the potential terms (which again have a central plaquette[@kivelson87]) now have fine-tuned couplings that depend on the nearby plaquettes. Explicitly, we have (cf. Fig. \[dimerconfig\]):

![(color online) Illustration to clarify the construction of the Hamiltonian . The central dimer pair is present in all configurations $C_i$ and $C_i'$, whereas the four surrounding dimers may or may not be present. The index $i$ enumerates the $2^4$ possible arrangements of the surrounding dimers. The configuration $C'$ differs from the corresponding $C$ by a single flip resonance of the central dimer pair.[]{data-label="dimerconfig"}](Ham_fig.eps){width="48.00000%"}

$$\begin{aligned}
  H_{d} &=& t\sum_i \Bigg[- \Big|C_i\Big>\Big<C_i'\Big| - \Big|C_i'\Big>\Big<C_i\Big| \nonumber\\
  &+&
  w^{R_{C_i'}-R_{C_i}}\Big|C_i\Big>\Big<C_i\Big|+w^{R_{C_i}-R_{C_i'}}\Big|C_i'\Big>\Big<C_i'\Big|\Bigg] \;, \nonumber \\
  &&
\label{Ham}\end{aligned}$$

where $R_{C_i}$ and $R_{C_i'}$ denote the number of pairs of present dimers formed in configurations $C_i$ and $C_i'$ respectively.

The Hamiltonian of Eq.  is designed in such a way that it annihilates any superposition of dimer-configuration states which have amplitudes that are of the form $w^{N_p}$, where $w$ is the parameter appearing in the Hamiltonian and $N_p$ is the number of pairs of neighboring dimers in the configuration.[@ardonne04] In this sense, the Hamiltonian is a sum of projection operators and consequently there is a unique ground state for each topological sector which must be composed 