One characteristic feature of strongly correlated electron systems that I think should be talked more is how sensitive they are to small perturbations. This is particularly true in frustrated systems. A related issue is that there are often several competing phases which are very close in energy. This can make variational wave functions unreliable. Getting a good variational energy may not be a good indication that the wave function captures the key physics.
Here are two concrete examples to illustrate my point.
Consider the spin 1/2 Heisenberg model on the isotropic triangular on a lattice of 36 sites, and with exchange interaction J. Exact diagonalisation gives a ground state energy per site of -0.5604 J and a net magnetic moment (with 120 degree order as in the classical model) of 0.4, compared to the classical value of 0.2.
In contrast, a variational short-range RVB wavefunction has zero magnetic moment and a ground state energy of -0.5579 J, which is only 0.5% larger than the exact value.
Yet, it is qualitatively incorrect because it predicts no magnetic order (and spontaneous symmetry breaking) in the thermodynamic limit. Furthermore, the energy difference is about 1/500 of J.
[For details and references see Table III in this paper].
The second example concerns the spin 1/2 Heisenberg model on the anisotropic triangular lattice, viewed as chains with exchange J and frustrated interchain coupling J'. For J ~ 3 J' this describes the compound Cs2CuCl4.
The figure below (taken from this paper) shows the calculated triplet excitation spectrum of the model, with a small Dzyaloshinski-Moriya interaction D, and without (D=0). It is striking that even though D ~J/20 it induces energy changes in the spectrum of energies as large as J/3, including new energy gaps.
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