Understanding lattice models for strongly correlated electron systems is a major challenge. Widely studied (and still poorly understood) models include the Hubbard and Heisenberg models. But some insight can be gained from studying model Hamiltonians on small clusters such as four lattice sites. Although, such a small system is far from the thermodynamic limit, these models can illustrate some of the essential physics associated with the interplay of strong electronic correlations, frustration, and quantum fluctuations. They illustrate the quantum numbers of important low-lying quantum states, the dominant short-range correlations, and how frustration changes the competition between these states.
Furthermore, understanding these small clusters is a pre-requisite for cluster extensions of dynamical mean-field theory and rotationally invariant slave boson mean-field theory which describes band selective and momentum space selective Mott transitions. See for example my earlier post on that.
I now give a concrete example which illustrates how frustration can change the quantum numbers of the ground and first excited states. This is taken from a 1996 PRL, Plaquette Resonating valence bond ground state of CaV4O9, by Ueda, Kontani, Sigrist, and Lee. They first consider a single
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