Who or what is RVB?

Friedrich August Kekule von Stradonitz (how could I not start with him?), in 1865, solved the structure of benzene, the molecule that would become the poster child for the idea of resonating valence bonds. But he had no such thoughts; he deduced the symmetrical six-membered ring from purely chemical data. It wasn’t until 1916 that G. N. Lewis came up with the idea that the valence bond was caused by a shared pair of electrons, one from each atom of the bond; the valence bond was first explained quantum mechanically by Walter Heitler and Fritz London in 1928. They showed that it could be due to the binding of a pair of electrons, one from each atom, by Werner Heisenberg’s exchange interaction. If we have two overlapping atomic orbitals ϕ₁ and ϕ₂, the exchange integral J defined by E _ex = J(S ₁ · S ₂), where E _ex is the exchange energy and S the electron spin, will be antiferromagnetic and will cause a singlet bond between the orbitals to be energetically favorable. Linus Pauling whisked that great idea out of his friends’ hands, and in the next few years he ran through most of chemistry showing that the pair bond was a useful heuristic tool. But not only is it heuristic, it has one property that is overwhelmingly common in chemistry, yet is not obvious in other theories—the bond is local and doesn’t depend much on what else is in the molecule or solid. So Pauling could take his pair bond and its component atoms from one substance to another and it would have the same length, strength, and so on.

However, the pair bond alone did not explain benzene. There are three “sigma” bonds (symmetric in the plane of the molecule) for each carbon, one to the hydrogen and one to each of its carbon neighbors; but that leaves only one more (“pi”—odd symmetry) valence electron per C. If that valence electron chooses one neighbor to bond to, the bonding will cause an asymmetrical structure of three C₂ pairs, what we would nowadays call a Peierls doubling, and that does not occur. So Pauling introduced the idea of resonance—the first initial of RVB—which proposes that there is an amplitude for the structure to form a quantum mechanically coherent mixture between the two different pairing schemes. Actually, there is also some admixture of other pairing schemes, where some pairs form right across the hexagon. Such resonances, like any other form of quantum-mechanical zero-point motion, lower the energy and enhance the strength of the bonding by an amount in good agreement with the experimental accuracy then available.

In the years 1931–33 Pauling added the concept of resonance to his heuristic toolkit and used that toolkit to produce his classic 1939 textbook, The Nature of the Chemical Bond . That remarkable book explains qualitatively the structures of most organic and metal–organic molecules and most insulating solids. He was able to parameterize the electronic quantities on which the bonding depends and thus quantified most of chemistry.

Molecular orbital theory

There is, of course, an alternative way of understanding the bonding of molecules and solids, called the molecular orbital theory, introduced by Friedrich Hund and Robert Mulliken a few months after the Heitler–London theory. In Pauling’s theory one assumes that an atom has a specific valency and that its electrons never really run free—in modern terms, the theory is based on starting with an atomic limit in which the atoms have fixed electron occupation numbers, enforced by their mutual Coulomb repulsion, U, and allowing the interactions of the atoms to perturb that. The MO theory takes the opposite tack: One treats the electrons as shared among all the atoms, calculating a set of electronic states or molecular orbitals, and takes the electron–electron interaction that keeps the electrons on their separate atoms into account only later—and often, in the early days, quite inadequately.

But it turned out that while Pauling had enormous success in the short run, he was the hare and the MO was the tortoise that caught up and passed him half a century later. Now even chemists tend to believe that the real electronic structure is the energy bands calculated using the so-called local-density approximation, devised by John Slater. Other useful tricks are pseudopotentials and, if necessary, other refinements on the pure MO scheme. One forgets the idea that relatively unchanging atoms have certain fixed valences and bonding electron identities. (Though, of course, everyone really thinks Pauling’s way at the beginning of a problem.) There is a way within band theory of understanding the locality of most chemistry, using Gregory Wannier’s wonderful transformation from band wavefunctions to atomic-like ones, but that wasn’t really understood until the late 1960s, and is still little used by many band theorists and quantum chemists.

The wavefunctions can be transformed into localized forms only if the electrons of interest are split off by a gap in energy from the empty states—that is, if the substance is insulating. Pauling ignored that restriction, and in the 1940s when I was learning solid-state physics, he tried to make a theory of metals using a quantum liquid of valence bonds—resonating valence bonds (RVB)—assigning fractional valences to the metal atoms and applying his usual bag of semiempirical tricks with which he could, as the saying goes, “fit an elephant.” But those tricks were not at all useful and could not possibly, for instance, account for all of the properties of metals that depend on the Fermi surface. Also, his RVB account of the physical sources of the binding energy was less satisfactory than that of band theory. It soon disappeared from view.

Using Pauling’s wrong idea

In the early 1970s, I began to wonder what kind of state might actually exemplify Pauling’s wrong idea. My motivation came from a subject I had long been involved in, antiferromagnetism. I now take another excursion into ancient history and also into parts of condensed-matter physics that may not be as widely taught as they should be. The line of thinking stems from another giant of early quantum mechanics, Hans Bethe, who in 1931 solved exactly a model for an infinitely long chain of atoms connected by antiferromagnetic Heisenberg exchanges—an infinitely long benzene ring, if you like—and left us with a dilemma: The result was not a metal, to be sure, but it also was not antiferromagnetic. He had found the first RVB! Later I will describe how fervently people came to hope it was the only one.

The word “antiferromagnetism” is one of those obvious constructions that seem to define themselves—yet someone must have been the first to use it. In 1936 it was used by Louis Néel to describe the ordered state of the spins of an actual substance—for which insight he received the Nobel Prize—and by Lamek Hulthén in reference to the linear chain Hamiltonian for which Bethe had found the ground state.

At an international meeting in 1939, Néel attempted to describe his experiments suggesting the existence of an ordered antiferromagnetic state. He met with such implacable opposition to its reality from the quantum theorists, on the basis that the only exact solution of the antiferromagnetic Hamiltonian was Bethe’s disorderly one, that he left the meeting in disgust and retained a lifelong prejudice against quantum theory.

It’s hard to believe, but true, that in spite of the neat and beautiful mean-field quantum treatment of antiferromagnetism that John Van Vleck published in 1941, there were still doubters, such as the great Lev Landau. However, in 1952 I pointed out that the difference between Bethe’s one-dimensional chain and real three-dimensional antiferromagnets (which by then had been observed unequivocally with neutron diffraction by Cliff Shull and colleagues) was that in 1D quantum fluctuations diverge, while in 3D they are finite. The method I used was to calculate the zero-point or thermal amplitude of the spectrum of spin-wave fluctuations—the equivalent of the Debye–Waller factor—and see if the amplitude came out finite.

Try two dimensions

My observations left partly open the intriguing case of two dimensions. My calculations said the ground state would be ordered in 2D, barely, but that without some added 3D help, or some anisotropy, the antiferromagnet would not be ordered at finite temperature T. (All the predictions turned out, in the end, to be beautifully verified in La₂SrCuO₄, but it was a while before we realized that!) Since my theory was the result of using a series in powers of 2/ZS, with Z the number of neighbors and S the spin, it didn’t really guarantee to be convergent, say, for the square lattice of Cu ⁺⁺ ions with S = 1/2 and Z = 4.

In 1972 I had been talking to Denis McWhan and Maurice Rice about some interesting data on the compound tantalum disulfide (which data, I believe, found some other explanation in the end). Those data turned out to be reasonably well modeled by a 2D triangular lattice of S = 1/2 spins. The triangular lattice has Z = 6, but the larger Z is compensated by the fact that you can’t set the three spins on a triangle antiparallel to each other. It’s “frustrated,” as we came to describe it; so the series converges as poorly as it does for the square lattice. With TaS₂ as an excuse, I set out to see if the Néel state was the correct ground state of the triangular S = 1/2 antiferromagnet or if it was indeed a fluid of resonating pair bonds like the Bethe 1D solution.¹ Not surprisingly, my natural optimism led me to conclude the latter; whether I was right is, I believe, still a matter of some controversy. Probably I was not literally right, according to recent work by Rajiv Singh, but neither is the Néel state; however, that is getting far, far ahead of the RVB story. What may be important about my paper is that it first posed the following questions: What is the nature of an RVB in two dimensions, if it exists? Does it have an order parameter? What is its relationship to Bardeen-Cooper-Schrieffer pairing of electron singlet pairs? How can you tell experimentally if you have one?

And, of course, what do you call such a state? RVB? The paper had a publishing history. Why is it in the Materials Research Bulletin? I have never before nor since used that journal. In fact, a letter had just arrived from the editor, Rustum Roy, soliciting papers for a Linus Pauling festschrift. Since the nature of the paper was a kind of “in-joke” on Pauling, suggesting that his theory of metals was a theory of something else, I thought it belonged in the festschrift. But I didn’t say so in the cover letter, and Rustum treated my paper as an “over the transom” submission—it appeared later in a miscellaneous issue of the journal, though with a nicely apologetic note.

During the period when that paper seemed irretrievably lost, a postdoc named Patrik Fazekas arrived unexpectedly from behind the Iron Curtain. He showed up with his luggage late one evening at the porter’s lodge of Jesus College, Cambridge, asking for me. The proverbially unflappable head porter found him a room and phoned me, and overnight I decided that RVB was Patrik’s problem if he wanted it. The paper that resulted from that work ² was much more thorough than mine and published in a more usual venue; ever since then both his name and mine have been associated with the idea—and Patrik has become a valued friend.

In a future column I hope to chronicle the rise and fall and rise again of RVB’s relevance to high T _c and other superconductors.

The stimulus for this note came from discussions with Shivaji Sondhi.