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24 CHAPTER 1. INTRODUCTION
(Haldane gap). In the one dimensional spin chain, a pure valence-bond 3 wavefunction
is clearly not the ground state but it gives indeed an energy that is even
better than the classical antiferromagnetic state:
• Classical antiferromagnetic state : E = −0.25J
• valence-bond of singlet states : E = −0.375J
• Exact ground state energy (Bethe Ansatz) : E = − √ 2log2J ≈−0.43J
Clearly the valence-bond state made of dimers is not close to the ground state of
the one-dimensional chain, though it were the case if we introduce the coupling
of the spin degrees of freedom to the lattice (there is an energy gain due to the
elastic energy that is reduced by forming singlets, this is the so called Spin-Peierls
instability). Nevertheless, it can still be expected, although the pure valencebond
state is not a good approximation of the ground-state, that a superposition
of many valence-bond state (resonating valence-bond) will give a good Ansatz
that keeps the key ingredient of the low energy physics. The valence-bonds are
expected to give a good variational basis to describe the low energy physics of
the model. This later argument is at the basis of the ideas of Anderson, and such
a proposal was made for the triangular lattice in 1975 [13].
1.4.3 Anderson’s Resonating valence-bond theory
The discovery of high-T c superconductivity, and the observation [14] that strong
correlations are important in connection with these compounds has led to a
tremendous interest in understanding strongly correlated electron physics. In
particular the two simplest models for strongly correlated electrons, namely the
Hubbard and t−J models, have been the subject of intensive studies. In a milestone
paper of 1987, Anderson proposed [14] that a resonating valence-bond
(RVB) wave function, which consists of a superposition of valence-bond states
(see Fig. 1.3), contains the ingredient to account for a consistent theory of the
Hubbard and t−J models. The t−J model on the square lattice has been discussed
extensively in the context of high-T c superconductivity, especially as a
framework for the implementation of the resonating valence bond (RVB) scenario
proposed by P.W. Anderson [15]. The t−J model can be viewed as the
large on-site repulsion limit of the Hubbard model, and therefore at half-filling
it describes an antiferromagnetic insulator. An RVB theory for the insulating
state predicts that the preexisting singlet pairs become superconducting when
the insulator is doped sufficiently. In that case the superconductivity would be
driven exclusively by strong electron correlations.
3 A valence-bond state is a paving of the chain with nearest neighbors singlets.