughly what I would have guessed, so let me note here for other casual readers that it is a state in which the pairing of neighboring spins is ordered in a way that breaks lattice symmetries, unlike the RVB which preserves the symmetries.) I'm also curious about the things you mention in the last paragraph; is there a good reference? –  Matt Reece Jan 15 '11 at 20:40
    
Dear Matt, they may be attracted by the need to fix errors in my comments; I couldn't have made their absence worse. ;-) The references to the new gauge theories are 11,12,15,16 cited in arxiv.org/abs/cond-mat/0206483 ... The spin-Peierls states are a topic of a comparable size as the RVB state itself, so I would choose not to comment on it haha. –  Luboš Motl Jan 15 '11 at 20:48
    
Ah. Without actually digging up those references, I would guess from the text of the paper you link that the large-N expansion here is actually a large $N_f$ expansion, and that $SU(N)$ and $Sp(2k)$ here are global rather than gauge symmetries. (I say this just based on the familiar case of ordinary QED$_3$ with $N_f \gg 1$ fermions, where such an expansion shows immediately that the IR theory is an interacting CFT with nontrivial anomalous dimensions, which is very unclear from ordinary perturbation theory.) In any case, I don't have time right now to look them up, but I'll keep them in mind. –  Matt Reece Jan 15 '11 at 21:18
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Thank you for the wonderful explanation @Lubos and thanks for the link to the RKS paper. I'm learning this subject slowly and missing pieces like the RKS paper are a great help. –  user346 Jan 15 '11 at 21:57
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Kivelson, Rokhsar, and Sethna discussed the RVB state with only nearest neighbor bonds that connect different sub-lattices. The constructed RVB state is an equal amplitude superposition of all the nearest-neighbour bond configurations. Such a RVB state is believed to contain emergent gapless $U(1)$ gauge field which may confine the spinons etc. A version of RVB state with deconfined spinons is the chiral spin state (see Phys. Rev. Lett., 59, 2095 (1987) and Rev., B39, 11413 (1989)). Later, another version of RVB state with deconfined spinons, the $Z_2$ spin liquid, is proposed (see Phys. Rev. Lett. 66 1773 (1991) and Phys. Rev. B44, 2664 (1991)). Both chiral spin state and $Z_2$ spin liquid state have RVB bonds that connect the same sub-lattice. In chiral spin state, different bond configurations can have complex amplitudes, while in $Z_2$ spin liquid state, different bond configurations only have real amplitudes. The RVB state on triangle lattice also realizes the $Z_2$ spin liquid (see arXiv:cond-mat/0205029), where different bond configurations only have real amplitudes.

(The above is adapted from spin liquid wiki page.)

The $Z_2$ spin liquid realizes one of the simplest topological order, described by $K=\left(\begin{array}[cc]\\ 0&2\\2& 0 \end{array}\right)$ in the K-matrix classification of 2D Abelian topological order. The chiral spin liquid realizes a different topological order, described by $K=\left(\begin{array}[cc]\\ 2 \end{array}\right)$.

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@ Xiao-Gang Wen Dear prof.Wen, what does the term 'sub-lattice' mean in your answer? For example, is a triangular lattice viewed as containing only one 'sub-lattice' or three 'sub-lattices'? –  K-boy Feb 12 at 14:17
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